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Stream: deprecated: topos theory

Topic: taking the category of sheaves

Jade Master (May 15 2021 at 14:00):

I've heard before that taking the category of sheaves on a site cocompletes your category with respect to a certain class of diagrams. I have two questions.

1. Is this true?
2. How does it work?

Nathanael Arkor (May 15 2021 at 14:19):

I can point you to some relevant references. First, note that a category of sheaves is a (lex) reflective subcategory of a presheaf category, hence a cocompletion under some class of colimits. Next, the sheaf condition can be described as an orthogonality condition for a presheaf: see the "Orthogonal Subcategory Problem (OSP)" subsection in this blog post, for instance. Orthogonality conditions are used to describe cocompletions that preserve some existing class of colimits (e.g. see Theorem 11.5 of Fiore's Enrichment and Representation Theorems for Categories of Domains and Continuous Functions), so this says essentially that the category of sheaves is a cocompletion that respects certain existing colimits (i.e. the covering colimits).

Jade Master (May 15 2021 at 14:22):

Interesting thanks.

Mike Shulman (May 15 2021 at 14:38):

The latter (a cocompletion that respects certain colimits) is the one I'm familiar with. What do you mean by the first one "hence a cocompletion under some class of colimits"?

Mike Shulman (May 15 2021 at 14:43):

The category of sheaves is also the free cocompletion respecting existing colimits "in the world of finitely complete categories". This is the statement that it's the classifying topos for cover-preserving flat functors. Other references include lex colimits and exact completions and small sheaves.

Mike Shulman (May 15 2021 at 14:43):

Of course, the "existing colimits" might not actually be colimits, if the site isn't subcanonical. And the "finite limits" might not actually exist in the site either.

Jade Master (May 15 2021 at 14:43):

I'm not sure about how this works but I was under the impression that the Grothendieck topology allows you to choose which colimits are in the category of sheaves.

Jade Master (May 15 2021 at 14:45):

I suppose it's true because of the first fact that Nathanael mentioned but I am curious about how it works in a bit more detail.

Mike Shulman (May 15 2021 at 14:48):

That's the second fact: it lets you choose which "colimits" in the site remain colimits in the category of sheaves. The category of sheaves always has all small colimits.

Jade Master (May 15 2021 at 14:50):

Oh I get it now.

Nathanael Arkor (May 15 2021 at 14:52):

Mike Shulman said:

The latter (a cocompletion that respects certain colimits) is the one I'm familiar with. What do you mean by the first one "hence a cocompletion under some class of colimits"?

I should have also mentioned that the right adjoint preserves some colimits, which is a characteristic property for cocompletions, as it intuitively expresses that these colimits are given freely. But really this first observation is just intended to give a clue that one might expect a sheaf topos to be characterised via a cocompletion condition; the formal argument is given by orthogonality.

John Baez (May 15 2021 at 17:43):

So Jade was thinking that by taking sheaves with respect to a Grothendieck topology we can decide which kinds of colimits to freely add to a category - but freely throwing in colimits destroys the colimits we had, and in fact what the Grothendieck topology does is let us decide which kinds of colimits we don't mess with.

John Baez (May 15 2021 at 17:44):

It's also pretty easy to do what Jade was originally talking about: take a category and freely throw in a certain class of coproducts.

John Baez (May 15 2021 at 17:45):

For example, if I have a category C and I want to freely throw in coproducts, I take the category of presheaves on C, and then take the full subcategory consisting of objects that are coproducts of representables. Voila!

John Baez (May 15 2021 at 17:47):

However, note that this works because coproducts of coproducts are coproducts.

John Baez (May 15 2021 at 17:48):

So by taking coproducts of representables, we get a bunch of objects that are closed under coproducts: a coproduct of coproducts of representables is a coproduct of representables.

John Baez (May 15 2021 at 17:48):

For other kinds of colimits it may not work so simply.

Mike Shulman (May 15 2021 at 20:52):

Yes, for other kinds of colimits what you need is the closure of the representables under those colimits, which you can build either by a transfinite iteration (or inductive predicate) or an impredicative intersection.

John Baez (May 16 2021 at 01:09):

By the way, in this answer on MathOverflow Matthieu Romagny demonstrates that finite colimits of finite colimits of representables are finite colimits of representables. You might naively think that finite colimits of finite colimits of any class of objects in any category can be re-expressed as finite colimits of objects in that class! But Romagny says

I want to emphasize that here we heavily used the fact that we're in a presheaf category.

I'm not sure if he just means his argument used it, or whether the result can fail in more general categories.

John Baez (May 16 2021 at 01:21):

I bet the result fails....

Jade Master (May 16 2021 at 03:10):

It seems true to me. A finite colimit of finite colimits in C is going to be a single colimit from a product diagram $J :A \times B \to C$. This is still finite because the product of finite categories is a finite category.

John Baez (May 16 2021 at 04:20):

Matthieu's argument was vastly more complicated, suggesting that either you or he are missing something. :upside_down:

Mike Shulman (May 16 2021 at 04:34):

The point is that a map between colimits isn't necessarily induced by a transformation between the diagrams of which they are a colimit. So if you have a diagram of objects, each of which is a colimit of objects in some class, it's not immediate (and probably not always true) that you can lift the maps in the first diagram to some diagram of maps between the original objects in the class.

John Baez (May 16 2021 at 05:01):

:thumbs_up: Yes, I guess the bulk of Matthieu's argument was a method of turning a diagram of colimits of diagrams into a bigger diagram!

Jade Master (May 16 2021 at 14:46):

If you have a map going out of a colimit then the universal property of colimits induces a transformation between the diagrams you used to construct the colimit right?

Reid Barton (May 16 2021 at 14:56):

Say you have two diagrams $X, Y : A \to C$ and form their colimits $\mathrm{colim}_{a \in A} X(a)$ and $\mathrm{colim}_{a \in A} Y(a)$. Then take an arbitrary map $f : \mathrm{colim}_{a \in A} X(a) \to \mathrm{colim}_{a \in A} Y(a)$. There's definitely no reason that it has to come from compatible maps $X(a) \to Y(a)$ for each $a$. For one thing, if $A$ is a discrete category, we could also build maps by sending each $X(a)$ to some $Y(b(a))$ instead.

Reid Barton (May 16 2021 at 14:58):

But more subtly, even if we look at one specific component of the map $X(a_0) \to \mathrm{colim}_{a \in A} Y(a)$, there's no reason why it has to factor through any $Y(a)$ at all. That depends on the object $X(a_0)$ and how it relates to the construction of colimits in our category.

Reid Barton (May 16 2021 at 14:58):

In a presheaf category, if $X(a_0)$ is representable, then it is true.

Jade Master (May 16 2021 at 15:17):

Oh I see now. Thanks

Jade Master (May 16 2021 at 15:19):

Wait what's a_0 and b though?

Reid Barton (May 16 2021 at 15:20):

Like there are maps $X_1 \amalg X_2 \to Y_1 \amalg Y_2$ which send both $X_i$ to $Y_1$, or $X_1$ to $Y_2$ and $X_2$ to $Y_1$.

Jade Master (May 16 2021 at 15:20):

Oh yeah, that makes sense

Reid Barton (May 16 2021 at 15:20):

Or which don't send either $X_i$ into either $Y_i$

Reid Barton (May 16 2021 at 15:21):

say in Set, if $X_i$ consists of multiple elements