2 Attractors in dynamical systems

We begin by recalling the definition of a dynamical system. A dynamical system on topological space is a continuous map that satisfies

1. for all ,

2. for all and ,

where is the time domain, either or . Next, we recall the definition of attractors and show how they form a bounded distributive lattice. We then illustrate this with a concrete example.

Let be a compact Hausdorff space. For a point , the orbit of describes how the system evolves in time starting from is In applications, differential equations yield examples of dynamical systems with continuous time . These continuous-time dynamical systems are called flows. We illustrate this correspondence with an example later on. Throughout this post, we focus on continuous-time dynamical systems, though many results extend to discrete-time systems. A subset is called invariant if it contains its whole orbit. Formally, is invariant if The collection of all invariant sets of is denoted by . Given a set , the maximal invariant set in is the union of all of the invariant sets that it contains. Formally, A subset is called an attracting neighborhood if eventually the orbit of every state in the closure of ends up in its interior. In other words, if there exists such that for all Now, is an attractor if it is the maximal invariant set of an attracting neighborhood. Formally, is an attractor if there exists a compact such that The collection of all attractors of a dynamical system, denoted , forms a bounded, distributive lattice with order given by subset inclusion, see [@Kalies2014]. The lattice operations are given by

3 Attractor lattices in decoupled product systems

Let and be two dynamical systems on compact, Hausdorff spaces. The product system is defined by .

4 Conclusion

In this post, we studied how attractor lattices behave when combining systems. The decoupled case highlights both the algebraic structure that persists and the limitations of product constructions. Moving forward, the coupled setting requires richer tools: cascade products and, more generally, sheaf-theoretic frameworks offer a natural way to extend these ideas to more general dynamical interactions. This next question will be my focus for the next couple of months.

Lastly, I would like to thank Sophie for her guidance and the entire Topos team for making my time there fulfilling and deeply inspiring. I learned a lot from everyone and look forward to building on these ideas in the months ahead. I would also like to thank my advisor, Dr. William Kalies, for suggesting this project and for his guidance, ideas, and numerous discussions that were instrumental in shaping this draft.

New features and updates

type Graph := [
  V : Entity,
  E : Entity,
  src : (Id Entity)[E, V],
  tgt : (Id Entity)[E, V]
]

type Graph2 := [
  V : Entity,
  g1 : Graph & [ .V := V],
  g2 : Graph & [ .V := V]
]

What next?

CatColab is now at the stage where we can start having serious conversations with experts from different disciplines and try to understand what they want and need from software. We have some exciting conversations planned already (for example, the AIM workshop on Formal scientific modeling: a case study in global health), but we’re entering into 2026 with more ideas and possibilities than we will ever possibly have time to work on to completeness.

But this is good news! Not only is it exciting for us to have a wide choice of possible projects, but it can also be exciting for you to be involved in helping steer the direction. If you’re interested in the software, the underlying mathematics, the ways in which it could fit into existing (or possible) real-world contexts, or have questions or desiderata, then we’d love to hear from you and chat about it all. The easiest way to join in the conversation is in our Zulip:

1 Algebra vs. coalgebra

Algebra is about operations that take multiple inputs and yield one output. For example, the usual addition operation takes two inputs (say and ) and produces one output (say ).

Dually, coalgebra is about co-operations, which take one input and yield multiple outputs. These multiple outputs are produced in a bunch, possibly without order. For example, in an undirected graph, each edge is assigned an unordered pair of vertices.

To put it another way: in algebra, a unique element is assigned to each operation filled with elements, whereas in coalgebra, a unique co-operation filled with elements is assigned to each element.

Before giving some examples of coalgebraic structures in more detail, first let’s discuss what “(co)operation” means.

2 Encoding (co)operations

Both operations and co-operations are expressions with slots capable of being filled with elements. This will be made precise by the concept of a functor . It may not be obvious at first, but such a functor is essentially equivalent to some expressions that can be filled with elements, subject to equations.

This is easiest to illustrate with an example. Suppose we are interested in a binary expression , subject to the equation .

This determines a functor as follows. Given any set , we obtain a set of expressions filled with elements of (modulo equations obtained from the given one by substitution). In this example, amounts to unordered pairs of elements in .

Expressions can be re-filled: given any (an expression filled with elements of ) and a map of sets , define (an expression filled with elements of ) by relabeling/substitution.

This indeed amounts to a functor . Formally,

translates to a presentation of this functor by generators and relations. That is, is the functor presented by

where denotes the term derived from by applying the function sending , , that is, .

Just like any other kind of algebraic structure, all functors may similarly be presented by generators and relations. (In more jargon, this is the “co-Yoneda lemma”: every -valued functor is a colimit of functors free on one generator.) That means any functor may be viewed in this way as some expressions subject to some equations.

A functor is sometimes called a -set for short. From here on, a functor will be called a -set for short.

2.1 Example: polynomial functors

Especially simple are expressions subject to no equations, which correspond to freely generated -sets (presented by generators and no relations).

For instance, suppose we are interested in a binary expression , a binary expression , a binary expression , and a unary expression , together satisfying no equations.

This corresponds to a -set where, as always, is the set of expressions filled with elements of .

In this example, may be described by the formula Here is the set of -tuples of elements of , that is, the set of functions , counting the possible ways of filling an -ary expression with elements. The functor with formula itself is the -set presented by a single -ary generator and no relations.

(This is the Yoneda lemma: representable -valued functors are free on one generator.)

In general, such freely generated -sets are called polynomials, since they are described by formulas

3 Examples of (co)algebras

The following seems a reasonable way of making precise the previously informal terms “operation” and “co-operation”: a (co)operation is a generator for a presentation of a -set. (It is an expression with many slots that can be filled with elements.) Whether we call this an operation or a co-operation depends on whether we are interested in algebra or coalgebra.

Given a -set , an algebra is a map (assigning to each operation filled with elements a unique element). A coalgebra is a map (assigning to each element a unique co-operation filled with elements). In the context of an algebra, the elements that fill the expressions act as inputs, whereas in the context of a coalgebra, the elements that fill the expressions act as outputs.

For example, an algebra for the -set described earlier presented by an subject to the equation is a set with a commutative binary operation, whereas a coalgebra is a set with a “commutative binary co-operation”, assigning each element an unordered pair of elements.

Likely most readers are more familiar with algebra/operations than coalgebra/co-operations, so here are some examples of the latter.

3.1 Directed multigraphs

Directed multigraphs (a.k.a. quivers) can be modeled as coalgebras, using an edge co-operation and a vertex co-operation.

The edge co-operation has two outputs ( and ), and the vertex co-operation has zero outputs. Together these two expressions and generate a polynomial (i.e. freely generated) -set:

Pictured below is a directed multigraph with four edges and four vertices, providing an example of a coalgebra . In this case the set has ten elements, four of which are assigned the edge co-operation, each thus outputting two elements, and six of which are assigned the vertex co-operation, each thus outputting no elements.

Not all coalgebras of this -set cohere as directed multigraphs: it is necessary to impose the additional condition that the two outputs of each edge are vertices, not edges. The the ability to specify such conditions on coalgebras is provided by the concept of a comonad.

NoteFurther reading

So far we have seen that the structure of a functor encodes expressions related by equations. A key aspect of universal (co)algebra is missing: relations between nested expressions. For example, in the structure of a -set generated by an operation , we cannot express the associativity equation , which we may wish to impose on algebras. Likewise, in the structure of a -set generated by “edge” and “vertex” co-operations and as above, we cannot express that the outputs of an edge are both vertices.

The additional structure of a monad or comonad expresses relationships between expressions and nested expressions (“expressions filled with expressions filled with elements of ”). Details are outside the scope of this post, but we refer the interested reader to (Mac Lane 1998) or (Riehl 2017) for prerequisite material on monads, then to (Adamek and Porst 2003) where it is shown that just as monads may be presented by operations and equations, comonads may be presented by co-operations and “co-equations”.

In particular, there is a comonad on for which coalgebras are directed multigraphs.

Note that a directed multigraph may be equivalently described as a functor -set (a functor ) where is the category:

In general, -sets for any category can be modeled as coalgebras of a polynomial -set. Namely, there is a co-operation for each object, whose outputs are all the arrows out of the object.

(In the directed graph example above, the identity arrows were omitted from the co-operations for simplicity. If they were included, then would become an operation with three outputs , , and , and would become an operation with one output . The corresponding polynomial -set has formula .)

Additional associativity and unit constraints are needed to ensure the elements and assigned co-operations cohere as a -set. For any category , there is a comonad on whose coalgebras are -sets. In fact, it was noticed only relatively recently (Ahman and Uustalu 2016) that comonads on given by polynomial formulas are precisely the same as (small) categories.

Viewing a -set as a coalgebra, the co-operation assigned to an element specifies the type of the element and produces at once all elements derived from that type. The same intuition may be applied to more general coalgebras of -sets.

3.2 Undirected multigraphs

There are coalgebraic structures besides -sets, for example undirected multigraphs.

In this case the two vertices output by the edge co-operation are unordered pair instead of an ordered pair. The generating co-operations and are as before, but this time with the relation . The undirected multigraph below is an example of a coalgebra for the presented endofunctor.

As in the case of directed multigraphs, there is a comonad on whose coalgebras are undirected multigraphs. Perhaps it is reasonable to call this a “generalized category” , for which “-sets” are undirected multigraphs.

3.3 Sheaves

This final example is intended to whet your appetite, so do not be disturbed if some aspects remain puzzling.

A sheaf is a way of gluing regions of a topological space together into a larger space that locally resembles the original space. It can be modeled as a coalgebra, using a co-operation for each point of , outputting the points in its “infinitesimal neighborhood”.

That is, sheaves over are coalgebras for a -set generated by a co-operation in for each point whose supports (see earlier section) are exactly the neighborhoods of . More specifically,

where is the neighborhood filter of . In terms of generators and relations, this -set is defined as follows: for each point , there is a -ary generator for each neighborhood , all of which are identified so as to only retain dependence on the elements that are within arbitrarily small neighborhoods of . This determines a co-operation that essentially outputs an infinitesimal neighborhood’s worth of elements.

Again, additional associativity and unit constraints on a coalgebra are needed to ensure that the coalgebra coheres as a sheaf, and again this is captured via comonad structure. For any topological space , there is a comonad on , with the above formula, whose coalgebras are sheaves over , and the topological space can be recovered from the comonad. In this way, like categories, topological spaces are identified with a certain class of comonads on .

