Why build a categorical model?

Topos Oxford Seminar

Tim Hosgood

2026-03-12

What are we going to do?

Context

In January, Nina Fefferman, Mary Lou Zeeman, and myself, supported by the Center for Analysis and Prediction of Pandemic Expansion and the American Institute of Mathematics, organised a workshop:

Formal scientific modeling: a case study in global health.

Theses

1. Building a formal categorical model is useful and also complex

2. Building a formal categorical model is made considerably easier with the right tools

Questions

1. How can formal categorical modelling help people to better understand scientific models outside their expertise?

2. What do we mean when we say things like “categorical modelling can help make assumptions explicit”?

3. How can we allow for somewhat informal-looking diagrams to convey precise quantitative information?

4. How can software support the process of building categorical models?

5. What does it look like, in practice, to add new functionality to CatColab?

The model

A. Hoyer-Leitzel, S.M. Iams, A.J. Haslam-Hyde, M.L. Zeeman, N.H. Fefferman, “An immuno-epidemiological model for transient immune protection: A case study for viral respiratory infections”. Infectious Disease Modelling 8 (2023), pp. 855–864. DOI: 10.1016/j.idm.2023.07.004

1. System of ODEs
2. Parameter table
3. Informal diagram

1. System of ODEs

\left\{ \begin{aligned} \dot{V} &= pI - cV - \mu VA - \beta VT \\\dot{T} &= gT\left( 1-\frac{T+I}{C_T} \right) - \beta' VT \\\dot{I} &= \beta' VT - \delta I - \kappa IF \\\dot{F} &= qI - dF \\\dot{B} &= m_1 V(1-B) - m_2 B \\\dot{A} &= m_3 B - rA - \mu' VA \end{aligned} \right.

2. Parameter table

The table of parameters

3. Diagram

An informal representation of the model

The exercise

We’re going to try to rebuild the model and explain it back, constructing a formal mathematical artefact that can be used to centre a discussion.

Then people can point at specific things that we say and respond with “this sounds wrong” or “this seems right”.

Doing the exercise

Infection

\left\{ \begin{aligned} \dot{V} &= pI - cV - \mu VA - \boxed{\color{purple}\beta VT} \\\dot{T} &= gT\left( 1-\frac{T+I}{C_T} \right) - \boxed{\color{purple}\beta' VT} \\\dot{I} &= \boxed{\color{purple}\beta' VT} - \delta I - \kappa IF \\\dot{F} &= qI - dF \\\dot{B} &= m_1 V(1-B) - m_2 B \\\dot{A} &= m_3 B - rA - \mu' VA \end{aligned} \right.

\left\{ \begin{aligned} \dot{V} &\overset{+}{=} VT \\\dot{T} &\overset{+}{=} VT \\\dot{I} &\overset{-}{=} VT \end{aligned} \right.

flowchart LR
  T("`**T**arget cells`") --> X[infection]
  V("`**V**irus`") --> X
  X --> I("`**I**nfected cells`")
  classDef transition fill:#53b6b2,color:#fff
  class X transition

\left\{ \begin{aligned} \dot{V} &\overset{+}{=} \beta VT \\\dot{T} &\overset{+}{=} \beta'VT \\\dot{I} &\overset{-}{=} \beta'VT \end{aligned} \right.

flowchart LR
  T("`**T**arget cells`") -->|𝛽'| X[infection]
  V("`**V**irus`") -->|𝛽| X
  X -->|𝛽'| I("`**I**nfected cells`")
  classDef transition fill:#53b6b2,color:#fff
  class X transition

Top: usual mass-action, but what’s missing?

Bottom: ah, coefficients! let’s just stick them here

(fun suggestion from an undergrad: no labels, just change the length of the arrow)

Assumption made explicit

Every interaction corresponds to a specific monomial, which gives a contribution to the ODEs governing both the inputs and outputs.

The coefficients of these contributions are specific to each input and output of the interaction, rather than being determined by just the interaction itself. In other words, every interaction is “mass-conserving up to a constant”.

