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Stream: learning: questions

Topic: exactness properties

Leopold Schlicht (Sep 01 2021 at 15:39):

Let $\Sigma$ be a set of exactness properties (properties of a category saying that certain types of limits or colimits exist, possibly together with some rules these limits and colimits have to satisfy, like "filtered colimits commute with finite limits" or "coproducts are stable under pullback"): https://ncatlab.org/nlab/show/exactness+property. I guess with a bit of work one could give a precise definition of "exactness property", although it's probably always open to dispute whether such a precise definition covers all aspects of the informal notion of "exactness property".

Now, consider the 2-category $\mathbf{Cat}_\Sigma$ of all categories having these exactness properties, all functors preserving the types of limits and colimits postulated to exist in $\Sigma$, and all natural transformations.

My question is: does the forgetful 2-functor $U\colon \mathbf{Cat}_\Sigma\to\mathbf{Cat}$ always have a left adjoint?

For instance, if $\Sigma$ is "(small) colimits exist", then the left inverse is the 2-functor sending a category to the category of presheaves on it. Also, if $\Sigma$ is "(small) filtered colimits exist", then the left adjoint is the ind-completion.

But is there something like "the elementary topos generated by an arbitrary category"?

How to construct the left adjoint? Why does it turn out that if one wants to freely add colimits, this amounts precisely to considering presheaves? (That's a bit mysterious.)

Morgan Rogers (he/him) (Sep 01 2021 at 16:01):

Hard to answer the first question when you're expressly asking about a vague definition, but there are a bunch of cases where that adjoint does exist, cf Kelly's On property-like structures and related papers.

Regarding the free elementary topos construction, I believe Freyd and Scedrov work up to this via Allegories in Categories, Allegories, but I can't remember the details of how that construction interacts with the existing features of the category.

Regarding presheaves, certainly this fact is a non-trivial result! It has been proved in various ways by various authors; hopefully someone other than me can point you to an explicit proof, at least.

Nathanael Arkor (Sep 01 2021 at 16:05):

I think the paper Lex colimits by Garner and Lack is very relevant to your question. You can consider KZ doctrines on 2-categories of categories with some class of limits, and these express exactness properties in some sense.

Nathanael Arkor (Sep 01 2021 at 16:07):

The fact that presheaves correspond to free cocompletion is mysterious indeed. Some work in that direction that I think is somewhat enlightening (but not the full story) is On the unicity of formal category theories by @Ivan Di Liberti and @fosco.

Mike Shulman (Sep 01 2021 at 16:25):

The 2-category of categories with any sort of limits and colimits and any sort of commutation laws between them is always pseudo 2-monadic over Cat. This can be proven by giving a presentation of the 2-monad by generators and relations. There's an informal sketch of the case of a terminal object in Lack's 2-categories companion, and some references to the literature.

Leopold Schlicht (Sep 03 2021 at 17:35):

Thanks all!

Leopold Schlicht (Sep 05 2021 at 17:42):

Are there some other opinions about why freely adding colimits precisely amounts to considering presheaves?

John Baez (Sep 05 2021 at 17:46):

Opinions? It's a categorification of how the free commutative monoid on a set S is the set of finitely supported functions $f: S \to \mathbb{N}$. $\mathbb{N}$ is the free commutative monoid on one generator, and $\mathsf{Set}$ is the free category with colimits on one generator.

John Baez (Sep 05 2021 at 17:49):

Colimits are the really nice categorification of sums. The obvious categorification is finite coproducts, or arbitrary coproducts, but while sets are discrete, categories are not, so we get more.

Leopold Schlicht (Sep 05 2021 at 18:12):

Thanks for that analogy! Can one prove a general theorem stating something like that the free object on $X$ is the object of (some kind of) functions from $X$ to the free object on $1$? How can one see immediately that "functor from $\mathcal C\to\mathbf{Set}$" is what corresponds to "finitely supported function $S\to\mathbb N$"? (Has this something to do with the fact that left adjoints preserve colimits?)

John Baez (Sep 05 2021 at 18:18):

There must be general theorems along these lines, and yes, if I wanted to prove one I'd try to use the fact that left adjoints preserve colimits. In the case of "the free commutative monoid on a set", I bet it's really important that every set is a colimit of copies of 1. In the case of "the free category with colimits on a category", maybe we should use an Cat-enriched version of this idea (like every category is a weighted colimit of copies of 1) or a more explicitly "categorifed" version of this idea (like every category is a pseudocolimit of copies of 1). But I don't have the energy to push this through - especially since there are people here who could do it much more easily.

Mike Shulman (Sep 05 2021 at 22:40):

In case this isn't obvious, the reason you have to use finitely supported functions rather than arbitrary ones is that in a commutative monoid you can only add up a finite number of things. Whereas in a category with colimits we include infinite colimits. If you want to freely cocomplete a category under finite colimits then you'll see something more like "finitely supported" functors, although the technical term is "finitely presentable".

John Baez (Sep 05 2021 at 22:51):

Right! It's actually mildly interesting, when thinking about these things, to consider a generalization of commutative monoids where you get to add up infinitely many terms, e.g. any cardinal's worth if you want to push your luck and fight with size issues.

The best-behaved of these are the suplattices, and if you impose that extra condition you don't suffer from size issues: the free suplattice on a set is just its power set.

John Baez (Sep 05 2021 at 22:53):

Thus, the suplattice example fits in nicely with the other examples we've been talking about: the free suplattice on $X$ consists of all functions $f: X \to 2$.

Leopold Schlicht (Sep 07 2021 at 17:27):

Thanks!
John Baez:

But I don't have the energy to push this through - especially since there are people here who could do it much more easily.

I'm waiting for them. :-)