In fact, topological spaces play the same role in relation to general comonads on as preorders do to categories. To learn about this analogy and related curiosities, stay tuned for the upcoming paper

COMONADS ON SET

by Kevin Carlson, Aaron David Fairbanks, and David Spivak

where we will push as far as we can the perspective that comonads on are a common generalization of both categories and topological spaces.

1 DOTS is DO ⦚ TS

There are two things to say before trying to explain what DOTS is.

1. It’s easier to explain what a DOTS is, and then say that DOTS is the study of DOTSs. This is entirely analogous to the ambiguity in the phrase “type theory”: we can define what a type theory is, and then say that type theory is the study of type theories. This means I get to write fun-to-say sentences like “Graphs is a DOTS”.

2. DOTS should be read as a definition with an attitude all in one, split very neatly down the middle: DO ⦚ TS. That is, the definition part is (an algebra over) a double operad, and the attitude with which to interpret this definition is as a theory of systems.

So the rest of this post will be split following this idea. First we’ll look at the DO part, and then the TS part, and finally we’ll explain one way that double theories can fit into the story. But here’s the brief takeaway from all that follows:

A DOTS is the data of three 1-categories, along with instructions for how to fit them together in such a way that we can think of them as “things”, “sub-things”, and “ways to glue things along their sub-things”. Furthermore, we can really say the word “system” instead of “thing”.

This means that you can come along with your favourite systems theory — say, Petri nets, or undirected wiring diagrams with little systems of ODES inside of them, or … — and ask “do I have a DOTS?”. But just like how you might come along with your favourite category and ask “do I have a symmetric monoidal category?”, the answer might be “I can’t really tell you unless you’re a bit more specific about some of your extra structure”.

2 What is a DO?

The definition of an algebra over a double operad is a categorification of that of an algebra over an operad: if we take the definition of the former (that we are still yet to give) and replace every category in sight by a set then we will recover the classical definition of the latter. For now we’ll just give a definition of the DO part with seemingly arbitrary notation and conditions, but these will be explained once we look at the TS part.

Here is the category of bags of objects in , defined as the lax slice of the core of finite sets over . This is essentially a categorification of the notion of multisets of objects, . Then an algebra over a double operad consists of

1. 1-categories , , and
2. a functor
3. a span
4. a functor , where is the pullback of the cospan
5. (some coherence data that I’m not going to describe)

such that the following diagram commutes:

3 What is a TS? (or: Why is a DO?)

So we’ve given a pretty unintuitive definition of an algebra over a double operad, but why? This is where the attitude comes in, by defining some terminology.

• Objects in are called systems, and the morphisms are called morphisms of systems. Don’t worry too much about what the word “system” means here. Think of this as a classic nLab-style definition: a system is just an object in the category of systems.

• Objects in are called interfaces; the functor takes a system to its interface, which can be thought of as its “boundary”, or as the part of the system that other systems can see and interact with.

• Objects in are called interactions, which take a whole bag¹ of interfaces as an input and a single interface as an output and tell us what the composition should be. (This part is just like how an algebra for an operad tells me about composition.)

• The functor is called the composition map. It takes a bag of systems and an interaction that can take their interfaces as input (i.e. the pullback ) and returns a system; the diagram commuting tells us that the system it returns has precisely the interface specified by the interaction pattern.

4 Double theories give DOTS

Recall (Lambert and Patterson 2023) that a double theory is simply a double category “with an attitude”, where that attitude is that we should be interested in its models, i.e. (lax) double functors .

It turns out that any double theory gives rise to a DOTS by looking at its models, in the following way:

1. The three categories:
   • is the category of pairs of models of
   • is the category of models of
   • is the category of multi-cospans in , i.e. an object is of the form for some .
2. The functor is given by projection onto the second component. This means that we might as well write the objects of as .
3. The left leg (resp. right leg) of the span is given by the source map (resp. target map) of the multi-cospans.
4. The functor is given by taking the colimit.
5. Everything just works I promise.²

This is a very particular kind of DOTS, known as a colimit DOTS (because ).

5 So which for what?

Although double theories give specific examples of DOTS³, studying them in their own right is still interesting. One thing that is particularly powerful about double theories and their models is that we can cleanly peel apart “shape” and “meaning” (syntax and semantics). For example, a really nice thing about basing CatColab to on the language of models of double theories is that we get this separation: I can build a causal loop diagram, and then afterwards decide if I want to interpret the objects and arrows as defining some linear system of ODEs.⁴

However, DOTS lets us specify interaction patterns (and thus compositions) much more flexibly, and talk about systems that don’t “look like categories”. This is something that double theories struggle with, since the simplest double theory — the point — already has categories as its models.

1 Introduction

Time was. Time is. Time will be.

Time is a concept embedded in our reality. Yet, when we consider the truth of statements, the logics we conceive might not have temporal aspects. Whether in propositional logic, predicate logic or epistemic logic, once a statement holds, it holds forever, no matter the time or the place.

But there are times and places when and where this is not true. Though the statement “it is raining” might not be true now, it might be true the next day. And if you are outside tomorrow, soaking in the rain, the statement “Rain is hitting you” might only hold until you arrive home.

Though many logics do not have the ability to describe these time-dependent statements, a family of modal logics do, known as temporal logics. There are many types of temporal logic: interval temporal logic (see Della Monica et al. 2011; also Moszkowski 2012), Lamport’s temporal logic of actions (see Lamport 1994; also Rabinovich 1998), computation tree logic (see Reynolds 2001), and other possibilities. Each temporal logic is grounded in a specific (philosophical) interpretation of time, contains differing modalities for time, and has numerous applications beyond mathematics and philosophy, such as in the formal verification of computational systems.

As with any logic, a natural question arises: is there a corresponding algebra to it? This question comes from mathematical logic, whereby given a logical theory , one asks whether one can construct the Lindenbaum-Tarski algebra induced by . When consists of propositional tautologies and the logic is classical, one finds that the Lindenbaum-Tarski algebra is the Boolean algebra. Likewise, if is the theory of intuitionistic logic, then the Lindenbaum-Tarski algebra is the Heyting algebra. We therefore ask: for which temporal logics can we apply Tarski’s method¹ and develop temporal algebras?

In this post, we’ll be focusing on a form of temporal logic called propositional linear temporal logic (PLTL), with the hopes of upgrading statements (that is, predicates) on interfaces that use classical logic into statements on systems that are time-dependent. To do this, we will work through four key parts:

1. We describe the axioms and rules of PLTL. From this, we can develop its theory.

2. We develop the category of PLTL algebras, describe the free PLTL algebra of a poset (such as a Boolean or Heyting algebra), and consider the PLTL algebras induced by a monad from this free-forgetful adjunction.

3. We tie this into José Siqueira’s recent work (see Siqueira 2025) on hyperdoctrines (which describe underlying logics such as classical or intuitionistic propositional logic), and how to temporalise them using PLTL.

4. We give the example of the stream comonad, which describes -sequences, and how to temporalise statements about these sequences.

2 PLTL

Let us introduce PLTL, as described in Reynolds (2001).

1 Example incremental solutions

2 Solution to the basic incremental search problem: purely additive rules

How can we unify and automate the intuition of the above examples, in the form of an algorithm that is a couple of lines long and self-evidently correct? The following dictionary will help map the intuitive concepts from the above examples onto basic concepts in category theory.

2.1 A dictionary for category theory concepts

Informal term	Categorical analogue
Setting for incremental search problem	An (adhesive2) category
Pattern / Query	An object,
State of the world / set of facts	An object
Pattern match of in	A morphism
Answer set to a query in state
An (additive) rewrite rule, with pattern and replacement	A monomorphism
Application of a rewrite rule to state with matched via	A pushout of and

2.5 Batch update from multiple rule firings

We can apply many additive rewrites at once with a single colimit, rather than as an artificially-ordered sequence of pushouts.

A match into the result of multiple simultaneous rewrites has a corresponding decomposition of the pattern into a colimit of subobjects of the same shape. The case induced adhesive ‘multicube’ is shown below:

Our algorithm generalizes: at compile-time enumerate possible nontrivial decompositions of with the above shape and all interactions of subobject morphisms with rewrite rules. At runtime, some finite family of rewrite rules (e.g. ) will have been executed with corresponding matches . We loop over possible partial multicubes and look for morphisms that lead to pullback faces above all of the matches, each of which uniquely results in a new match .

WarningOptimized version when has complements

There is also an optimized batch algorithm that requires just looking for maps that commute with all and rather than filtering for ones which furthermore satisfy a pullback property. In brief, we require : this codifies that we want each interaction to be as small as possible.

2.5.1 Batch example

Let’s consider a pattern that looks for paths of length three, rather than two. This example has a batch two applications of the triangle-introducing rule. This leads to a match which incorporates material from the old graph as well as newly introduced material from both rewrites:

The resulting match, , is ultimately found starting from the decomposition of on top, which asserts parts of must overlap with , , and . The middle edge in the pattern comes from the loop of the old graph, whereas the first and last vertices and edges are respectively created via the two rule applications.

3 Conclusion

Adhesive categories are a general setting for reasoning about pattern matching. We assume some computational primitives can be implemented in any domain of interest, thought of as an adhesive category:

1. computing decompositions of an object into subobjects
2. computing small (co)limits
3. extending a partial map into a set of total ones.

The efficiency of this approach derives from its systematic transformation of a subgraph isomorphism problems into a collection of rooted subgraph isomorphism problems. Another aspect that makes this algorithm special is that it leverages information beyond the extensional difference of an update in order to speed up incremental answer set update.

Solving the base problem can be straightforwardly generalized to non-monic matches, non-connected patterns, and batch updates. In cases where one can compute complements to subobjects, we can be even more efficient.

There is an implementation in AlgebraicRewriting.jl in arbitrary -Set categories (2022). is a special case of this. There is more to be said about deletion and merging in future work!

1 Unrelatedly, substitution is sometimes pushout

We’ll demonstrate this phenomenon by focusing on two categories, and , as examples.

1.2 Second example category: Datalog

Let’s attempt to understand the logic programming language Datalog categorically. The key concepts here are relation, fact, constant, variable, rule, and query. For example one can write some facts about a binary edge relation and constants v1, v2, and v3.

edge(v1,v1). edge(v3,v3). edge(v1,v2). edge(v3,v2).

When someone writes a Datalog program, there is a starting set of ground facts (which are facts about constants). There are also rules, which have two parts, known as the head and the body (written HEAD :- BODY.⁵). We can think of the body as the preconditions of the rule and the head as the postcondition(s).⁶ An example of a rule (which involves both variables and constants) is:

edge(v1,X) :- edge(X,X), edge(X,v2).