Clouds and no links

\left\{ \begin{aligned} \dot{V} &\overset{-}{=} \mu VA \\\dot{A} &\overset{-}{=} \mu'VA \end{aligned} \right.

flowchart LR
  T("`**T**arget cells`") -->|𝛽'| X[infection]
  V("`**V**irus`") -->|𝛽| X
  X -->|𝛽'| I("`**I**nfected cells`")
  V -->|𝜇| Y{{antibody-virion binding}}
  A("`**A**ntibodies`") -->|𝜇'| Y
  classDef transition fill:#53b6b2,color:#fff
  classDef death fill:#ddd,color:#555
  class X transition
  class Y death

Now things look v different from the diagram in the paper; theirs shows antibodies as inhibiting virus production, whereas ours shows antibodies and viruses interacting in some process as sort of equal players

Here I’m showing that, to me, the antibody response interaction is of the same sort of shape/flavour/vibe as the infection interaction: it takes inputs and gives outputs, and the interaction itself is not modulated by some any parameter, or affected by any other variables. By building this diagram we are expressing our understanding of what we intend the model to represent, by essentially saying things like “I don’t believe in modulated/parametrised interactions, only direct agent-to-agent interactions”.

\left\{ \begin{aligned} \dot{I} &\overset{-}{=} \kappa IF \\\dot{F} &\overset{+}{=} qI \\\dot{F} &\overset{-}{=} dF \end{aligned} \right.

flowchart TB
  I("`**I**nfected cells`")
  F("`Inter**f**erons`")
  X{{I-F interaction}}
  Y[interferon production]
  Z{{interferon degradation}}
  I -->|𝜅| X
  F -->|0| X
  I -->|0| Y
  Y -->|𝑞| F
  F -->|𝑑| Z
  classDef transition fill:#53b6b2,color:#fff
  classDef death fill:#ddd,color:#555
  class X death
  class Y transition
  class Z death

Interferons are named as such because they “interfere” with virus production (so Wikipedia tells me). In this model, it seems like they do so only by means of reducing the number of infected cells as \dot{I}\overset{-}{=}\kappa IF.

In particular, for example, our diagram now says that interferons do not get used up in the interferon-infected cell interaction.

Assumption made explicit

We are not distinguishing between “zero-consumption” and “parametrised-rate” interactions.

We are led to consider the important question: what is the difference between an interaction that takes some input X but doesn’t “consume” it, and an interaction that does not take X as an input but instead accepts it as a parameter to modulate its rate?

Merging or separating interactions

\left\{ \begin{aligned} \dot{V} &\overset{+}{=} pI \\\dot{I} &\overset{-}{=} \delta I \\\dot{F} &\overset{+}{=} qI. \end{aligned} \right.

flowchart LR
  V("`**V**irus`")
  I("`**I**nfected cells`")
  F("`Inter**f**erons`")
  X[1]
  I -->|𝛿| X
  X -->|𝑝| V
  X -->|𝑞| F
  classDef transition fill:#53b6b2,color:#fff
  class X transition

\boxed{?}

\left\{ \begin{aligned} \dot{V} &\overset{+}{=} pI \\\dot{I} &\overset{-}{=} \delta I \\\dot{F} &\overset{+}{=} qI. \end{aligned} \right.

flowchart LR
  V("`**V**irus`")
  I("`**I**nfected cells`")
  F("`Inter**f**erons`")
  X[1]
  I -->|𝛿| X
  X -->|𝑝| V
  X -->|𝑞| F
  classDef transition fill:#53b6b2,color:#fff
  class X transition

flowchart LR
  V("`**V**irus`")
  I("`**I**nfected cells`")
  F("`Inter**f**erons`")
  X{{2a}}
  Y[2b]
  I -->|𝛿| X
  I -.-> Y
  Y -->|𝑝| F
  Y -->|𝑞| V
  classDef transition fill:#53b6b2,color:#fff
  classDef death fill:#ddd,color:#555
  class X death
  class Y transition

flowchart LR
  V("`**V**irus`")
  I("`**I**nfected cells`")
  F("`Inter**f**erons`")
  X{{3a}}
  Y[3b]
  Z[3c]
  I -->|𝛿| X
  I -.-> Y
  I -.-> Z
  Y -->|𝑝| F
  Z -->|𝑞| V
  classDef transition fill:#53b6b2,color:#fff
  classDef death fill:#ddd,color:#555
  class X death
  class Y transition
  class Z transition

Assumption made explicit

We don’t have to group together all contributions with the same monomial form. In fact only do so when we want to suggest that these contributions arise from the same interaction or process or physical phenomenon or …

At this point, it might seem like what we’re doing is entirely the opposite of building some flexible model that will be useful for collaboration. We keep having to lock in assumptions and make choices and commit to them, and we’re not allowed any wriggle room. But this isn’t true.

