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Stream: learning: questions

Topic: motivating "sites"

David Egolf (Sep 25 2024 at 18:22):

I eventually want to work on understanding smooth manifolds in terms of sheaves on some site, in #learning: questions > A topos containing manifolds . However, I first want to learn what a "site" is!

I'm currently reading the motivational introduction for sites in "Sheaves In Geometry and Logic". It sounds like part of the historical motivation involved maps $\rho:W \to X$ that "should" be fiber bundles, but failed the "local triviality" condition for at least one point $x \in X$, contained in some open set $U \subseteq X$. Apparently this situation could be rescued by using a "suitable unramified covering $U' \to U$".

David Egolf (Sep 25 2024 at 18:23):

I have been unable to decipher what is meant by a "suitable unramified covering $U' \to U$". It may be that I don't know enough algebraic geometry to figure this out (?). But it would be great if someone would like to elaborate on this point!

David Egolf (Sep 25 2024 at 18:32):

Setting aside "unramified" for the moment, we presumably have some covering space $\pi:U' \to U$. But I still don't see why this should help things!

David Egolf (Sep 25 2024 at 18:36):

I am assuming that this pullback square is relevant, somehow:
pullback square

Graham Manuell (Sep 25 2024 at 19:00):

I'm not sure how to address your precise questions, but I think it would be helpful to start by looking at sites where the underlying category is thin. These will give toposes that are essentially the same as the ones you know but from a different perspective. Sites give a way to 'present' the topological space. Then it is a small step replace the thin category with a more general category.

David Egolf (Sep 25 2024 at 19:03):

Thanks! That does sound like a good approach. (And it would probably be wise of me to avoid getting caught up in the details of this particular motivation from "Sheaves in Geometry and Logic"! :sweat_smile: )

David Egolf (Sep 25 2024 at 19:04):

Before moving on from that, I did want to briefly note one idea I had in that direction. I believe this bundle is not a fiber bundle, because of the "branching" at $y$:
not a fiber bundle

David Egolf (Sep 25 2024 at 19:06):

But if create a second copy of $U$ and place both copies in a space $U'$, then I can get a related bundle that I think probably is a fiber bundle:
probably a fiber bundle

David Egolf (Sep 25 2024 at 19:07):

Hmm, that's not quite right!

David Egolf (Sep 25 2024 at 19:08):

The idea I had in mind was to have one "branch" be projected to one copy of $U$ and the other branch be projected to the other copy of $U$. However, I now see I have a problem - I don't know where to send the non-branching part (to the left of $y$). And, actually, I don't even know what should happen to $y$, which was the problematic point I was trying to deal with!

John Baez (Sep 25 2024 at 19:12):

David Egolf said:

I have been unable to decipher what is meant by a "suitable unramified covering $U' \to U$". It may be that I don't know enough algebraic geometry to figure this out (?). But it would be great if someone would like to elaborate on this point!

That's algebraic geometry, using the algebraic geometers' definition of 'ramified' = 'branched' covering. E.g the map from the complex plane to itself given by

$z \mapsto z^2$

is a branched covering that's branched, or ramified, at $z=0$. If you draw it or look it up you'll see it has two sheets that come together at $z=0$.

John Baez (Sep 25 2024 at 19:16):

I don't think you want to learn algebraic geometry before learning sites, so we can stick with topology where you already know what a covering space is - and those are by default assumed to be unbranched, aka unramified - which is the case Mac Lane and Moerdijk are discussing here.

David Egolf (Sep 25 2024 at 19:17):

That sounds reasonable to me! Thanks for explaining, and giving that example of a "branched" covering.

Chris Grossack (they/them) (Sep 25 2024 at 19:18):

Here's a quick motivation for sites which you might find appealing. Hopefully you're sold on the idea that sheaves solve problems, so that it would be nice to have a notion of sheaf that works in as many situations as possible. As a quick tl;dr for why sheaves solve problems, the idea is that if you want to solve some problem and your problem is "sheafy" then you can solve that problem locally (where the problem is often easier), check that the local solutions "agree on overlaps", and then the sheaf condition ensures you actually have a solution to your whole problem!