This says that: “if you give me any vertex with a self loop and edge into v2, I’ll give you an edge from v1 into ”. If we ran the Datalog program with this rule and the ground facts above, we would derive the fact that edge(v1,v3) because, if one queries the preconditions, one can get an answer X=v3 which then gets plugged into the postcondition. One can imagine making these moves until there are no further changes in the set of facts: this is called the Herbrand model of the Datalog program. The sets of facts which we encounter along the way to the Herbrand model are called Herbrand interpretations (hereafter: interpretations).

We want to think of the initial collection of ground facts as an interpretation, an update that brings us closer to the Herbrand model as a morphism of interpretations, the rule as a morphism preconditions -> postconditions of interpretations, and the finding of a set of terms that satisfies the preconditions as a morphism of interpretations preconditions -> current_interpretation. Furthermore, we want this update morphism to shake out as the result of taking the pushout in this category. Let’s craft a definition to make this happen!

Tip definition (concrete)

Fix some signature assigning arities to a set of relation symbols. Let be some set of constants. We define (hereafter just and will be implicit) to be the category of interpretations and interpretation morphisms, where:

• An interpretation consists of a set of ‘variables’, as well as relations on for every relation symbol of arity .
• A morphism of interpretations is a function , assigning variables to either constants or variables,⁷ that preserves whatever relations are in .

So to interpret the example rule above as a morphism , we must take to be edge(X,X), edge(X,v2) and to be edge(v1,X), edge(X,X), edge(X,v2). Note that, because these rules tell us how to add facts (never removing them), we should always think of the preconditions as implicitly included in the postconditions. We have and with the morphism data the function sending , which preserves the relations in because extends the set of facts of .

It will be easier to show that this category has pushouts when defined in a more abstract way.

Tip definition (abstract)

Given a signature we define the category to contain an object as well as morphisms for each . There is an associated finite limit sketch which adds limit cones asserting that its models have, for each , that are jointly monic.⁸

We define to be the the category of models of , which is subcategory of . As a category of limit sketch models, is a reflective subcategory whose reflector quotients each , merging together elements that agree on all values of . Reflective subcategories of cocomplete categories are cocomplete, with colimits computed by first including into the larger category, performing the colimit there, and then applying the reflector to coerce the result back into the subcategory.

Constants are handled via coslicing: let be , where is the functor with and for all . Note that coslices of categories with pushouts have pushouts, computed in the base category.

Consider interpretations that have just a binary edge relation. These are pretty similar to graphs as defined in the previous section; however, there is no notion of multiple edges. Also the vertices can have fixed, stable identities: although the graph is isomorphic / not meaningfully different from because the labels are not meaningful, an interpretation whose only fact is edge(v1,v2) will not be isomorphic in to one whose only fact is edge(v2,v1), if v1 and v2 are constants.

To see these pushouts in action, consider applying the rule above, using the depicted match morphism (i.e. answer X=v3 to the query of the rule’s preconditions). Note we distinguish variables from constants by using red for variables:

Note that X=v1 was also a valid answer to the query. However, in , the pushout does nothing, rather than adding another loop to v1:

Each answer to the query of a Datalog rule’s preconditions is a potential way to ground the rule, which can potentially add a new fact and bring one’s interpretation closer to the Datalog program’s Herbrand model.

A common extension, Datalog, allows for the heads of rules to introduce new terms. For example, the rule “if you gave me an edge, I could give you path of length two which has the same source and target” is expressed as:

edge(X,Z), edge(Z,Y) :- edge(X,Y). # note "Z" not in body of rule!

This is actually the natural interpretation of , though we could restrict to a subcategory which has no ability to introduce terms: to do this we would require all rule morphisms to have their underlying function be of the form .

2 Takeaway

We considered categories where objects play dual roles of being queries as well as data. In such categories, it is natural to think of elements of as answers to the query of shape in data of shape . We can also think of an element of as a query of shape that depends on results to a query of shape . Taking an answer to and using it to construct a canonical answer to is a kind of substitution (or grounding), which can be characterized as a pushout.

3 Special thanks to Minnowbrook logic programming seminar

Making explicit this connection between graph rewrite rules and datalog rules was one of many thoughts which came about talking to Michael Arntzenius and others while hiking in Adirondacks with a great group of logic programmers. This workshop was organized by Kristopher Micinski in May 2025.

2 Which axioms should we take?

A work in synthetic mathematics begins by laying out some axioms which hold in the topos we intend to work in. However, it hasn’t always been clear exactly which axioms we need in order to prove the things we want to prove. For example, while the first axiom of synthetic differential geometry presented above is sufficient for some basic facts about derivatives, in order to argue well about higher order derivatives it needs to be extended to higher order infinitesimals. Even with a pretty comprehensive list of axioms (such as those at the beginning of Bunge et. al.’s Synthetic differential topology), we often run into cases that require more axioms.¹⁰ It’s not tenable to constantly fiddle with foundations in the course of a proof.¹¹

The main axioms generally fall in the following three sorts:

• (Duality): There is a duality between “affine schemes” and “finitely presented algebras”. In the algebraic setting, this is what Ingo Blechschmidt called synthetic quasi-coherence in his thesis. But it also includes the Kock-Lawvere axiom, which asks for a duality between Weil algebras and “infinitesimal varieties”¹², and Phoa’s principle as emphasized in Sterling and Ye’s recent paper (see Section 7). It also appears as Axiom 10 in this paper on synthetic higher category theory.
• (Local choice): Any surjection into an “affine scheme” admits sections on an “open cover”. This was isolated by Cherubini-Coquand-Hutzler in the algebraic setting as Zariski local choice, but it also includes the Bunge-Dubuc Covering Property from synthetic differential geometry, as well as the projectivity of the simplices which Mitchell Riley and I assume of our simplicial cohesion in order to give a synthetic proof of the nerve theorem.
• (Tinyness): Some objects are tiny: the functor has¹³ a right adjoint. This appears in synthetic differential geometry with the assumption that the infinitesimal varieties are tiny, which allows for the definition of differential form classifiers. Mark Williams has used presentability of (tiny) representables to show that dualities present for presheaves descend to sheaves.

But this scheme is more of a suggestion than a formula. What we need is a systematic approach to choosing axioms for synthetic mathematics.

3 Blechschmidt’s generalized nullstellensatz

In an unpublished but widely circulating note, “qcoh”, Ingo Blechschmidt has put forward a general axiom scheme which should work for any topos. I’ll work towards describing his idea here (using my own notation, for those following along in qcoh). From this point on, this blog post will assume that you know a fair bit of topos theory.

Lawvere’s functorial semantics for algebraic theories in cartesian categories and finite-product-preserving functors may be extended to a functorial semantics for positive, infinitary first-order theories (“geometric” theories) in toposes and “geometric morphisms” between them. A “geometric” theory is a theory in first-order logic which allows for equality, finite conjunctions, and infinite disjunctions (i.e. disjunctions indexed by an arbitrary set). For any such theory , there is a classifying topos so that models of in any topos correspond to geometric morphisms . The identity morphism therefore corresponds to a model of in , and this is the universal model of .

Furthermore, every topos of sheaves is the classifying topos for some theory; namely, if is the topos of sheaves on some site, then classifies “flat, -continuous functors” out of by Diaconescue’s theorem. Therefore, no matter what topos we intend to work internal to, we should at least start by assuming that we have a model of a theory it classifies. For example:

• If we want to do synthetic algebraic geometry in the big Zariski topos, we may first appeal to the fact that the big Zariski topos classifies the theory of local rings¹⁴ with the affine line being the universal local ring itself. We may then begin our axiomatization by asking for a local ring .
• If we want to work with simplicial sets, we may appeal to the fact that the topos of simplicial sets classifies total orders with distinct top and bottom elements, with the 1-simplex being the universal model (and with the order relation). We may then begin our axiomatization by asking for a total order with distinct top and bottom elements.

However, satisfying this assumption only guarantees that we are working in some topos equipped with a geometric morphism to the topos we intended to work in; we need something else to guarantee that this geometric morphism is the identity. Now, inverse images of geometric morphisms preserve the interpretations of all positive formulas (because they preserve finite limits and infinite colimits, and these suffice to interpret any positive formula), so that any positive formula which is true of the universal model is true of all models in all toposes; conversely, every positive formula which is true of the universal model is positively provable (see, e.g. Theorem 2.4 of this paper. Therefore, whatever extra axioms we need to take, they can’t be positive. The special features of the universal model are negative.¹⁵

Some of these negative properties have been known for a long time. For example, Kock remarked in 197? that the affine line in the big Zariski topos (the universal local ring) is in fact a field: every non-zero element has an inverse. This is a negative property because it is an implication with an implication in the antecedent:

In his thesis, Bechschmidt extended the Kock-Lawvere axiom to his synthetic quasi-coherence axiom and observes that it suffices to prove all known negative properties about the affine line, including that it is a field. But Blechschmidt doesn’t stop there. In qcoh, he isolates a general axiom scheme which suffices to prove every special property of the universal model of a geometric theory . He calls this his general nullstellensatz, though I believe it should be called quasi-coherent definability.

The idea is simple and obviously wrong: what if every proposition concerning the universal model were positively definable, and every true proposition provable? Any true proposition about the universal model — even the weird negative ones — would follow from this axiom scheme and the assumption that was a model of .

Of course, this axiom scheme can’t hold. Blechschmidt’s genius in qcoh is to find a restricted version of it which does. Suppose is a positive theory (in ); we can then pull it back to get a positive theory in the classifying topos . We can then form a new theory, the slice theory in by adding in constant terms for every (of the correct sorts, when is multi-sorted). This is the theory of -homorphisms out of ; its classifying topos is, externally, the morphism from the classifying topos for -homomorphisms to the classifying topos of which classifies the domain of the universal homomorphism.

This is a special sort of -theory, since it only ever uses -indexed disjunctions.¹⁶ Blechschmidt calls these “geometric* theories”, but I will call them quasi-coherent theories on account of the following analogy: a theory is coherent when it only uses finitary disjunctions, and a sheaf of modules is coherent when it is locally of finite presentation¹⁷; a sheaf is quasi-coherent when it is locally of -indexed presentation; so, a theory is quasi-coherent when it only uses -indexed disjunctions.¹⁸

Blechschmidt’s updated axiom scheme (see Theorem 5.2 of qcoh) is then that (in the internal logic of ):

1. For every context of , and any suboject of the interpretation of that context at the identity morphism (which is a model of the slice theory ), there is a quasi-coherent formula in the slice theory whose interpretation at the identity is : every subset of the universal model is quasi-coherently definable in the slice theory.