Regaining flexibility

flowchart LR
  subgraph source
    direction LR
    I1("`**I**nfected cells`")
    V1("`**V**irus`")
    F1("`Inter**f**erons`")
    X1{{3a}}
    Y1[3b]
    Z1[3c]
    I1 -->|𝛿| X1
    I1 -.-> Y1
    I1 -.-> Z1
    Y1 -->|𝑝| F1
    Z1 -->|𝑞| V1
  end
  subgraph target
    direction LR
    V2("`**V**irus`")
    I2("`**I**nfected cells`")
    F2("`Inter**f**erons`")
    X2{{2a}}
    Y2[2b]
    I2 -->|𝛿| X2
    I2 -.-> Y2
    Y2 -->|𝑝| F2
    Y2 -->|𝑞| V2
  end
  source --> target
  classDef transition fill:#53b6b2,color:#fff
  classDef death fill:#ddd,color:#555
  class X1 death
  class Y1 transition
  class Z1 transition
  class X2 death
  class Y2 transition
  classDef box fill:#fff,stroke:#555,opacity:0.5,color:#000
  class source box
  class target box

Future topics

By studying the category (as opposed to the mere set) of these categorical objects, we gain the ability to express how different assumptions relate to one another.

The importance of brackets

\dot{B} = m_1V(1-B) \tag{$\dagger$} \dot{B} = m_1V - m_1VB \tag{$\ddagger$}

\dot{B} = m_1V(1-B)

flowchart TB
  subgraph dagger ["Diagram (†)"]
    direction LR
    V("`**V**irus`")
    B("`**B**-cells`")
    X[X]
    V -.-> X
    X -->|"𝑚₁(1-𝐵)"| B
  end
  classDef transition fill:#53b6b2
  class X transition
  classDef box fill:#fff,stroke:#555,opacity:0.5,color:#000
  class dagger box

\dot{B} = m_1V - m_1VB

flowchart TB
  subgraph ddagger ["Diagram (‡)"]
    direction LR
    V("`**V**irus`")
    B("`**B**-cells`")
    X[X]
    Y[Y]
    V -.-> X
    X -->|𝑚₁| B
    V -.-> Y
    B -->|𝑚₁| Y
  end
  classDef transition fill:#53b6b2
  class X transition
  class Y transition
  classDef box fill:#fff,stroke:#555,opacity:0.5,color:#000
  class ddagger box

Future topics

Using the theory of graded categories, we can allow for more complex labels on the arrows. Through a sort of DOTS approach to migrations of theories, we could allow different parts of the diagram to follow different labelling rules. Finally, by machine composition, we can substitute parameters for (functions of) state variables.

Assumption made explicit

“Interactions in the diagrams” and “monomials in the system of ODEs” are in a one-to-one correspondence.

Future topics

We can black-box (and un-black-box) parts of a model in a compositional (or “decompositional”) way.

flowchart LR
  V("`**V**irus`")
  subgraph boxed ["V–B interaction"]
    X[X]
    Y[Y]
  end
  B("`**B**-cells`")
  V -.-> X
  X -->|𝑚₁| B
  V -.-> Y
  B -->|𝑚₁| Y
  classDef transition fill:#53b6b2,color:#fff
  class X transition
  class Y transition
  classDef box fill:#fff,stroke:#555,opacity:0.5,color:#000
  class boxed box

flowchart LR
  V("`**V**irus`")
  T["V–B interaction"]
  B("`**B**-cells`")
  V -.-> T
  T --> B
  B --> T
  classDef transition fill:#53b6b2,color:#fff
  class T transition

