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Stream: deprecated: topos theory

Topic: coverage of Top

Matteo Capucci (he/him) (Jan 16 2021 at 12:39):

On the nLab is often mentioned that one can extend the usual site structure on a topological space $X$ to the whole category of topological spaces, and often this latter category is referred to as a classical example of 'large site'.
That said, what's exactly a covering sieve in $Top$?
Some candidates:

1. Jointly epimorphic maps
2. Jointly epimorphic monomorphisms
3. Local homeomorphisms
4. ??

Matteo Capucci (he/him) (Jan 16 2021 at 12:41):

(If it makes things easier, instead of $Top$ I accept any sufficiently nice subcategory of $Top$. For instance, I will probably work with $Top_\kappa$ for some regular cardinal $\kappa$ for size reasons)

Fawzi Hreiki (Jan 16 2021 at 12:45):

Maybe this will help.

Matteo Capucci (he/him) (Jan 16 2021 at 12:48):

Thanks! So the answers is apparently 'jointly surjective open immersions' + their saturation

Matteo Capucci (he/him) (Jan 16 2021 at 12:49):

It makes sense: monomorphisms could have a closed or even worse image, which doesn't fit the feng-shui

Fawzi Hreiki (Jan 16 2021 at 13:28):

An interesting fact which may or may not be related: the category of topological spaces is extensive which means that the category of product preserving presheaves on it is a topos

John Baez (Jan 16 2021 at 17:16):

Are there are some interesting facts about this topos of product preserving presheaves on Top?

John Baez (Jan 16 2021 at 17:20):

What we're doing here is thinking of $\mathsf{Top}^{\text{op}}$ as an algebraic theory, which is something I've never thought about, and looking at its category of models. (I've also never thought about treating $\mathsf{Top}$ as an algebraic theory.)

David Michael Roberts (Jan 17 2021 at 09:33):

There are a bunch! Here are some that in some cases are only equivalent to the usual open cover topology for a subcategory $T$ of $Top$. Here "cover" means a collection of subsets whose union is the whole space

1. Numerable open covers ($T$ = paracompact spaces)
2. Finite open covers ($T$ = compact spaces)
3. Countable open covers ($T$ = Lindelöf spaces)
4. Something about point-finite open covers (chosen so that $T$=normal spaces)
5. Covers by regular closed sets (=closures of their interiors) such that their interiors form a cover ($T=?$)
6. Covers by arbitrary subsets whose interiors form a cover ($T=?$)

We can give up equivalence to the usual topology at all, and ask for the extensive topology, where a covering family consists of a bunch of disjoint clopen subsets that cover.

We can drop the condition that we are using subsets and consider

1. Jointly surjective open maps
2. Jointly surjective universally closed maps

Then there are things like the topology generated by the extensive topology and the biquotient maps, and the same with biquotient replaced by triquotient.

Extra pullback-stable conditions can be added like separatedness (https://stacks.math.columbia.edu/tag/0CY0), which is probably superfluous when dealing with $T$ = Hausdorff spaces.

And so on.

Jens Hemelaer (Jan 17 2021 at 12:20):

If you take as covering sieves on $\mathsf{Top}$ the sieves that contain a finite jointly surjective family, then you get condensed/pyknotic sets.
This is the same thing as presheaves on $\mathcal{C}$ that send finite coproducts to finite products, where $\mathcal{C}$ is the full subcategory of $\mathsf{Top}$ consisting of those spaces $\beta S$ that are Stone–Čech compactifications of sets $S$.
(Here the explanation by Clausen).

If I understand correctly, this gives another item on @David Michael Roberts's list above.

7. Finite disjoint open covers ($T$ consists of the spaces $\beta S$ or more generally Stonean spaces).

Fawzi Hreiki (Jan 17 2021 at 13:26):

@John Baez I think the idea is that while presheaves in general have the ‘wrong’ colimits (geometrically speaking), product preserving presheaves on an extensive category (ie sheaves with respect to the extensive topology) will at least have the ‘correct’ sums.

John Baez (Jan 17 2021 at 16:45):

Hmm, I have no intuition for this, so I don't know what the 'wrong' or 'correct' sums are.

Fawzi Hreiki (Jan 17 2021 at 16:59):

Well 'correct' and 'wrong' in the sense that the Yoneda embedding preserves limits but not colimits (which, when dealing with sites consisting of spaces, often express some sort of geometric content). Extensive categories have fairly 'geometric' coproducts (a bundle over a sum $E \rightarrow A + B$ decomposes nicely into a pair of bundles over the summands $E_1 \rightarrow A$ and $E_2 \rightarrow B$). The extensive topology basically preserves (at least) this geometry.