Now usually we make this notion of "local" precise with open covers. We say that a sheaf

1. gives data attached to every open (with compatible restriction maps)
2. given an open cover and compatible data on each member of the open, we can glue that data to data defined on the whole open.

If we want to mirror this for a general category, what do we need? Well, we need a way to assign data to every object, compatible with restriction maps. That's easy, we just look at a contravariant Set valued functor. To talk about "intersections" of elements of a cover, it's nice for your category to have pullbacks (do you see why a pullback is something that you might reasonably call an intersection?) but I think there's more technical versions that relax this condition.

Lastly, how do we talk about "open covers"? Well Grothendieck realized we can just declare some families to be covers! So you get this thing called a Grothendieck Topology which says which families of arrows "cover" which objects. And this has to satisfy certain axioms, like if you restrict a covering family to a smaller object (read: pull back along an arrow) then it should still cover. If you cover every element of a covering family then the union covers the original thing, etc.

With this notion in hand, you can literally just copy/paste the old definition of sheaf and it works! We say that a category $\mathcal{C}$ equipped with a grothendieck topology $J$ is called a Site, and you can think of it as a combinatorial way of presenting a topos.

Chris Grossack (they/them) (Sep 25 2024 at 19:20):

A topos is a big object. Indeed, the topos of sheaves on a point is already the whole category of sets! So we can't possibly do concrete computations on the topos as a whole, it's too big. Instead, just like if you want to do really concrete computations in an algebra you fix a presentation, if you want to do really concrete computations in a topos you fix a site presenting it! These are usually finite, so you can use them to answer questions like "does the topos think that all functions are continuous?" or "does de morgan's law hold?"

John Baez (Sep 25 2024 at 19:22):

Great, @Chris Grossack (they/them)!

That's basically all there is to the definition of 'site' for a first pass, @David Egolf: a category C with a Grothendieck topology. This is just the info you need to say when a bunch of morphisms to some object of C are decreed to 'cover' that object, allowing you to copy the usual definition of when a preheaf is a sheaf.

Don't be scared by Chris' word 'combinatorial': I would have omitted that myself, since we're not doing combinatorics in any obvious way.

Chris Grossack (they/them) (Sep 25 2024 at 19:24):

First, I'll shamelessly plug someone else's work. If you've heard of Olivia Caramello's "bridge technique", this is really all that's going on. The idea is that you find two different sites $(\mathcal{C}, J)$ and $(\mathcal{D}, K)$ which present the same topos. Then you see what happens if you compute the same invariant in two different ways. On the one hand, you get some description in terms of objects of $\mathcal{C}$ and objects of $\mathcal{D}$ separately. But on the other hand, you have to get "the same answer" since they're presenting the same topos! So this tells you how to relate properties of $\mathcal{C}$ to properties of $\mathcal{D}$, when they might look like very different categories at the beginning.

Chris Grossack (they/them) (Sep 25 2024 at 19:28):

Second, I'll even more shamelessly plug my own work. I have a lot of blog posts where I go through and use sites in order to answer questions about various topoi... I don't have a single blog post that really motivates and explains sites (though maybe I should write one...) but you might be interested in

1. this post which computes very basic things like cardinalities in a topos of $G$-sets and a topos of sheaves on a topological space.

2. my series on Johnstone's Topological Topos. Especially the third post where I use a site presentation in order to compute all sorts of facts about the topos! And this really needs the framework of sites to get off the ground, rather than (1) which you can get by with presheaves and locales (which already have a good notion of open cover)

Chris Grossack (they/them) (Sep 25 2024 at 19:31):

John Baez said:

Don't be scared by Chris' word 'combinatorial': I would have omitted that myself, since we're not doing combinatorics in any obvious way.