2. Let be a quasi-coherent formula in the slice theory. Then its interpretation at the identity of holds (that is, ) if and only if is provable in the slice theory.

These two axioms suffice to show that the sheaf of quasi-coherent formulas in context in the slice theory, modulo provable equivalence, is isomorphic to the sheaf of subsheaves of the interpretation of at the identity of . This really is a universal axiom scheme (Theorem 1.2 of ibid.) for first-order statements about the universal model: if is a first-order formula, then we can interpret it as a subobject of the universal model; it is therefore quasi-coherently definable by the axiom scheme, and is true if and only if it is provable.

Blechschmidt uses a delicate forcing argument to prove that this axiom scheme holds for any theory . The aim of this blog post is to liberate quasi-coherent definability from these sorts of forcing arguments.

4 Dreaming of homotopy types

The difficulty with forcing argument is that they are highly reliant on syntax. This is not so awful when we’re trying to prove theorems about 1-toposes where syntax means first order logic (though, it’s a little awful). But it becomes haltingly difficult when we want to reason about higher toposes, where syntax means “homotopy type theory” and general coherence issues must be taken into account.

Let’s start by sketching a generalization of Blechschmidt’s scheme from logic to types. The main idea of propositions as types is that propositions are the -truncated types of mathematical objects. Accordingly, we have the following analogy:

Level	Syntax	Semantics
	Formula	Proposition
	Proof	Truth
	Definition	Type
	Term	Element

I am using “definition” and “term”¹⁹ here for the syntactic analogue of “type” and “element”, taking the role of “formula” and “proof” in relation to “proposition” and “truth”. We might say then that the quasi-coherent definability axiom scheme, in a highly conjectural setting, is that for any positive homotopy type theory internal to the -topos of homotopy types (“positive” in the sense of involving only -indexed higher inductive types (which include the unit, , , and all -indexed colimits), and no negative types like functions or universes),

1. For any context of and any type family , there is a quasi-coherent definition in the slice theory which interprets to .

2. For any element , there is a quasi-coherent term which interprets to .

Now, a definition should be quasi-coherent if it only uses -indexed higher inductive types, and not -indexed higher inductive types.

Mitchell Riley and I have been chatting for years about quasi-coherent definability. Mitchell observed that it really looks like some sort of path induction principle (itself a form of the Yoneda lemma): To do something for a general map (model of the slice theory), it suffices to do it at the identity. Mitchell and I call this conjectural induction principle quasi-coherent induction, and we’ve been dreaming about a type theory which would interpret into all toposes and for which the axiom scheme would become a rule. More about this later in the blog post.

For now, the scaffolding to work syntactically with internal homotopy type theories doesn’t quite exist (though Uemura-Nguyen -type theories might help). For that reason, let me turn to semantics.

5 Liberating synthetic quasi-coherence from syntax

In this section, I want to liberate quasi-coherent definability from syntax. In fact, I will show that it follows from a delightfully simple lifting property first observed by Ivan Di Liberti in his enlightening exegesis of coherent toposes and ultrastructures. This lifting property will also make it clear that we really are dealing with a form of directed path induction for toposes.

The lifting property we will need appears as Theorem 1.3.1 of Di Liberti’s paper on coherent toposes. It is so clean that I will prove it here in its entirety; this will also make it clear that it should hold just as well for -toposes.²⁰

We work over an arbitary base topos , which I will just call the topos of “base types”. In the world of higher toposes, Martini and Wolf have been working out a theory of internal higher toposes. I do not want to fuss with any details (including actually subtle size issues, and careful internalization) in this blog post, so please regard all the rest as conjectural.

Let be a -closed sub-universe of a — a “regular cardinal”.

We now prove Di Liberti’s theorem:

Now, let’s free our minds from various internalization anxieties such as the fact that regular cardinals aren’t stable under inverse image and define a quasi-coherent topos.

We might say that a quasi-coherent -topos is -locally of constant presentation. Note that if , then the definition trivializes: every -topos is -quasi-coherent, because any -filtered -category has a terminal object and so -filtered colimits are just given by evaluation at that terminal object.

Now for any -topos , we can form the arrow topos (the arrow object in the 2-category of -toposes). When classifies models of , then is the classifier for -homomorphisms. Over , we have an adjoint triple where classifies the domain of the universal -homomorphism, the codomain, and the identity of the universal model of . This in particular means that and over .²²

However, we want to conclude a statement in ’s internal logic, so we will need to consider as a -topos. I will always consider as a -topos via , and as a -topos via . Note that can view itself as an topos, since : is an -point of .

On the other hand, doesn’t exist over or . But it still leaves a trace. All that remains of , from ’s point of view, are the following two facts:

1. preserves -indexed colimits (because it has a right adjoint over ).

2. takes values in -presentable objects of for some -cardinal (because over , and is a right adjoint between -presentable categories and so is -accessible).²³

We are now ready to prove quasi-coherent induction.

Let’s now deduce quasi-coherent definability from quasi-coherent induction.

Corollary (Quasi-coherent definability). For any -theory , let . Then it is true internally to that

1. Every object of is quasi-coherently presentable (i.e. definable) in the slice theory .

2. Every element of every object is quasi-coherently definable in the slice theory .

6 The internal logic of all toposes

Quasi-coherent induction resembles path induction in that both express a lifting property of some class of display maps against the inclusion of constants into a sort of path object:

As in cubical approaches to homotopy type theory, is a space of functions from an interval object: specfically, the Sierpinski topos which classifies -propositions.

If we had a type theory where contexts were interepreted as toposes (and not just objects of a fixed topos), then we could potentially express quasi-coherent induction as an induction principle. If we could also express all the ordinary type theory constructions in a particular topos (considered as the toposes etale over that particular one), then we could have a type theory where synthetic quasi-coherence was provable and not an axiom, for any sort of synthetic mathematics what-so-ever. Mitchell and I have been calling such a potential type theory “theory type theory” or “”.

Such a type theory would have a number of benefits for synthetic mathematics. First, of course, is that synthetic quasi-coherence should become provable, rather than an axiom. But also, the ability to jump between toposes in the middle of an argument promises a resolution to one of the curious problems of homotopy type theory: despite the fact that all types are now implicitly homotopy types, much of homotopical mathematics (“brave new algebra”) becomes impossibly difficult because combinatorial methods become infeasibly littered with coherence conditions. With a type theory that interprets into all toposes, we could just jump into the topos of simplicial objects of our current topos whenever we wanted to make a simplicial (or categorical) argument.²⁵ I want to give a very rough sketch of what such a type theory could look like.

Here at the Topos Institute, we are interested in declarative modelling: first, the modeller schematizes their models, defining the theory what it means to be a model (this is also known as metamodelling). Then they develop models within that schema. And then they analyze those models computationally. By making the model description explicit, it becomes clear how we could change the model when analysis conflicts with our data or intentions; by making the theory explicit, we extend this fleetness of thought to a change in modelling paradigms.

Declarative modelling underlies the modelling approach of the Algebraic Julia project, where one first schematizes using a generalized algebraic theory (GAT), potentially including base-type valued attributes. Generalized algebraic theories correspond to the lex fragment of positive logic whose classifying toposes are the cofree toposes of presheaves on a finitely complete category .²⁶

This approach to declarative modelling also underlies the design of CatColab, where a theory is a cartesian double theory and a model within that theory is expressed as a notebook in a structure editor (left below) and analyzed using general computations (right below):

At a very high level, then, the modelling task consists of defining a (positive) theory for models, presenting a model within that theory (via positive definitions), and then analyzing that model using general computation (using the negative notions of streams and functions). I have a lot more to say about this point of view, especially concerning the development of models and the ecology of models (to take a term from Nate Osgood), but I think I’ll save this for another blog post.

Our own Owen Lynch has come up with a 2-level type theory, “Element Model Type Theory (EMTT)” for defining lex theories and their models, named after its four basic judgements. The theory has a simple presentation in the natural model style which I will now include:

This type theory is highly reminiscent of the multi-level type theories in András Kovács’ extraordinary thesis, which goes much further in the exploration of type theories for theories and their models.

Toposes over are a model of EMTT, as presented above. We take the theories over a topos to be the bounded geometric morphisms and we take the types over a topos to be the etale morphisms . The existence of -theories over types comes from the exponentiability of etale maps, proved in B4.3.1 of the Elephant.

We could then add to the above rules all higher inductive types to , since these are all stable under inverse image of geometric morphisms (they are positive). Indeed, such higher inductive types eliminate into all toposes, since colimits of etale maps are etale. To get negative types, and to express quasi-coherence, we need to keep track of the difference between a general extension and an etale extension. Etale maps preserve both positive and negative types; they are logical. Indeed, in higher topos theory, preserving functions and universes characterizes the etale maps (see Proposition 5.10 of ibid.).

Rather than add negative types in one-by-one, we can add in a single negative type former that suffices to construct all of them (I contend): model universes . If is a theory, them is the type of models of that theory. Geometrically speaking, this is the (core of the) category of points of the classifying topos of . This certainly suffices to give us universes and functions:

• The type of models of the theory of types is a type universe.

• Given a dependent type , the type of models of the theory of elements of indexed by is a type of functions .

The difficulty is that is only stable under etale substitution. It is not clear to me how to handle this syntactically in a 2-level type theory. However, in the natural model style, it should be described by a pullback as follows:

where the natural models have been restricted to the category of contexts and etale substitutions.²⁷

Actually expressing the quasi-coherent induction rule in such a setting is difficult for a number of reasons; not the least of which being that defining quasi-coherent is difficult. The notion of quasi-coherence is a bit like “crispness” in Shulman’s cohesive type theory; it means that a theory extension doesn’t use types defined from that theory.

For example, the theory is quasi-coherent in the empty context, but the theory is not quasi-coherent in the context of . This examples shows that a naive inductive approach to defining quasi-coherence will have to be modified. For this reason, Mitchell and I have also been playing around with a much more involved type theory that includes judgements for definition and term; but nothing has firmed up yet.

The second difficulty concerns the slice theories (and Hom-theories). In principle, these should be observable from the structure of the theory itself. If this is the case, it may be better to understand quasi-coherent induction not as an induction principle, but rather as a fibrancy condition on quasi-coherent theory extensions which enables certain transports.

I’ll leave the elaboration of these ideas to the future. My goal in this blog post is only really to express the possibility of a type theory for all toposes which is complete for all negative properties of all universal models.

