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Stream: learning: questions

Topic: Colimit in terms of a localisation

fosco (Jan 07 2021 at 21:46):

I need to tweak this seemingly well-known fact about fibered category theory, so I'm looking for a reference that spells out the proof, or at least the sketch: let $F : A \to Set$ be a functori, and $\Sigma_F : E \to A$ the associated discrete fibration; then

the colimit of $F$ coincides with the category $E[C_F^{-1}]$, the localization of $E$ at the $\Sigma_F$-cartesian morphisms of $E$

Using the fact that $E$ is nothing but the category of elements of $F$, one gets a very explicit cocone $E\to \text{colim } F$ sending a pair $(a,x\in Fa)$ into the image of $x\in Fa$ under $\iota_a : Fa \to \text{colim } F$; yet, I am still wrapping my head around what are the $\Sigma_F$-cartesian morphisms of $E$, in order for them to yield a discrete localization!

This is certainly elementary, but that's exactly why I don't want to fight invain if the result is written somewhere else; algebraic geometers, save me! :smile:

Matteo Capucci (he/him) (Jan 07 2021 at 23:07):

Aren't they all cartesian?

Matteo Capucci (he/him) (Jan 07 2021 at 23:09):

In fact it makes intuitive sense: the colimit of $F$ is obtained by glueing along elements as prescribed by the category of elements $E$. But localizing here means exactly glueing, so to get the same result you need to localize the same morphisms.

Matteo Capucci (he/him) (Jan 07 2021 at 23:21):

This makes me think: maybe you want horizontal, not cartesian? Another hint of this is that one level up you localize with respect to horizontal arrows.

fosco (Jan 08 2021 at 10:06):

I don't know, there's something that does not convince me: if all morphisms are cartesian, then localising at them yields the maximally connected groupoid over the set $\text{colim } F$, not just $\text{colim } F$!