struct FinSet
n::Int
end
struct FinFunction
dom::FinSet
codom::FinSet
values::Vector{Int}
endLecture 13: Products and typed products
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Products
In previous lectures, we covered colimits, which allowed you to take disjoint unions and to glue things together. In this lecture, we are going to cover limits, which allow you to make tuple types and to filter.
Let’s start with an example.
Suppose we are working in the category \mathsf{FinSet}, and we implement it with
Given two finite sets A and B, we want to make a finite set A \times B, such that an element of A \times B consists precisely of an element of A along with an element of B. Well, the elements of any finite set in our representation are just natural numbers. So how do we make a natural number represent two natural numbers? This is the same problem encountered when trying to store a matrix in linear memory, and we will use a similar solution.
Namely, if A = \{1, \ldots, n\} and B = \{1, \ldots, m\}, then we can make A \times B = \{1, \ldots, mn\}. Then for k \in A \times B, we get its two components as \mathrm{div}(k, m) + 1 \in A and \mathrm{rem}(k, m) + 1 \in B. Conversely, given i \in A, j \in B, we get (i - 1) m + j \in A \times B.
We formalize this construction using the idea of a product. Just like we characterized coproducts using maps out of them, we characterize products using maps in to them.
In order to do this properly, we need to talk about duals. But it will be easier to motivate duals once we have products, so let’s talk about products first.
Let \mathsf{C} be a category, and let A,B be objects of \mathsf{C}. The product of A,B (if it exists) is then any object A \times B along with morphisms \pi_1 \colon A \times B \to A, \pi_2 \colon A \times B \to B, such that for any object C with morphisms f \colon C \to A, g \colon C \to B, there is a unique map \langle f, g \rangle \colon C \to A \times B such that the diagram below commutes.
(fill in diagram)
If products exist for all pairs of objects, then we say that \mathsf{C} has products.
The “category” of Julia types and functions between them has products. The product of types A and B is Tuple{A,B}, whose elements are of the form (a,b) for a::A, b::B.
The category of graphs has products. The product of two graphs G and H has
(G \times H)(V) = G(V) \times H(V) (G \times H)(E) = G(E) \times H(E) (G \times H)(\operatorname{src})((e_1, e_2)) = (G(\operatorname{src})(e_1), H(\operatorname{src})(e_2)) (G \times H)(\operatorname{tgt})((e_1, e_2)) = (G(\operatorname{tgt})(e_1), H(\operatorname{tgt})(e_2))
(product of interval graphs)
(product of path graphs on interval)
The poset of subsets of a given set has products. The product of two subsets is their intersection.
Typed objects
We’re now going to consider products in a special type of category.
Let T be a set. A typed set with types in T is a set A along with a map t \colon A \to T. If (A,t), (A', t') are typed sets, then a morphism between them is a function f \colon A \to A' such that for all a \in A, t(a) = t'(f(a)).
Or in other words, the following commutes
Let T = \{a,b,c\}. Then …
Given a category \mathsf{C} and an object T \in \mathsf{C}, let \mathsf{C}/T be the category where the objects are pairs (A, t \colon A \to T) and the morphisms are commuting triangles
RegNets as typed graphs
Typed Petri nets
Typed products
Let’s work out what the product in \mathsf{FinSet}/T is.
Formal definition of typed product/pullback