1 Weak equivalence of diagrams

We’ve been deriving equations by chaining together initial functors between diagrams, going in either direction. Let’s give a name to this equivalence relation on diagrams.

Definition 1 A weak equivalence from a diagram to another diagram is an initial functor making the triangle commute:

Two diagrams in a category are said to be weakly equivalent if they can be connected by a zig-zag of weak equivalences.

Why call this relation weak equivalence? First, it is not equivalence in any familiar 2-category of diagrams. Moreover, though initial functors are closed under composition and any isomorphism of categories is initial, as follows directly from the definition, initial functors are most interesting when they’re not invertible. So, in order to make weak equivalence be an equivalence relation, we have to consider zig-zags of initial functors.¹

WarningWeak equivalence versus commutativity

By definition, weak equivalence of diagrams preserves limits whenever they exist. Sometimes this property can be used to extract commutation relations from diagrams, as we saw last time. However, we caution that weak equivalence does not preserve commutativity:² if a commutative diagram is weakly equivalent to another diagram , it need not be the case that commutes.

The following weak equivalence of diagrams is a minimal counterexample.

The functor shown is initial because the limit of both diagrams is the equalizer of and , whenever it exists. However, if we suppose that

so that limit of both diagrams is , then the first diagram commutes but the second does not.

It can be a useful feature of weak equivalence that it works just as well regardless of whether the diagrams involved commute. As it happens, I first got into this topic by studying “Tonti diagrams,” which present partial differential equations in physics (Patterson et al. 2023). These diagrams do not commute; if they did, the equations would already be solved!

2 E-diagrams

We now attempt to make contact with e-graphs. That’s less straightforward than it might seem since, taken at face value, the concrete descriptions of e-graphs offered by Nelson (1980; see also Detlefs, Nelson, and Saxe 2005, sec. 4.2) and by Willsey et al. (2021, sec. 2.1) look rather different. We reproduce features of both without claiming to be fully faithful to either.

Definition 2 An e-diagram over a diagram is a factorization of through an initial functor, comprising a category , an initial functor , and a diagram such that .

The pieces constituting an e-diagram have interpretations in e-graph jargon:

• The target category is a theory, most clearly when is finitely presented, so that its generating graph is a signature, defining types and (unary) operations, and its path equations are equational axioms.
• The diagram is a term graph in the theory , particularly when the indexing category is freely generated by a finite graph. The objects of are e-class IDs.
• The category is something like a hash cons. Its objects are canonical e-class IDs.³ Below, we give conditions on the functor that make the morphisms of encode the mapping of a hash cons.
• As for the functor , the object map assigns each e-class ID to a canonical e-class ID, like in a union-find, while the morphism map just exists to witness the initiality of .

We now define the category of all e-diagrams over a diagram.

Definition 3 A morphism of e-diagrams over a common diagram is a functor making the two triangles commute:

We do not assume that the functor is initial but, as we will see shortly, a cancellation property of initial functors ensures that is automatically initial. So there is not actually a choice to make.

In other words, the category of e-diagrams over a diagram is the full subcategory of the factorization category of spanned by factorizations whose left component is initial. The factorization category construction goes back at least to Lawvere (1986).

3 The trivial e-diagram

Any diagram trivially has an e-diagram over it, given by the factorization . This e-diagram is initial in the category of e-diagrams over :

The terminology is apt because the initial e-diagram is, in practice, how an e-diagram will be initialized from an existing diagram (term graph).

4 The ideal e-diagram

What is not so obvious is that the category of e-diagrams over a fixed diagram also has a terminal object. This follows from the foundational theorem by Street and Walters (1973) that initial functors form the left class of morphisms in an orthogonal factorization system (OFS) on , called the comprehensive factorization system.

Recall that a functor is a discrete opfibration, here denoted⁴ , if, for every morphism in and every object with , there exists a unique morphism in such that .

The existence of the comprehensive factorization system is a powerful result with many consequences, such as:

• the cancellation property of the left class in an OFS implies that any functor constituting a morphism of e-diagrams is initial, as noted above;
• a closure property of the left class states that initial functors are stable under pushout, as used below.

Moreover, comprehensive factorization implies that the factorization of a diagram as an initial functor followed by a discrete opfibration is terminal in the category of e-diagrams over :

The existence and uniqueness of a functor filling the square is a direct application of initial functors being left orthogonal with respect to discrete opfibrations.

The e-diagram produced by comprehensive factorization is ideal in the sense that the “hash cons invariant” (Willsey et al. 2021, Definition 2.7) is satisfied for any canonical e-node that can be formed. Rephrased in e-graph jargon, the functor being a discrete opfibration says that for every canonical e-class ID and every operation in with domain —that is, for every canonical e-node “”—there exists a unique lift of through to a morphism . The codomain of this unique lift is the canonical e-class ID associated with the canonical e-node “”. Thus, in the ideal e-diagram, the relation from canonical e-nodes to canonical e-class IDs is a function, and evaluating this function decides whether canonical e-nodes are equivalent.

So, is the ideal e-diagram the only e-diagram we’ll ever need? No, because in general it cannot be computed. As we’ll have occasion to study in a future post, the comprehensive factorization of a functor is constructed by left Kan extending the terminal copresheaf on along . Left Kan extensions generally cannot be computed, for two reasons. First, even when all the input data is finite, the output may be infinite. Second, even when the output is finite, it still might not be possible to compute it. Left Kan extension is formally uncomputable: it includes solving the word problem for categories, hence also the word problem for groups, an early and famous undecidable problem. Thus, to suggest that comprehensive factorization or left Kan extension “solves” the problem of reasoning with e-graphs would be entirely backward; rather, it is because left Kan extension is uncomputable that we need a flexible e-graph data structure supporting a variety of reasoning heuristics.⁵ Iteratively approximating a left Kan extension is just one important application of e-graphs.

5 Rewriting e-diagrams

Having codified the objects of interest as e-diagrams, we turn to manipulating e-diagrams using the formalism of pushout-based rewriting. The goal is to show that calculations similar to last time’s can be carried out by applying rewrite rules, where each rule application is guaranteed to preserve weak equivalence of diagrams. Note that this scheme is not intended as a guide to implementation. Though e-graphs can be implemented using rewriting techniques, it should likely not be in terms of initial functors between categories. Rather, having proved the procedure correct, we can forget about initiality and compile to rewrite rules operating directly on the graphs generating the diagram shapes. As a proof of concept, my colleague Kris Brown has implemented e-graphs using hypergraph rewriting (Brown 2025).

Rewrites of an e-diagram may affect just the category or both and . Below we imagine that the “term graph” only ever grows by introducing new terms, whereas the “hash cons” is also reduced through equation introduction and congruence closure. Other schemes are also possible.

5.1 Rewriting over a fixed diagram

The simplest rewrites change the e-diagram without changing the diagram it is over. Given a rewrite rule of the form

along with a match of its left-hand side, namely any map in , we take a pushout

to obtain a new e-diagram over . We use the fact that initial functors are stable under pushout.

TipRule: congruence reduction

Each morphism in generates a rewrite rule

that can be applied in this fashion to perform congruence reduction in .

TipRule: equation introduction

Any path equation in , say , generates a rewrite rule

where the checkmark indicates that the square in the indexing category itself commutes. Thus, at the expense of using diagram shapes that are not free, we can apply an equation in using a single initial functor, rather than a span of initial functors as in previous post.

5.2 Rewriting the diagram itself

Given a rewrite rule in as above along with a match , now in , we produce a compatible match in by post-composing with . We then apply the rule along both matches by taking two pushouts, which the pasting law for pushout squares allows us to express as:

The result is a new e-diagram over the new diagram and, by construction, is weakly equivalent to the original diagram . Again, we use that initial functors are stable under pushout.

TipRule: term introduction

Each morphism in generates a rewrite rule

that can be applied as above to introduce a new term in both and .

6 Outlook

In this second post on diagrammatic reasoning, we filled in aspects of the formalism neglected in the first post, proposing a definition of e-diagram and showing how to rewrite e-diagrams by applying rules in the form of initial functor motifs. While this is all “just” formalism, with the key ideas already present in the examples and calculations from last time, it has been a good excuse to review the comprehensive factorization system on , a powerful theorem that, among other consequences, points to the ideal object of e-graph computation.

Having set out the basic formalism, we are now free to embelish it by considering categories with extra structure. Next time we’ll study diagrams and initial functors for categories with finite products, recovering the standard setting of e-graphs that allows constants and operations of arity greater than one. More ambitiously, in future posts, we hope to show the benefits of a clean categorical semantics by generalizing to new settings where e-graphs are not standardly applied.

7 References

2 CatColab

Our first foray into user-facing technology is CatColab, a collaborative environment for formal, interoperable, conceptual modeling.

Now at an early alpha stage, CatColab supports modeling, qualitative and quantitative, across a range of domain-specific languages. Its key components include a library for its core logic, written in Rust; a backend server, also written in Rust; and a web frontend supporting real-time collaborative editing, written in TypeScript. At the same time, CatColab is based on, and is being developed continuously with, new mathematical approaches to categorical logic and categorical systems theory. Thus, CatColab is at once a tool intended to be used by modelers in science, engineering, and beyond, and a means to ground and validate new mathematics being created at the Institute.

While we’re proud of what we’ve built so far, we have a long way yet to go. Much of our vision for CatColab is only implicit in the tool as it exists today. A few directions that I’m particularly excited about are:

• Compositional models: The only reliable way to build or understand a complex model is to decompose it into a smaller pieces. CatColab will support importing existing models and fusing them together into a new model, a fully declarative approach to modularly constructing complex models.
• Data instances: It will be possible to create not just data schemas but also instances of tabular data over a schema. Schema and instance will be editable live and in parallel, offering much of the fluidity of a spreadsheet but with the guardrails and type safety of a database system.
• Specs and verification: In engineering, systems are designed to meet external specifications or requirements. CatColab will support defining specifications and aid in verifying that a proposed design satisfies the specs.

Ultimately, we aim to make a CatColab a next-generation tool for collective inquiry that seamlessly integrates models, data, analysis, learning, specification, and verification.

3 Growing the team

To realize this ambitious vision, we need to grow our team of technologists. I’m excited to announce two new positions on the Topos tech team:

• Software Engineer (US): call for applications
• Software Engineer (UK): call for applications

These will be the first full-time software engineers that we hire at Topos, complementing the research scientists and research software engineers currently on our technical staff. As such, this is an unique opportunity to shape the design and implementation of CatColab at a critical juncture in the project, and to both learn from, and advance toward practice, the mathematical research that CatColab is based on.

