Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: A diagram in context

Fawzi Hreiki (Sep 27 2020 at 22:53):

At first approximation, a mathematical structure $S$ in a category $\mathscr{C}$ consists of:

• Data/structure $D$ (certain objects and morphisms in $\mathscr{C}$).
• Axioms/properties $A$ (the commutativity of certain diagrams, usually in $\mathscr{C}$ but sometimes elsewhere).

For example, a monoid $M$ in a category $\mathscr{C}$ with products consists of objects and maps $1 \xrightarrow{e} M \xleftarrow{.} M \times M$ such that the usual associativity and identity diagrams commute.

So I want to say that a structure $S$ in $\mathscr{C}$ consists of:

• A (small) category $K$ and a functor $D: K \rightarrow \mathscr{C}$.
• A (small) category $J$ and a functor $A: J \rightarrow \mathscr{C}$ (lets keep the axioms in $\mathscr{C}$ for simplicity)

We can package all the data into one functor by taking coproducts in $\textbf{Cat}$, and likewise for the axioms. Usually $K$ has some further categorical structure (e.g. products, power objects, etc..) which $D$ is required to preserve, and likewise for $J$ and $A$.

The problem I'm having is that the functor $A$ is meant to be able to reference the functor $D$, so $A$ needs to be a functor into $\mathscr{C}$ 'in context $D$'. So either $A$ needs to be from $J$ into some sort of comma category relating $D$ and $\mathscr{C}$ or it needs to be from some sort of exponential category relating $J$ and $K$ into $\mathscr{C}$. I'm sure theres an easy fix but its going over my head. Any help would be much appreciated.

Morgan Rogers (he/him) (Sep 28 2020 at 08:53):

Have you heard of sketches? They package data in approximately the way you're describing. The workaround is that the functor $A$ you mention should have codomain $K$ rather than $\mathcal{C}$, so the diagrams commute in $K$ already (commutativity is preserved by any functor, in particular by $D$).

Fawzi Hreiki (Sep 28 2020 at 09:18):

I'll read into it. Thanks.