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Stream: learning: questions

Topic: Sketch of a given category

Jonathan Arnoult (Jan 28 2025 at 16:22):

I read that every accessible category is sketchable. So, I'm looking to construct the sketch for simple categories, based on the presentation given in Sketches of an Elephant.
If we consider the category with two objects and no nontrivial morphisms (discrete category), what should its sketch be? More generally, for a free category on a given graph, is there an algorithm to get the corresponding sketch?
The proof of the general result in prop. 2.3.8 does seem too abstract to be usable in practice as a construction mechanism

Morgan Rogers (he/him) (Jan 28 2025 at 16:46):

Here's a sketch for the two-object discrete category. It has objects $L,R,T$, arrows $L \to T \leftarrow R$, $T$ is required to be a limit (terminal object), $L \to T \leftarrow R$ a colimit cone. Then a model of this sketch must send $T$ to a terminal object, while $L,R$ are sent to sets whose coproduct is $T$ (there are two options). That gives a category of models equivalent to the two-object discrete category, and you can see how to generalize it to an $n$-object category. I think it generalizes a little further to accessible posets, but I'll let you figure out the extent to which that's true.