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Stream: learning: questions

Topic: UMP of models of a sketch

Jonas Frey (Oct 18 2022 at 00:40):

Hi! I'm looking for a reference for the following statement:

Let $D : \mathbb J \to \mathcal C$ be a small diagram in a small category, and let $\kappa:\Delta C\to D$ be a limit cone over $D$. Let $\mathfrak U \subseteq[\mathcal C,\mathsf{Set}]$ be the full subcategory on functors $F:\mathcal C\to\mathsf{Set}$ such that $F\circ\kappa$ is a limit cone. Then the restricted Yoneda embedding $\mathcal C^{\mathsf{op}}\to\mathfrak U$ sends $\kappa$ to a colimiting cone.

More generally, if $\mathcal C$ is the category underlying a limit sketch then $\mathsf{Mod}(\mathcal C)$ is the cocompletion of $\mathcal C^{\mathsf{op}}$ subject to the side condition that designated cones in $\mathcal C$ are sent to colimits.

Does anybody know a reference for that? Thanks!

Jonas Frey (Oct 18 2022 at 00:43):

Ahh, I just saw this, that might be the answer!