Yeah, I almost always use the word "combinatorial" to mean "something you can actually compute with". Like, you have some easily described set of things, and rules for pushing those things around, and you can sit down and practice using these rules to do computations! With effort, you could even get a computer to do the computations for you, which I think some people in the AlgebraicJulia scene are working out as we speak!

David Egolf (Sep 25 2024 at 19:49):

Chris Grossack (they/them) said:

As a quick tl;dr for why sheaves solve problems, the idea is that if you want to solve some problem and your problem is "sheafy" then you can solve that problem locally (where the problem is often easier), check that the local solutions "agree on overlaps", and then the sheaf condition ensures you actually have a solution to your whole problem!

Oh, thanks for spelling that out! I recently did a proof where I was aiming to show that some function was continuous. The proof strategy I used was to check that the function was continuous in some open neighborhood of each point. Then I used the fact that "gluing together" continuous functions defined on open sets (provided they agree on overlaps) gives us a continuous function. I can imagine that having access to this pattern of reasoning in any "sheafy" setting could be quite handy!

David Egolf (Sep 25 2024 at 19:52):

(As as side note, because I hope to use this stuff to think about medical imaging reconstruction algorithms at some point in the future: I wonder if it could be interesting to consider which image reconstruction algorithms are "sheafy" in the sense that "local" agreement on what's present can be patched together to get agreement over a larger regions of observing parameter space.)

David Egolf (Sep 25 2024 at 20:00):

Chris Grossack (they/them) said:

To talk about "intersections" of elements of a cover, it's nice for your category to have pullbacks (do you see why a pullback is something that you might reasonably call an intersection?) but I think there's more technical versions that relax this condition.

If we work in the category $\mathcal{O}(X)$ where objects are open sets of a space $X$ and morphisms are inclusions, then I suspect that the pullback of $U \subseteq X$ and $V \subseteq X$ is $U \cap V \subseteq X$.

If that's true, then that would explain why we might want to call a pullback an intersection, perhaps especially in the case where the interesting morphisms in our original diagram (the diagram which our pullback is the limit of) are monomorphisms.

David Egolf (Sep 25 2024 at 20:19):

Chris Grossack (they/them) said:

Instead, just like if you want to do really concrete computations in an algebra you fix a presentation, if you want to do really concrete computations in a topos you fix a site presenting it!

Oh! So can every topos be viewed as a topos of sheaves on some site? The nLab seems to suggest that this is true, at least for a "Grothendieck topos"!

Equivalently, a Grothendieck topos is any category equivalent to the category of sheaves on some small site.

Very cool!

Chris Grossack (they/them) (Sep 25 2024 at 20:38):

That's right! This is the primary difference between "grothendieck topoi" and "elementary topoi". Both are interesting, with the main difference being that grothendieck topoi have a bit more structure by default, and can be fruitfully viewed as geometric spaces (in a way that I find compelling and interesting). The price you pay is that this definition by sites relies on a "base topos" (usually the universe of ZFC sets, etc) in which you define your sites, so every definition is relative to this base topos. Elementary topoi, by contrast, can be defined from nothing via a first order axiomatization in exactly the same way ZFC is (this is why they're called elementary).

Chris Grossack (they/them) (Sep 25 2024 at 20:41):

And this is not to say you can't do computations with elementary topoi! You don't have a site at your disposal, but usually you have some other gadget that plays a similar role. I think this related to the tripos to topos construction, but my knowledge of elementary topoi is really sparse, so I'll leave it for someone else to talk about this if you're interested.

