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Stream: learning: questions

Topic: weighted colimit is an evaluation

fosco (Sep 28 2020 at 10:23):

This should be well-known, but I can't find a reference.

Given a functor $F : {\cal C} \to {\cal D}$, if $\cal C$ has a terminal object $t$, $\text{colim } F$ coincides with the evaluation of $F$ at $t$.

What about weighted colimits? Let's say $F : {\cal C} \to {\cal D}$ is a $\cal V$-functor and $W:{\cal C}^{op} \to {\cal V}$ a weight. If ${\cal V}={\sf Set}$ it's easy to see that the following conditions are equivalent:

• the category of elements of $W$ has a terminal object
• $W$ is a representable functor, say $\hom(-,A)$.

Thus, $\text{colim}^W F\cong FA$.

But what if there is no category of elements construction for $\cal V$?

Martti Karvonen (Sep 28 2020 at 14:51):

The fact about terminal objects corresponds to the inclusion of the terminal object into $\mathcal{C}$ being a final functor. I think Kelly discussess final functors in passing (and has some results about weighted colimits too) in section 4.5. in the enriched cats book - perhaps this will satisfy your needs.

fosco (Sep 28 2020 at 17:43):

Ah, yes, it is a good idea to rephrase the property into a finality request. Thank you!