If you have any questions about the positions or the process, please don’t hesitate to reach out at the email addresses linked in the calls for applications. Thank you for your interest in Topos Institute!

1 Representing equations using diagrams

If you’ve encountered any abstract algebra, you’ve likely seen how category theorists represent equations between morphisms by drawing commutative diagrams. For example, saying that a square of morphisms

in some category commutes means that equation holds.

Let’s recall the formal definitions, which are standard if not without fine distinctions.

Note that a diagram may well be both free and commutative: freeness is a property solely of the diagram’s shape, whereas commutativity also crucially concerns the diagram’s image in the target category.

In practice, we work with diagrams that are free and finitely presented. That means that the diagram’s shape is freely generated by a finite graph, a data structure easily implemented on a computer. When doing computer algebra, the target category will itself be finitely presented by generators and relations. Initially, the only equations that we “know” are the ones explicitly given as relations. Our aim is to derive further equations in the target category by manipulating diagrams in it.

But wait: if a diagram in a finitely presented category can be described using only graph-theoretic concepts, familiar to any computer scientist, what is the point of introducing categories and functors? We’ll see that category theory is indispensable for concisely describing the valid manipulations of diagrams. These manipulations take the form of morphisms between diagrams, which are in the first instance functors.

2 Initial functors

Reasoning with diagrams, or “diagram chasing,”¹ is the art of transforming a given diagram, say , into another diagram , following certain rules of inference. In ordinary mathematical practice, nobody bothers to say what these rules are. We aim to be more systematic.

Let’s consider the obvious moves we could make. Given a diagram , we could pull it back along a functor to obtain another diagram . Conversely, we could extend the diagram by finding a functor and along with a diagram such that :

Without assuming more, neither contemplated rule of inference is valid in the sense (to be made precise) of exactly preserving solutions to the equations presented by diagrams. For that, we need to be a special kind of functor, an initial functor, characterized by the following definition/theorem.

2.1 Definition

Theorem 1 (Initial functors) A functor is initial, here denoted , if any of the following equivalent conditions hold:

• Pullback along preserves limits: if is a category and is a diagram in , then the canonical morphism between limits is an isomorphism, whenever the limits in question exist.
• Pullback along preserves limits of sets: if is a diagram of sets, then the canonical function between limits is a bijection.
• Slices relative to are connected: for any object , the comma category is non-empty and connected.²

For a proof of (the dual of) this equivalence, see (Riehl 2014, Lemma 8.3.4).

With a good intuition for limits, the limit characterization of initiality is useful because you can often see at a glance whether a given functor is initial just by imagining what will happen to a limit of sets. On the other hand, the slice characterization is useful in being concrete and syntactical. Recall that, given a functor and an object , the comma category has

• as objects, an object together with a morphism in ;
• as morphisms , morphisms in making the triangle commutes:

In the typical case that the categories involved are free and finitely generated, proving initiality—checking that each comma category is non-empty and connected—thus reduces to a combinatorial problem.

3 Examples and calculations

The remainder of the post will illustrate the concept of initial functor through commonly occurring examples and explicit calculations. The aim is to suggest how initial functors can be deployed for mechanical reasoning with diagrams, deferring to next time a further theoretical development.

3.2 Pasting commutative squares

We’re ready to do our first calculation, proving the pasting law for commutative squares: that two commutative squares with a common edge can be juxtaposed to produce a commutative rectangle. The notation makes this seem obviously true—that’s what makes it good notation!—but it still needs a proof. We’ll present the proof in a highly mechanical style, where each diagram in the sequence is related to the previous one by an initial functor or, when introducing an equation, by a span of initial functors.

Assume: We are given the two commutative squares

By this we mean, the target category is generated by six objects, called , and seven morphisms, called , subject to those two equations.

Construct: Start with the singleton diagram on , whose limit is trivially . Introduce the left- and right-hand sides of the path equation we want to prove:

Recognize the path as the left-hand side of the first square, then introduce that equation:

Apply congruence reduction to the occurrences of :

Now recognize the path as the left-hand side of the second square, then introduce that equation:

We’re already done, but we might as well tidy up by congruence reducing the occurrences of and then :

Conclude: The outer rectangle commutes, since the limit of the diagram is still .

3.3 A monoid presentation

Consider a monoid with an element satisfying the equations

That is, the target category is a generated by a single object, say , and an endomorphism, , satisfying those two equations.

Start with the singleton diagram and introduce both defining equations, noting that since each equation has an identity on one side, they appear as cycles:

We neglect to label the arrows since they are all over the generating morphism . We’ll now compute the congruence closure, iteratively applying congruence reduction until that is no longer possible. Arrows affected by a congruence reduction are colored the same on either side of the reduction, starting with the two blue arrows above. The first reduction is:

Next:

Congruence reducing three more times gives the sequence:

We could stop here, but we might as well reduce twice more, yielding the diagram that is the self-loop over . We have proved the equation

since the limit of the diagram is the one we started with, namely the unique object . In other words, the monoid presented is actually the trivial monoid. You might remember this is as a special case of the fact in group theory that if a group element satisfies for relatively prime numbers and , then .

4 Discussion

Category theory is often said to be abstract, even “abstract nonsense,” but it is the interplay between abstract and concrete that gives category theory its power. Initial functors, with their complementary intrinsic and syntactical descriptions, exemplify this interplay.

We’ve begun to explore how initial functors are the basis for an operational semantics for e-graphs. Rather than defining the individual steps and then chaining them together, as in the usual “small-step” operational semantics, we define the valid transformations invariantly via initial functors and then look for useful specializations. Beyond its conceptual content, this flexibility matters because practical systems tend to batch reductions together to improve efficiency. For us, that will often just amount to composing initial functors.

We haven’t yet made a direct connection with e-graphs, though there have been many hints. Another hint is that our final example is lifted directly from Nelson and Oppen’s early work (Nelson 1980; Nelson and Oppen 1980), where it is a running example. In future posts, we’ll tighten up the connection with e-graphs. We’ll also explain how the formalism generalizes to categories with finite products, the categorical counterpart to algebraic theories.

5 References

Schopenhauer’s View

A favorite answer of mine to the questions up top is Schopenhauer’s. He says things are said to be in two ways: first, as Will (capitalized á l’Allemand), in which there are no things at all, really, just the surging, chaotic flow of pure existence that lies behind all appearances.

The second, which Schopenhauer calls the world as Representation, is the familiar world of subjects-regarding-objects. What’s interesting is that, even in this world of Representation, Schopenhauer subdivides into four distinct ways in which things are grounded, or in which things can be: cognitive objects, grounded by reason (if all men are pigs and Socrates is a man, then Socrates is a pig), material objects (through causation), moments and places in time and space (grounded by other nearby places and times), and the ground of motive (the connection between intention and action).

Models, for real

Why am I inflicting this recapitulation of Schopenhauer on you? Well, let me get back to that…First: I’m interested in world modeling—what it is, how to do it, and whether AI can help. Here are some examples of world models:

• GDP = C+I+G+X
• If GDP’ > 0, then all in all the economy is doing well.
• God is love.
• The vibes are off with that guy.
• The rich are, by and large, talented, hardworking, and probably way cooler than you.
• The rich are, by and large, greedy, lucky, and probably ugly.

• 

A world model is at bottom nothing more or less than a representation in the sense of Schopenhauer: a network of objects: a subject organizing (part of) the world to itself. Modernity is the state of a civilization that can develop precise language and intrepid norms of discourse to permit more-or-less free competition between publicly expressed world models. Postmodernity is the observation that grand public world models seem to often throw the baby (the illegible, fractally complex, organic system determining individual flourishing) out with the bathwater (naive adherence to tradition).

The fundamental problem of today’s civilization is finding what comes after postmodernity, which correctly teaches that excessive reliance on legible public models leads to runaway harm beyond the edges of the model, but then gets mired in inescapable negativity. Some individuals find a way out of nihilism to what might be called a meta-modern attitude of understanding and using modern systems (plural) without identifying with them, but our civilization hasn’t advanced to this way of being at scale. (If you like this paragraph, you’ll love David Chapman’s writings on meaning and metarationality.)

The World Is Not An Object

My point in juxtaposing Schopenhauer’s metaphysics and the dream of meta-modern world modeling is this: a world model is a claim, made by some particular subject, about what is to be attended to (perhaps, what is to receive care), which is pragmatically indistinguishable from a claim about what is real. The middlebrow attitude is that a model approximates the world. This is wrong, because the world is not an object. There is no “ground truth” to compare models to, not because we’re solipsists, but because the world-in-itself is a chaotic surging mess in which no grounds are admitted.

The world is not an object. The only objects are our models of the world. This doesn’t mean models can’t be wrong—if you believe “vaccines cause autism,” I can produce observations that (defeasibly!) contradict it. But it wouldn’t be by “looking at the ground truth”; I’d observe that your qualitative model is inconsistent with what seem to me to be reasonable quantitative models, and we’d debate, presumably, until exhaustion.

A Story About the Sun

Let me tell a story about a day when Alice and Bob, a couple, were discussing the ideal frequency of sunscreen use. Alice thought people should use sunscreen every day. Bob opposed this, citing studies showing most deadly skin cancer risk comes from childhood sunburns, with moderate sun exposure possibly beneficial (via Vitamin D). Alice responded citing anecdotes of contacts who tanned and later died of skin cancer.

This seemingly rational debate concealed deeper models. Alice’s unstated world model factors included “if Bob really cares about me, he’ll listen without being annoying about statistical details.” Bob’s hidden model factors included “I’m highly sensitive to the perception of being forced into something, even when the point is correct” and deeper still, something like “the Sun is Good”—a quasi-religious view.

This pattern—where legible, objective modeling on the surface masks emotional, relational, and spiritual paradigms below—is not special to small-scale, intimate interactions, but characterizes human disagreements generally. Consider NAFTA debates: quantitative economic models may mask deeper motivations about the decay of childhood hometowns, manufacturing jobs’ reality-grounding nature, or—of course—attitudes toward foreigners.

We often have this cargo cult of public conversation where we, complicated bundles of stories and feelings, drape ourselves in black cloth and hold out crisp graphs as the only focus of discussion, hoping no one will look under the drape at the messy, emotional, willful grounds of our complete world model. One clear sign of reaching into meta-modernity would be when the sad, angry, hopeful creatures under the cloth start explaining their real models in all their incommensurability, and only then seek common ground as necessary.