Eventually I'll take some time to learn how elementary topoi work, how to do computations with them, and what interesting examples might be (I only know to care about realizability topoi). When that happens I'll definitely write a blog post on it, and probably share that here in the zulip

John Baez (Sep 25 2024 at 20:56):

@David Egolf - yes, keep this in mind:

• A Grothendieck topos is a category of sheaves on a site.
• every Grothendieck topos is an elementary topos, but not vice versa.
• The simplest case of an elementary topos that's not a Grothendieck topos is the category of finite sets. A Grothendieck topos, i.e. a category of sheaves on some site, always has all limits and colimits. Quite obviously the category of infinite sets doesn't have all coproducts, e.g. $1 + 1 + 1 + \cdots$ is not finite.

John Baez (Sep 25 2024 at 20:58):

When I was struggling to learn the very simplest basics of topos theory it used to bug me immensely that 'topos' means two things: elementary topos and Grothendieck topos. It's as if there were a subject called 'group theory' where experts used 'group' to mean two different things! But I've gotten used to it.

Mike Shulman (Sep 25 2024 at 23:52):

To be fair, there are other fields where unadorned nouns have implicit adjectives that depend on the context and the speaker. Is a "graph" simple? Is it directed? Is a "ring" commutative? Does it have a multiplicative identity? And let's not even get started on what a "space" is...

John Baez (Sep 26 2024 at 02:47):

For some reason I was never confused by any of these as much as with 'topos'. I guess I read little facts about toposes here and there, never seriously studying the subject, and was very confused by the two quite different stories... oh yes, and I was also confused by 'gros' versus 'petit' toposes.

Somehow none of the others bothered me: I could always tell what was meant.

Of course 'space' is deliberately ambiguous: math needs flexible words like 'space', 'structure', 'number'.

Madeleine Birchfield (Sep 26 2024 at 05:09):

Every Grothendieck topos is only an elementary topos if you believe in power sets in your foundations / category of sets. If you don't have power sets - say you work in CZF or something similar, then your Grothendieck toposes are only going to be Heyting pretoposes or $\Pi$-pretoposes.

Morgan Rogers (he/him) (Sep 26 2024 at 13:29):

Chris Grossack (they/them) said:

If you want to do really concrete computations in a topos you fix a site presenting it! These are usually finite, so you can use them to answer questions like "does the topos think that all functions are continuous?" or "does de morgan's law hold?"

That's not exactly true: toposes of sheaves on finite sites are always equivalent to presheaf toposes, so you miss out on a lot of interesting stuff that way! But the site itself often has a presentation that is finite (or at least nicely indexed) from which the site is generated in one of many ways.

John Baez (Sep 26 2024 at 15:36):

And we should also note that many of the classic sites - the first sites introduced by Grothendieck and others introduced to do algebraic geometry - are not finite: for example the [[Zariski site]], the [[etale site]] and the [[fppf site]]. These come from Grothendieck topologies on the categories of schemes, or affine schemes, or either of those over a fixed scheme.

John Baez (Sep 26 2024 at 15:39):

I don't understand algebraic geometry very well, but I'm trying, so these examples are always on my mind. For example there's a site such that a scheme is a sheaf on this site with an extra property.

John Baez (Sep 26 2024 at 15:40):

This site, called the [[Zariski site]], comes from the Grothendieck topology on $\mathsf{CommRing}^\text{op}$ called the 'Zariski topology'.

John Baez (Sep 26 2024 at 15:45):

I mention this mainly because if anyone wants to learn about topos theory and also algebraic geometry simultaneously, following this route seems like a lot of fun, in part because that's how topos theory originated. There's a bit of this stuff in Mac Lane and Moerdijk's book.

Chris Grossack (they/them) (Sep 26 2024 at 15:57):

Something about this that John already knows, but might be interesting for other people is that you can think of objects of a topos as being "objects from the site glued together nicely". For instance, a presheaf topos is the [[free cocompletion]] of the base category, so its objects are literally built by gluing objects of the site together (with no other restrictions, since it's "free"). Then adding a grothendieck topology basically amounts to restricting attention to a subcategory of objects that are glued together "nicely", where here "nice" is mediated by the grothendieck topology and basically means "in a way that respects open covers".