Our final diagram

We still have some issues… but

flowchart LR
  T("`**T**arget cells`")
  V("`**V**irus`")
  I("`**I**nfected cells`")
  F("`Inter**f**erons`")
  A("`**A**ntibodies`")
  B("`**B**-cells`")
  X0[⚠️ target-cell growth ⚠️]
  X1[infection]
  X2{{interferon interference}}
  X3[virus production]
  X4[interferon production]
  Z3{{F death}}
  Y1[B-cell activation]
  Y2[antibody response]
  Y3[antibody-virion binding]
  Z1{{V death}}
  Z2{{I death}}
  Z4{{A death}}
  Z5{{B death}}
  T --> X0
  I --> X0
  X0 --> T
  T -->|𝛽'| X1
  V -->|𝛽| X1
  X1 -->|𝛽'| I
  I -->|𝜅| X2
  F -.-> X2
  I -.-> X3
  X3 -->|𝑝| V
  I -.-> X4
  X4 -->|𝑞| F
  V -.-> Y1
  Y1 -->|"⚠️ 𝑚₁(1-𝐵) ⚠️"| B
  B -.-> Y2
  Y2 -->|𝑚₃| A
  V -->|𝜇| Y3
  A -->|𝜇'| Y3
  V -->|𝑐| Z1
  I -->|𝛿| Z2
  F -->|𝑑| Z3
  A -->|𝑟| Z4
  B -->|𝑚₂| Z5
  classDef transition fill:#53b6b2,color:#fff
  classDef death fill:#ddd,color:#555
  class X0 transition
  class X1 transition
  class X2 death
  class X3 transition
  class X4 transition
  class Y1 transition
  class Y2 transition
  class Y3 transition
  class Z1 death
  class Z2 death
  class Z3 death
  class Z4 death
  class Z5 death

CatColab?

Mistakes I made

• I write down the equations generated by a part of the diagram and then realise that I’ve drawn an arrow between the wrong boxes, or in the wrong direction, or gotten two parameter labels swapped around, or …

• I count the number of interaction boxes and the number of monomial terms and see they don’t match up, and realise that I’ve missed a term

• I realise that I’ve just made a typo and this has lead to two things being swapped around, or a box called “inteferon” being created alongside one called “interferon”

• I make some mistake in the Mermaid code that means it won’t render

flowchart TB
  T("`**T**arget cells`")
  V("`**V**irus`")
  I("`**I**nfected cells`")
  F("`Inter**f**erons`")
  A("`**A**ntibodies`")
  B("`**B**-cells`")
  X0[⚠️ target-cell growth ⚠️]
  X1[infection]
  X2{{interferon interference}}
  X3[virus production]
  X4[interferon production]
  Z3{{F death}}
  Y1[B-cell activation]
  Y2[antibody response]
  Y3[antibody-virion binding]
  Z1{{V death}}
  Z2{{I death}}
  Z4{{A death}}
  Z5{{B death}}
  T --> X0
  I --> X0
  X0 --> T
  T -->|𝛽'| X1
  V -->|𝛽| X1
  X1 -->|𝛽'| I
  I -->|𝜅| X2
  F -.-> X2
  I -.-> X3
  X3 -->|𝑝| V
  I -.-> X4
  X4 -->|𝑞| F
  V -.-> Y1
  Y1 -->|"⚠️ 𝑚₁(1-𝐵) ⚠️"| B
  B -.-> Y2
  Y2 -->|𝑚₃| A
  V -->|𝜇| Y3
  A -->|𝜇'| Y3
  V -->|𝑐| Z1
  I -->|𝛿| Z2
  F -->|𝑑| Z3
  A -->|𝑟| Z4
  B -->|𝑚₂| Z5
  classDef transition fill:#53b6b2
  classDef death fill:#ddd,color:#555
  class X0 transition
  class X1 transition
  class X2 death
  class X3 transition
  class X4 transition
  class Y1 transition
  class Y2 transition
  class Y3 transition
  class Z1 death
  class Z2 death
  class Z3 death
  class Z4 death
  class Z5 death

The meta-mistake

I often make errors in the process of turning the diagram into equations.

CatColab model and analysis