A case at larger scale: in problems like “design a zoning policy for Berkeley,” I want everyone with a stake to potentially contribute. There are serious challenges:

1. Normal people would rather socialize than specify policy preferences
2. Models range from “Berkeley should never change” to complex quantitative projections
3. There are many stakeholders with stakes of many, contested, sizes

Sortition (jury-duty style participation) could help with problems 1 and 3, but number 2 is deeper. How do you funge the infungible, commensurate the incommensurable? Current options are:

1. Establish a top-down modeling paradigm producing scalar ratings that can be mechanically aggregated
2. Contributions are in plain text, recombined unpredictably and illegibly by decision-makers to produce a conclusion

Both have problems: Option 1 excludes those who can’t express themselves in the approved language, leads to overconfidence, and lacks consensus mechanisms to update. Option 2 determines the course of civilization by who yells loudest nearest the leader.

What Is To Be Done?

People haven’t really learned to communicate their world models to each other; modernity was built on narrow exceptions that postmodernity has shown unsatisfactory. How do we get people explaining honestly, fully, how they see the world? This problem has spiritual aspects—getting people to notice how they see the world may require a lot of meditation, for instance. But that part really isn’t my brief.

What do I offer, then? Structures for models: organizing, translating, comparing them. In the sunscreen example, the conversation would have improved if Alice and Bob had made their models explicit. Both could have seen the sensitive emotional points they were approaching instead of debating factual details. I think this would really help, modulo the UX problem of figuring out how to develop these models in the midst of a real, human conversation.

For those UX problems at the very least, AI support is a critical activator, smoothing over what would otherwise often become a grind of manual model refinement disincentivizing the kind of deep iterated reflection that seems worthwhile even for this kind of quotidian discussion. We need low barriers to entry but high legibility and reliability.

My suggestion, instantiated in the CatColab software being built at Topos and used to make the models shown above, is not to pick a single modeling language but a single way of constructing languages, formal enough to describe translations between approaches. AI can help by scanning possibilities and showing candidate jumps in the space of models. If candidate models have to be expressed in a well-specified language, that solves some AI-slop problems. If humans pick from candidate refinements, rather than relying on the AI to provide a consensus world model out of its broad averaging over humans-whose-writing-is-on-the-internet, that solves more problems.

The key to all this is to use AI to help clarify people’s world models, not press people to align with algorithm-generated models. We must preserve and enrich individuals’ and communities’ abilities to see the world clearly and express their vision as it is, enhanced, never constrained, by new technological affordancesssss

Thumbnail credit: Jan Buchholtz on flickr

If you liked this and want to full-length and slightly unhinged version, including rambling footnotes on the dating of the beginning of modernity and all, it’s here, below the short version.

1 Graded categories, concretely

There are several quick, conceptual ways to define a category graded by a monoidal category. I prefer to start with a fully concrete description.

2 Graded categories, double-categorically

A category graded by a monoidal category is succinctly described as a category enriched in , the presheaf category with its monoidal structure given by Day convolution. While brief, this characterization is somewhat complicated since the formula for Day convolution involves a coend. The following characterizations are equally brief but avoid mention of any colimits.

Since the only arrow in is the identity, a lax double functor can equally be described as a lax functor from the delooping bicategory of to the bicategory of spans. As usual in such cases, the advantage of working double-categorically rather than bicategorically is that we get the right morphisms. It is straightforward to show that a (tight) natural transformation between lax double functors is just a graded functor between graded categories, in the sense of (Lucyshyn-Wright 2025, Definition 3.5).

Pursuing this further, the above Proposition 1, stated somewhat loosely as a correspondence between objects, would be better upgraded to an equivalence between virtual double categories.¹ Such an equivalence almost certainly holds, though I will not check the details here.

3 Examples

You can easily find examples of graded categories once you know to look for them. Two large classes of examples are -enriched categories and -actegories, each of which embed fully faithfully into -graded categories (Lucyshyn-Wright 2025, Examples 3.8 & 3.9). Let’s look at more specific examples instead.

A step up in complexity to grade by a thin monoidal category, i.e., a monoidal preorder. In progamming jargon, the morphisms in such a graded category have types that are implicitly converted to a common supertype when composed.

NoteGrading by semilattices

Promonads capture the situation that a common set of objects admits two category structures, with one type of morphism mapping functorially into the other. Such situations occur often in mathematics, and not necessarily with just two types. Generalizing, we can take to be any join-semilattice. For example, functions between real vector spaces, or subsets thereof, may obey any of the properties recorded in the following lattice (Patterson 2020, Examples 2.4.2 & 2.4.3):

In such a -graded category, an affine map can be composed with a conic-linear map to obtain a convex-linear map. As another example, Bonchi et al define an effectful triple to be, among other things, a category graded by the linear order , where the three types of morphisms model programs with pure functions, local effects, and global effects (Bonchi, Di Lavore, and Román 2024).

Finally, categories with “parameterized” maps are often viewable as categories graded by the parameter spaces, with reindexing in the graded category being reparameterization. The nLab page on the Para construction lists many examples inspired by machine learning and optimization. The Para construction is taken to produce a bicategory, but making the reparameterizations be 2-cells in a bicategory obscures their fibered character, which is made explicit in a graded category. In terms of Proposition 1, the bicategory of parameterized maps can be recovered as the elements construction of a span-valued double functor or, equivalently, as the total double category in a discrete double fibration.

For the sake of variety, let’s see a different flavor of parameterized map, inspired by the Moore path category from topology.

4 Bigraded categories

A careful study of graded categories from the perspective of double category theory (which I have not done!) would likely yield many insights. I’ll mention just one, concerning bigraded categories, that jumped out to me while skimming Lucyshyn-Wright (2025)’s paper.

Given a monoidal category , write for the monoidal category with reversed multiplication, so that .

The bigraded product is a construction that takes a -graded category and a -graded category and produces --bigraded category. It admits a simple description using double functors, which follows directly from the definitions.

5 Outlook

If a -graded category is a lax functor on the delooping double category, then why not drop the restriction to monoidal categories and define a category graded by a double category to be a lax presheaf on or, equivalently, a discrete double fibration over ? That is precisely what Michael and I called a model of the simple double theory in our paper on cartesian double theories (Lambert and Patterson 2024, sec. 3).

However, grading is a concept with an attitude, suggestive of new examples that we did not consider. For example, a category graded by the core of the double category of finite sets and spans would be a higher-dimensional combinatorial species, having both objects and morphisms indexed by finite sets. I hope to revisit such ideas in a future post.

6 References

1 Introduction

Euclid’s elements has an almost mythical role in the history of modern mathematics. In particular it had significant influence on the notion of axiom and proof systems. Establishing a fixed set of rules and axioms and then seeing what can be built from those using proofs. Yet it is also a practical text, the propositions are often not just proofs, but operations that use straight edge and compass to create geometric figures, from the equilateral triangle to the regular polyhedra¹.

In modern mathematics the notion of proof has been well developed to be independent of the notion of construction. What happens if we instead explore ideas of construction and abstract models of tools independently? In doing so we will show how wiring diagrams provide a model to structure manufacturing methods and add detail to David Spivak’s notion of plausible fiction as a model for manufacturing.

2 Euclidean construction

Euclidean construction begins with two tools, the straight edge and compass. The axioms establish a mapping between physical actions with these tools and abstract notions. For example a straight edge can be placed next to two points on a piece of paper and the line between them drawn. This is abstracted as the axiom that any two points lie on a straight line. Similarly the point of a compass can be placed at one point and the circle through the second traced. Which gives the abstract definition of a circle.

At this stage it is worth noting that the rules allow our actions to have abstract meaning as well as physical content. The points we begin with will not be perfect points, but dots drawn with area. The line between them will thus not really be unique, even if there is sufficient skill to draw the line so it does actually pass through them. Yet, abstractly we can consider the small dots to represent pure points, and then the line we draw represents the line between them. Similarly the surface we are working on, for example a sheet of paper, is clearly never the infinite Euclidean plane. It has boundary, and if we zoom in far enough it will no longer be flat. Yet in many cases the assumption it is just Euclidean is incredibly useful.

The beautiful aspect of this abstraction is that these issues can often be ignored, and thus the physical drawings can help understand the world created by the axioms and in contrast more advanced propositions can create useful physical processes. In the case of geometry we are quite comfortable with this relationship and often do not even reflect on it. It is therefore worth considering a case where the abstract and physical traditions of geometry diverge, in the construction of regular polygons. We therefore have a conversation between the abstract system and the physical where each can inform and help understand the other.

3 The challenge between theory and practice

The relationship between the two can be quite subtle and not always what is expected. For example, take the powerful results of Gauss and Wantzel showing exactly which regular polygons can be constructed with straight edge and compass. In particular the heptagon can’t be and the 17-gon can. Yet in Islamic art, that used straight edge and compass extensively, we find heptagonal tilings. These were constructed not by bringing in an additional tool but by a process of approximation:

1. Take the circle you want to construct a heptagon, and guess the right length for the edge.
2. Run the compass round the circle making points at that distance.
3. Adjust based on the final error.
4. Repeat 2 and 3 until the error is small enough.

In practice, therefore, the lack of an exact construction for the heptagon is not a significant barrier. In addition the slight blurring between the physical and the abstract, mentioned above, ensures that the resulting heptagon is no less accurate than a pentagon constructed perfectly by Euclidean methods. When considering the 17-gon this becomes even clearer, as the “perfect” construction is technically challenging and complex. When I have tried it, my initial guess of the correct edge length is often more accurate than the length I obtain following the supposedly exact method. The approximation method is also significantly simpler. Thus, even when a perfect abstract method exists, it might not be the best option in practice.

This illustrates a key challenge in considering abstract models of manufacturing. Especially as they are developed these might seem to be ill-defined and non-rigorous to mathematicians, on one hand and a lot of work for little practical benefit to engineers and practical makers. Yet it is already happening informally, especially with digital manufacturing, which turns computer designed models into physical objects with CNC and 3d printing. Creating good abstractions to look at the systems as a whole has the potential to make the processes more visible and thus help with communication of ideas from prototype to manufacturing, reveal significant challenges and enable more direct application of research into practice.

A good candidate for this abstraction is the wiring diagram. One of the ways these can be implemented is in data flow programming, for example Grasshopper (part of the CAD software Rhinoceros 3d). The following examples will be implemented in Grasshopper to keep a consistent visual language, even when there is no computational content.