Chris Grossack (they/them) (Sep 26 2024 at 15:58):

From this perspective, it shouldn't be surprising that schemes are nice sheaves on the site of affine schemes (read: of $\mathsf{CommRing}^\text{op}$) since we already know that every scheme is built by gluing together affine schemes!

Chris Grossack (they/them) (Sep 26 2024 at 15:59):

You can see this clearly with [[simplicial sets]], which is a topos of objects you get by gluing simplices together. And of course, this is the presheaf topos on the category of simplices!

David Egolf (Sep 26 2024 at 16:06):

Wow, that's really cool! That sounds like a powerful pattern:

• start with some category containing objects of interest
• to be able to "glue together" objects "freely" use the Yoneda embedding and move to the category of presheaves on our original category
• and if one wants to glue together objects in some "nice" way, respecting certain structure as described by a Grothendieck topology, then we can use the category of sheaves on our original category, provided that we equip our original category with some Grothendieck topology

David Egolf (Sep 26 2024 at 16:13):

To try and make an analogy:

• we start with some lego blocks, together with understanding of how the blocks relate to one another (this is our base category; for example $\Delta$)
• moving to the presheaf topos allows us to connect the lego blocks to one another, in a way compatible with how the blocks were described to relate to one another (for example, we get $[\Delta^{\mathrm{op}}, \mathsf{Set}]$)
• adding a Grothendieck topology and then working only with the resulting sheaves restricts the ways in which we can connect the blocks in some way (?)

Chris Grossack (they/them) (Sep 26 2024 at 16:14):

Exactly! If you meditate on this you'll also start to uncover the differences between "big topoi" and "little topoi". If you start with a site that's already a bunch of spaces (eg. affine schemes, euclidean spaces, etc) then the objects of the topos look like "generalized spaces" (eg. schemes, generalized manifolds, etc). This is emblematic of a "big topos" -- its objects are spaces. The topos is a universe in which these spaces naturally live.

If you start with a site that looks like opens of a particular space (eg. some locale/topological space) then the objects of the topos look like "generalized open subsets" of that space (eg. etale opens of that locale). This is emblemeatic of a "little topos" -- the objects look like opens of some space (that maybe didn't exist before). In this case, the topos can be viewed as a space in its own right!

David Egolf (Sep 26 2024 at 16:47):

I see! Based on this perspective, I think I can begin to understand how to describe a smooth manifold as a sheaf on some site. The objects of the underlying category should be presumably like "little pieces of $\mathbb{R}^n$ for some fixed $n$. The morphisms should relate these pieces of $\mathbb{R}^n$ in some suitable way (which probably would depend on the flavour of manifold we want to end up with?).

David Egolf (Sep 26 2024 at 16:47):

Then the topos of presheaves on this category gives us the category's free cocompletion: we have a new category where objects are like a bunch of little pieces of $\mathbb{R}^n$ that have been glued together "freely". That starts to sound a lot like a manifold!

David Egolf (Sep 26 2024 at 16:48):

Then, if we want to restrict consideration to certain nice versions of these "manifolds" perhaps this can be achieved by putting a Grothendieck topology on our original category and then considering the topos of sheaves on the resulting site.

David Egolf (Sep 26 2024 at 16:48):

(To figure out how to do this in detail, though, I'd need to probably need to first work on some examples/puzzles relating to all this!)

John Baez (Sep 26 2024 at 17:52):

Note that Chris is implying we can think of an etale space over a topological space X as a 'generalized open subset' of X.

John Baez (Sep 26 2024 at 17:57):

This is a nice perspective I'd never thought of, though earlier in this thread I was pointing out that the inclusion of any open set U of X is an etale space over X.

John Baez (Sep 26 2024 at 17:58):

In general an etale over X looks locally just like an open subset of X.

