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Stream: theory: category theory

Topic: strict localizations

Jonas Frey (Dec 13 2022 at 23:19):

Is there anything written about "strict" localizations, ie the universal solution to mapping a set of arrows in a category to identities?

Specifically I'm looking for references for the following two statements.

1. $2$-categorical universal property: given a small category $\mathbb C$ and a set $I$ of arrows in $\mathbb C$, there exists a functor $E:\mathbb C\to L(\mathbb C, I)$ which which sends $I$-maps to identities, and such that for every category $\mathbb X$, precomposition with $E$ induces an isomorphism of categories between $[L(\mathbb C, I),\mathbb X]$ and the full subcategory of $[\mathbb C,\mathbb X]$ on functors sending $I$-maps to identities.
2. Given a diagram $D:\mathbb J\to\mathbf{Cat}$ of small categories, $\mathsf{colim}(D)$ can be computed as strict localization of $\int D$ (the covariant Grothendieck construction) at the cocartesian arrows.

Thanks!

Mike Shulman (Dec 14 2022 at 00:20):

(1) can be constructed easily as a Cat-enriched colimit, of course. But I can't recall having seen it written down.

Jonas Frey (Dec 14 2022 at 01:13):

Mike Shulman said:

(1) can be constructed easily as a Cat-enriched colimit, of course. But I can't recall having seen it written down.

Ahh yes, as enriched pushout against $\Delta(I)\times[1]\to\Delta(I)$.

Jonas Frey (Dec 14 2022 at 01:31):

I'm trying to come up with a slick argument that for $D:\mathbb J\to \mathbf{Cat}$, the canonical arrow $E : \int D\to\mathsf{colim} (D)$ is final. One can show that

• a functor $F$ is final whenever it is "co-fully faithful", ie precomposition by $F$ is fully faithful, and
• a strict localization functor is co-fully faithful whenever it satisfies the 2-dimensional universal property.

It remains to show that the localization by cocartesian arrows is actually the colimit, but I think that's straightforward by writing out the definitions.

Jonas Frey (Dec 14 2022 at 01:48):

From this, one gets a nice proof of the colimit decomposition formula [1], which says that for $C:J\to \mathsf{Cat}$ and $D:\mathsf{colim}(C)\to\mathbb X$ with $\mathbb X$ cocomplete, one has

$\mathsf{colim}(D) = \mathsf{colim}_{j\in J}\mathsf{colim}_{c\in C_j} D_{\sigma_j(c)}$,

where $\sigma_j:C_j\to \mathsf{colim}(C)$ is the colimit injection.

The proof is as follows: We have $\mathsf{colim}(D) = \mathsf{colim}(D\circ E)$ since $E$ is final, and this colimit can be computed by left-Kan-extending along $\int C\to J\to 1$. By this , the first Kan-extension is computed by taking colimits over fibers, which recovers exactly the stated formula.

[1] Peschke, George, and Walter Tholen. "Diagrams, fibrations, and the decomposition of colimits." arXiv preprint arXiv:2006.10890 (2020).

Jonas Frey (Dec 15 2022 at 15:41):

Jonas Frey said:

I'm trying to come up with a slick argument that for $D:\mathbb J\to \mathbf{Cat}$, the canonical arrow $E : \int D\to\mathsf{colim} (D)$ is final. One can show that

• a functor $F$ is final whenever it is "co-fully faithful", ie precomposition by $F$ is fully faithful, and
• a strict localization functor is co-fully faithful whenever it satisfies the 2-dimensional universal property.

It remains to show that the localization by cocartesian arrows is actually the colimit, but I think that's straightforward by writing out the definitions.

In case anybody is reading this, one correction: $\mathsf{colim}(D)$ is not the "strict localization" (better: coidentifier) of $\int D$ at all cocartesian arrows, but at the cocartesian arrows of a split cleavage of the split opfibration $\int D\to \mathbb J$. Contracting all cocartesian arrows would kill all the automorphisms, which is obviously wrong.