4 Example: Euclid Book 1 Proposition 1

Let us begin by going back to Euclid for inspiration. Here is the construction from Proposition 1 of book 1 of Euclid’s elements rendered in Grasshopper:

Note that this is a functioning operation that gives a triangle for a given base. This implementation immediately reveals two limitations in Euclid’s version, for example the lack of any axiom to say when two circles intersect (that box is labelled with a question mark). There are also two intersections and both are drawn here, giving two triangles.

Video

After the construction, however, the proof of Proposition 1, continues, showing that the resulting triangle is indeed equilateral. So the large green region can be seen as a component in its own right, taking in a line and returning an equilateral triangle with that line as base. In Euclid this proposition is used as a component in many later proofs starting with Proposition 2.

5 Proposal: Augmented Wiring Diagrams

The key proposal here is to augment each component in the wiring diagram with assertions about the nature of the result. For a full wiring diagram these assertions could be used to prove properties of the output, perhaps with additional assertions on the input. In the case of Proposition 1, above, the base level components are the axioms and definitions of the elements.

At the lowest level these assertions would be just that, clear assumptions about the nature of the inputs and outputs. They can then be combined into standard workflows (just like propositions) where the behavior of a collection of components has an argument (or even proof) that it will work a certain way, given the assumptions of the sub-components. The behavior of more complex components thus rests on the hopefully simpler assertions of its components, rather than just an assertion that the whole thing works.

The goal might be best described as creating a more local mathematics. Instead of proof being needed for powerful universal statements within a broadly accepted axiom system or field of study, the proofs would just be of interest to a specific case, the axioms required just being assumptions about how certain tools work (or should work). Below I give an example of how the process of milling wood can be considered in this frame. One distinct aspect of this local mathematics is that the axioms are far more lightweight. The solution to a problem might be including a new tool, and thus axioms about its use, rather than clever working with what is already on hand.

In developing a techniques for unfamiliar or new problem, this process connects directly to David Spivak’s notion of plausible fictions. An overall claim is made that can be broken down into a series of steps. At each stage the steps can be asked if they are plausible. Any step that is perhaps plausible but certainly not straightforward—maybe on the verge of “magical”—must be broken down further. In this case instead of building up the wiring diagram from already accepted components we start with a component we want to create. To achieve this much of the work might take place through standard processes, so the magic unknown component can then be pushed down into a simpler subcomponent. Over several rounds everything is perhaps reduced to known work and the whole process can be tested. Alternatively a key step might still be missing but new software, machine or tool can be developed to solve that issue.

The multi-level nature of wiring diagrams, combined with the notions of plausibility and proof also allows for details to be blurred out an revealed as needed. For example in the planning stage a task that has not been done can be deemed plausible to achieve and left until it needs to be done. An example might be work holding of material in a machine. This always requires a little bit of work, with standard techniques like vises, but can generally be left until the time comes to machine the part.

In this picture there are obviously multiple levels of plausibility. In mathematics for example, everything must be reduced to an accepted axiom set as the only possible plausible set. For manufacturing plausibility is allowed to rest on concrete practical claims about the abilities of certain tools, but the list of tools available is flexible, so new axioms can be created. This is still more concrete than the examples of social change in Spivak’s work that allow plausibility to include predictions about reasonable future acts.

6 Example: Milling Lumber

We now consider a more practical example, turning rough cut lumber into a dimensioned board². Here is the diagram:

This process begins by using a jointer which is able to create a flat surface on one side of the board, and a second flat surface at ninety degrees to the first. This creates a reference edge. Then a planer is used to make the other side of the board parallel to the first. Finally a table saw makes the far edge parallel to the jointed edge and cuts the board to length. These operations also give the height, width and length of the board.

Each tool here behaves like an axiom, we asserted its abilities. Much like the Euclidean axioms assert the abilities of straight edge and compass. Combining these assertions we are able to prove that the resulting board can be considered abstractly as a cuboid with the given dimensions. A rather complex assertion is thus reduced to assertions about the abilities of individual machines. At each stage a different machine might substitute as long as it is capable of the equivalent operation. A CNC router for example can be used to flatten a face, and give a straight edge, replacing the jointer. Similarly a mitre saw can replace the table saw in crosscut to cut the board to length.

It is worth noting that the assertions made here can also be augmented, for example to add claims about the time spent, or the quality of the finish that a tool gives. These extra assertions might mean that a previously possible replacement tool cannot be used.

7 Example: Curved Metal Sculptures

The examples given so far are more illustrative of the idea, placing classic ideas of mathematical proof or practical process into the language discussed here. To conclude I have a final example where versions of these ideas were used in a research setting to develop a new technique; the zipform system I developed with Emily Baker, initially to create the Gearhart Courtyard Curvahedra sculpture (above) at the University of Arkansas.

The nature of practical examples means that the process has a larger technical load that might be of general interest, but is not all directly relevant to the specific discussion here. I give a more detailed discussion in an appendix. Here is the wiring diagram:

In order to make this sculpture the challenge was to create curved metal beams. In order to achieve this we already had a powerful tool, a plasma cutter that can cut sheet metal into nearly any shape we ask of it. Sheet metal can also be bent in many ways, as long as its intrinsic geometry is not changed. Mathematically these bent sheets can be considered as developable surfaces. In the physical space we can therefore assume we can make arbitrary shapes and then find ways to bend them precisely. This created two smaller challenges, firstly could we create developable surfaces with the properties we wanted, and secondly could we bend the cut sheets into the right form.

The first problem was solved with differential geometry, and the Bishop or zero-torsion frame which creates developable surfaces, at right angles containing a given curve. The physical solution also used this frame, building a jig to keep the two flat sheets at right angles as they were bent together to recreate the curve in physical space.

This theoretical process, with its assertions, was also being tested in practice. Our first tests were not even in steel, but in a different material (paper) cut in a different process (laser cutting). Like sheet metal this can be assumed to be a developable surface. We then made scale models in metal. For both these test models we used augmented reality to compare the designed beam on the computer with the physical model. Abstractly this gives a different mapping from the computer model to the physical world. We therefore achieve a form of commuting diagram to validate that the beam built is the one we think it is. The accuracy was rather necessary for the large sculpture as the beams formed loops and needed to be rigid, so the tolerances needed to be tight enough that the loops could be connected without bending.

The use of paper implies a different abstract relationship. As both paper and sheet metal can be considered as developable surfaces a paper model can be scaled up to one from sheet metal. Here, though, one does have to be careful. Not all properties are shared, for example paper is very elastic when bent, and so in combining the two pieces hand holding and glue were sufficient for the integration phase. This is why, while the paper model was encouraging, the scale model in metal was necessary. This is an example of how making assertions clear is useful even when the assertion is incorrect or not complete. The argument here that gives a path from a paper model to metal just uses the developable surface assertions. These do not say anything about the elastic properties of the materials. Understanding this reveals where work might need to be done. The use of paper models as a design space for metal ones can then take into account the precise differences as well as the relationships.

After validation, and just like the propositions of Euclid, this technique can now be used in many different situations. For example we have used it to create a completely different shape a hyperboloid, rather than a sphere (shown below). Part of the mathemalchemy exhibit. The zipform technique was also used to create precise developable surfaces to cast optimised concrete beams. In each of these cases additional geometry was required to create the curves involved, but the method described then smoothly recreated this geometry in physical form.

8 Conclusion

The ideas described here are still very much in development. To me, however, they reveal some powerful aspects of flexibly mixing abstraction into physical manufacturing. In particular I feel there are several potential benefits, such as:

• Breaking problems down into component pieces. (like the milling of wooden blocks, or the propositions in Euclid’s elements).
• Combining tools and processes to create new reliable technologies (like zipform).
• Clearly describing the power and limitations of moving between different physical modeling spaces (like the difference between constructing polygons in theory and practice, or moving between paper and metal models).

The Author

Edmund Harriss is an assistant professor at the University of Arkansas holding a joint appointment between mathematics and art. Much of his work considers the ways mathematical ideas interact with the physical world other than in the classic notion of scientific models. This includes methods of illustrating mathematics to reveal the power and beauty of the ideas, but also open up questions for mathematical research. To do this he often works to create illustrations and artworks using digital manufacturing, leading to an interest in the right ways to conceptualize those processes, and thus the work discussed here. He can be reached at eharriss@uark.edu and as Gelada on Mastodon and Bluesky. Other examples of his work to bring mathematics into the physical world include his two coloring books and the construction toy curvahedra.

9 Appendix: Zipform details

The heart of zipform uses the rigidity of a T-beam, but creates one that is curved, rather than straight. In fact where the joint between the vertical and horizontal followed an arbitrary smooth curve in 3d. In the resulting wiring diagram (below) information about the bending of the curve is split into two curvature functions (using the zero torsion or bishop frame). These functions can then be used to created two curves with that curvature. Thickened versions of these curves are then cut out of sheet steel. The actual curve is the centre line of one piece that will form the crossbar of the T and the edge of the other (forming the vertical). The two flat sheets are then placed together in a T-shaped jig (shown above) and welded together. They are then pulled a little way through, in each case the thin direction of the metal allows it to bend to match the curve on the other sheet. As they are pulled, bent and welded together the curve along their intersection recreates the input curve. Here is the diagram:

We therefore start with an assertion about this process as a whole. The curve that is actually made is the input curve. In development this was broken down into smaller assertions that could be combined with proof to create a theoretical process.

We began with a commonly used assertion, that sheet metal cut on a plasma cutter, and then carefully bent is a realisation of a developable surface (a surface with constant Gaussian curvature 0). This assertion for a sheet material (thinking of it as having no thickness) is quite close to the assertion that a straight edge creates lines, or a compass draws circles.

With this assertion, on the abstract side we can therefore create developable surfaces knowing they can be turned into physical objects. The zero torsion frame of the curve is perfect for this. Taking any line at right angles to a point on the curve the zero torsion frame sweeps it out along the curve in a unique way so that it never twists around the curve’s tangent (this twisting is torsion). The animation above shows curves and the resulting surfaces for different initial lines. Note that they twist along the whole length of the curve at once. The resulting surfaces are always developable. These developable surfaces can then be flattened onto the plane. When this happens the intersection with the original curve becomes a 2d curve. The curvature of the two curves produced are called the geodesic and normal curvatures. These two curvature functions determine the original curve up to rigid motion.

The resulting flat shapes can now be cut out. They then need to be bent into shape, to reverse the flattening process and recover the initial curve. In this case we began not with a process we had used before but the assertion we wanted. Essentially we needed a physical process that would work like the integration of geodesic and normal curvature. The jig described above was the result of this development.