David Egolf (Sep 26 2024 at 18:07):

That makes sense with the perspective that a sheaf can describe a "gluing together" of a bunch of "building blocks". In this case, I think the building blocks are open subsets! This helps me better understand the equivalence between sheaves on a topological space $X$ and etale spaces over $X$.

Mike Shulman (Sep 26 2024 at 18:59):

In fact, given a site $C$, it's actually possible to define (a category equivalent to) the topos of sheaves on that site to have objects that are manifestly "objects of the site glued together". Specifically, one of its objects consists of:

• A small family of objects $\{x_i\}_i$ of $C$. These are the objects we intend to "glue together".
• For each $i,j$ a small family of spans $\{x_i \leftarrow x_{ij\ell} \to x_j\}_\ell$. These specify how to glue them together: each $x_{ij\ell}$ parametrizes one "bit of glue" between $x_i$ and $x_j$.
• These families of spans are an "equivalence relation relative to the Grothendieck topology". This means that:
  • For each $i$, there is a covering family $\{y_{in} \to x_i\}_n$ such that each span $x_i \leftarrow y_{in} \to x_i$ factors through some span $x_{ii\ell}$ (reflexivity).
  • For each $i,j,\ell$ there is a covering family $\{y_{ij\ell n} \to x_{ij\ell}\}_n$ such that each induced span $x_i \leftarrow y_{ij\ell n} \to x_j$ factors through the reversal $x_i \to x_{jik} \to x_j$ of one of the spans associated to $j,i$ (symmetry)
  • For each $i,j,k,\ell,m$ there is a covering family $\{y_{ijk\ell m n} \to x_{ij\ell}\times_{x_j} x_{jkm} \}_n$ such that each span $x_i \leftarrow y_{ijk\ell m n} \to x_k$ factors through some span $x_i \leftarrow x_{ikp} \to x_k$ (transitivity). (This can also be formulated if $C$ doesn't have pullbacks, but it's even more annoying.)

So, for instance, if $C$ is the category of open subsets of $\mathbb{R}^n$ and smooth functions, then we can describe a manifold $M$ by taking the $x_i$ to be the domains of the charts in an atlas of $M$, and the $x_{ij\ell}$ to be the opens corresponding to the intersections in $M$ of the images of each pair of charts. Similarly, we can describe schemes by explicitly gluing together affine schemes (i.e. objects of $\rm Ring^{op}$).

However, the morphisms between these "congruences" are more difficult to describe. Several ways of doing it are described in my paper Exact completions and small sheaves. Unfortunately it's rather technically presented there, not really at the level of an "introduction to toposes". But the technicalities do avoid the tedious manipulation of indices that you can see happens above.

One nice thing about this construction is that it works just as well even if the site $C$ is not small (like, for instance, $\rm Ring^{op}$). It doesn't yield a topos in that case, but an "infinitary-pretopos" that has all the exactness properties of a Grothendieck topos, though not a generating set or power-objects. And you also get useful constructions if you restrict the cardinality of the families in a uniform way.

David Egolf (Sep 27 2024 at 17:57):

I really like this "gluing" intuition for thinking about sheaves on a site! However, the way in which the choice of Grothendieck topology impacts the gluing process is intuitively unclear to me. I suspect what @Mike Shulman said above spells this out in detail, but currently those details feel a bit intimidating to me!

David Egolf (Sep 27 2024 at 17:57):

I've found working exercises/puzzles and drawing pictures to be quite helpful for developing intuition in #learning: reading & references > reading through Baez's topos theory blog posts . So I'll probably try to dream up a puzzle to work on in this thread, to try and develop some intuition for Grothendieck topologies. Ideally, I'd like to see how changing a choice of Grothendieck topology impacts the "gluing together" performed by the sheaves on the resulting site. (Alternatively, if someone has a puzzle in mind relating to Grothendieck topologies, I'm all ears!)

