# du : ℝⁿ -> ℝⁿ is the vectorfield
# init : ℝⁿ is the initial state
# dt : ℝ is the time increment
# m : ℕ is the number of steps to run for
function euler(du::Function, init::Vector{Float64}, dt::Float64, m::Int)
n = length(init)
x = zeros(Float64, (m + 1, n)) # a matrix to store the path
x[1,:] = init # initialize the first row
for i in 1:m
x[i+1,:] = x[i,:] + dt * du(x[i,:]) # write the next row
end
x # return the path
endAlgebraicJulia Tutorial
Applied Category Theory 2023
Example: diffusion in two chambers
\[\begin{align*} \dot{c}_1 &= \rho(c_2 - c_1) \\ \dot{c}_2 &= \rho(c_1 - c_2) \end{align*}\]
\[\begin{align*} X &= \{1,2\} \\ I &= \{1\} \end{align*}\]
\[ \mathrm{expose}: 1 \mapsto 1 \]
Undirected wiring diagrams (UWDs)
The simplest combinatorial operad is undirected wiring diagrams. An undirected wiring diagram consists of boxes, ports, outer ports, and junctions.
UWDs as cospans of finite sets
Undirected wiring diagrams as C-sets
An undirected wiring diagram is a functor into \(\mathsf{FinSet}\) from the category \(\mathsf{UWD}\) freely generated by:
So a UWD is a cospan with a grouping of the “ports” on one side into “boxes.”
Composing systems using operads
In compositional systems theory, we want to…
put systems inside the bubbles of a composition pattern
![]()
“push a button” to get a larger, composite system
In the case study from Part 2:
- the systems are dynamical systems as resource sharers
- the composition patterns are undirected wiring diagrams
“Composition patterns” are formalized as the operations of an operad.
Operads and SMCs
(Typed) operads, aka symmetric multicategories
- have objects (“types”)
- have morphisms (“operations”) whose domains are finite lists of objects rather than single objects
- formalize “hierarchical” composition
- generalize symmetric monoidal categories
Any symmetric monoidal category \((\mathsf{C},\otimes)\) induces an operad \(\mathrm{Op}(\mathsf{C})\) with types given by the objects of \(\mathsf{C}\) and with operations \((A_1,\ldots,A_n) \to B\) given by the morphisms \(A_1 \otimes \cdots \otimes A_n \to B\) of \(\mathsf{C}\).
Why talk about both?
- Operads are more practically convenient for specifying compositions
- SMCs are more mathematically convenient
Category theory of UWDs
There is a symmetric monoidal category \(\mathsf{Csp}\) whose
- objects are finite sets
- morphisms are (equivalence classes of) cospans, composed by pushout
The symmetric monoidal product is \(+\) (coproduct in \(\mathsf{FinSet}\)).
So a UWD is an operation in the induced operad \(\mathsf{UWD} = \mathrm{Op}(\mathsf{Csp})\):
\[ A_1 + \cdots + A_n \longrightarrow X \longleftarrow B \]
The operad algebra of resource sharers
Let’s define a lax symmetric monoidal functor \(\mathrm{RS} \colon (\mathsf{Csp}, +, 0) \to (\mathsf{Set}, \times, 1)\)
\(\mathrm{RS}(I)\) is the set of resource sharers with interface \(I\)
For the laxators, given resource sharers \((S_1,\mathrm{expose}_1, v_1), \ldots, (S_n,\mathrm{expose}_n, v_n)\) on interfaces \(I_1, \ldots, I_n\), we construct a resource sharer
\[(S_1 + \cdots + S_n, \mathrm{expose}_1 + \cdots + \mathrm{expose}_n, v_1 \times \cdots \times v_n)\]
on the interface \(I_1 + \cdots + I_n\).
Finally, for the action on morphisms, given a cospan \(I \rightarrow J \leftarrow I'\) and a resource sharer \((S, \mathrm{expose}, v \colon \mathbb{R}^S \to \mathbb{R}^S)\) on \(I\), we make a new resource sharer \((S', \mathrm{expose}', v')\) on \(I'\) by taking
![]()
and \(v' := f_\ast \circ v \circ f^\ast \colon \mathbb{R}^{S'} \to \mathbb{R}^{S'}\)
Unpacking the composition of resource sharers
Via the pushout, any two state variables in \(S\) connected to the same junction in \(J\) by a variable in the interface \(I\) are merged:
Then, to make the vector field \(v'\), we precompose with \(f\) to go from \(\mathbb{R}^{S'}\) to \(\mathbb{R}^S\), apply the vector field \(v\), and finally pushforward along \(f\) to go from \(\mathbb{R}^S\) to \(\mathbb{R}^{S'}\), via
\[ f_\ast(s)_{x'} := \sum_{x \in f^{-1}(x')} s_x. \]
Intuitively, when state variables from multiple resource sharers are merged, we sum each resource sharer’s “pull” on that state variable to get the overall derivative.
Example setup
dotr(u,p,t) = p.α*u
dotrf(u,p,t) = [-p.β*u[1]*u[2], p.γ*u[1]*u[2]]
dotf(u,p,t) = -p.δ*u
rabbit_growth = ContinuousResourceSharer{Float64}(1, dotr)
rabbitfox_predation = ContinuousResourceSharer{Float64}(2, dotrf)
fox_decline = ContinuousResourceSharer{Float64}(1, dotf)
rf = @relation (rabbits,foxes) begin
growth(rabbits)
predation(rabbits,foxes)
decline(foxes)
end
to_graphviz(rf, box_labels=:name, junction_labels=:variable)
Bibliography
Baez, John C., and Kenny Courser. 2020. “Structured Cospans.” arXiv. https://doi.org/10.48550/arXiv.1911.04630.
Brown, Kristopher, Tyler Hanks, and James Fairbanks. 2022. “Compositional Exploration of Combinatorial Scientific Models.” arXiv. https://doi.org/10.48550/arXiv.2206.08755.
Halter, Micah, Evan Patterson, Andrew Baas, and James Fairbanks. 2020. “Compositional Scientific Computing with Catlab and SemanticModels.” arXiv. https://doi.org/10.48550/arXiv.2005.04831.
Libkind, Sophie, Andrew Baas, Micah Halter, Evan Patterson, and James Fairbanks. 2022. “An Algebraic Framework for Structured Epidemic Modeling.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 380 (2233): 20210309. https://doi.org/10.1098/rsta.2021.0309.
Libkind, Sophie, Andrew Baas, Evan Patterson, and James Fairbanks. 2022. “Operadic Modeling of Dynamical Systems: Mathematics and Computation.” Electronic Proceedings in Theoretical Computer Science 372 (November): 192–206. https://doi.org/10.4204/EPTCS.372.14.
Patterson, Evan, Andrew Baas, Timothy Hosgood, and James Fairbanks. 2022. “A Diagrammatic View of Differential Equations in Physics.” Mathematics in Engineering 5 (2): 1–59. https://doi.org/10.3934/mine.2023036.
Patterson, Evan, Owen Lynch, and James Fairbanks. 2022. “Categorical Data Structures for Technical Computing.” Compositionality 4 (December): 5. https://doi.org/10.32408/compositionality-4-5.
Vagner, Dmitry, David I. Spivak, and Eugene Lerman. 2015. “Algebras of Open Dynamical Systems on the Operad of Wiring Diagrams.” arXiv. https://doi.org/10.48550/arXiv.1408.1598.