John Baez (Sep 27 2024 at 19:43):

David Egolf said:

I really like this "gluing" intuition for thinking about sheaves on a site! However, the way in which the choice of Grothendieck topology impacts the gluing process is intuitively unclear to me. I suspect what Mike Shulman said above spells this out in detail, but currently those details feel a bit intimidating to me!

Let me boil it down to the basics. You may already know this. A sheaf is a presheaf with a special property: under certain conditions we can glue together elements of the presheaf to get another element. But we need the Grothendieck topology to say when we're allowed to do this, since it says when a bunch of morphisms to some object count as 'covering' that object.

Now let me say it again in more detail. It's a lot like the case we've already discussed: a sheaf on a topological space, so let me review that first.

As we've seen, a sheaf on a topological space is a presheaf with a special property: if you have an open set $U$, open sets that $U_\alpha$ coverin $U$, and elements

$s_\alpha \in F(U_\alpha)$

whose restrictions to the intersections $U_\alpha \cap U_\beta$ agree, there exists a unique element

$s \in F(U)$

that restricts to give each of these $s_\alpha$. We then say that 'gluing together' the $s_\alpha$ give the element $s$.

Note how the concept of 'cover' is crucial here.

A sheaf on a category $\mathsf{C}$ works the same way. It's a presheaf

$F: \mathsf{C}^{\text{op}} \to \mathsf{Set}$

with a special property. A Grothendieck topology is simply a rule that tells us when a bunch of morphisms

$f_\alpha : c_\alpha \to c$

can be said to 'cover' an object $c$. If they do, and we have a bunch of elements

$s_\alpha \in F(C_\alpha)$

that 'agree on the intersections' (i.e. pullbacks), then there exists a unique element

$s \in F(c)$

that 'restricts' to give each of these $s_\alpha$. The quoted phrase is a sloppy way to say

$F(f_\alpha)(s) = c_\alpha$

John Baez (Sep 27 2024 at 19:50):

Note that in the classic case - sheaves on a topological space - the category $C$ is the poset of open subsets of that topological space. Since it's a poset, if there's a morphism from $U_\alpha$ to $U$ there's only one such morphism. This lets us talk what it means for a bunch of objects $U_\alpha$ to cover $U$ without mentioning the morphisms - the inclusion maps $f_\alpha : U_\alpha \to U$.

When we study a Grothendieck topology on a general category, we don't have this luxury! So we don't talk about objects covering an object: we talk about morphisms covering an object.

Chris Grossack (they/them) (Sep 28 2024 at 00:58):

You can think of a presheaf topos as a sheaf topos for a kind of stupid topology -- the only covering maps are isomorphisms! You can also go to the other extreme and say that EVERY (non-empty) family is covering

Chris Grossack (they/them) (Sep 28 2024 at 01:01):

So this means if you have a lattice of opens of a space, there's three different topologies you can do computations with to gain intuition. The trivial topology (only isos cover), the atomic topology (every arrow covers), and the "canonical topology" (open covers in the usual sense)

Chris Grossack (they/them) (Sep 28 2024 at 01:02):

For a simple topological space, it might be fun to look at sheaves for all three of these topologies and see how they compare!

ww (Sep 28 2024 at 10:50):

Trying to follow and might have a misaligned understanding but...

Suppose I have some category of stochastic processes. If I take two copies a processes and do them both (tensor them together; i guess it is a symmetric monoidal category) then I get the same thing as one copy running twice as fast -- providing we insist on exponential rates. If I take two different processes then the pair of rates just adds with independent rates as long as the processes are independent. If they are not independent, the overlap (for whatever definition of overlap), we had better want it to behave like tensoring it with itself, otherwise there will be problems. It sounds like there is a sheaf there.

If we do not insist on exponential rates but allow more general ideas of how timing of events should work, where it is unclear how they should add, then we only have a presheaf. Do the hazard functions in AlgebraicABMs behave in a way that produces a sheaf?

John Baez (Sep 28 2024 at 15:11):

Hi! If something is a presheaf it's a presheaf on some category, called the 'site', which might be the poset of subsets of some space, or some more general category. So the first question I've got is: what category are we talking about in this stochastic process example?

John Baez (Sep 28 2024 at 15:12):

I know I'm supposed to be helping you answer that question.

John Baez (Sep 28 2024 at 15:54):

(In my own personal experience, very few things I study turn out to be sheaves, except when I'm working on algebraic geometry or complex manifolds - the topics that sheaves were invented for in the first place. Then sheaves are great. But maybe I'm not looking hard enough. Presheaves, on the other hand, are everywhere: they're fundamental to AlgebraicJulia under the name 'C-sets', and we use them a lot in our epidemiological modeling software.)

ww (Sep 29 2024 at 09:50):

I will need to think about it more (and maybe this risks derailing this particular conversation so we could move it to a different topic) but...

Maybe we have a category where objects rewriting rules, *PO or whatever, with rates. Not sure what the morphisms are, probably a partial order where there L of one rule overlaps the R of another. Get the identity morphisms from the I. Refine the partial order of rules to break ties by comparing the rates which themselves would need to have a partial order. If a partial order is what we want there.

What I'm calling stochastic processes are sets of these rules. Actually, maybe they are coproducts because they end up as a sum in the Hamiltonian. At least when the rates are nice, maybe a different kind of colimit when they are not nice. I don't think this detail matters here.

I also have sets of observables which are little graphs, much like the ones in the L, I, R of rules, that I use to count things that I can also measure out in the world (which get used to fit the model to data) or am interested in (which I don't know and are why I am bothering to go to the trouble of making a model).

What I am trying to do is explore the landscape of rulesets that preserve the set of observables I use to count things that I can measure. This is very different from freely generating rulesets and I think is where the gluing comes from.

So, I think the site is the observables. It is easy enough to make a category of them, the morphisms are probably a partial order, as you say. They constrain the space of stochastic processes to the surface consistent with things I can count and compare to measurements.

David Egolf (Sep 29 2024 at 16:45):

John Baez said:

A Grothendieck topology is simply a rule that tells us when a bunch of morphisms

$f_\alpha : c_\alpha \to c$

can be said to 'cover' an object $c$. If they do, and we have a bunch of elements

$s_\alpha \in F(C_\alpha)$

that 'agree on the intersections' (i.e. pullbacks), ...

This sounds helpful! I don't quite follow the parenthetical mention of pullbacks, so let me see if I can figure that out.

David Egolf (Sep 29 2024 at 16:45):

We have a bunch of morphisms $f_\alpha:c_\alpha \to c$ that are in some sense "covering" some object $c$. We then have some $s_\alpha \in F(c_\alpha)$ for each $\alpha$ and we want to know if they "agree on overlaps". (I assume that $F$ is a presheaf $F:C^{\mathrm{op}} \to \mathsf{Set}$).

David Egolf (Sep 29 2024 at 16:45):

Pullbacks were mentioned above as a way to generalize intersections of open sets. So if we think of $f_\alpha:c_\alpha \to c$ and $f_\beta:c_\beta \to c$ as "generalized open subsets of $c$", then we might try using this pullback square to talk about the "intersection" of those generalized open subsets:
pullback square

David Egolf (Sep 29 2024 at 16:45):

We might think of $f_\beta \circ p_\beta = f_\alpha \circ p_\alpha:c_\alpha \times_c c_\beta \to c$ as the "intersection" of the two arrows to $c$ that we started with.

David Egolf (Sep 29 2024 at 16:47):

Now, let's assume we have some $s_\alpha \in F(c_\alpha)$ and some $s_\beta \in F(c_\beta)$, and we want to decide if these agree on the "intersection" of $f_\alpha:c_\alpha \to c$ and $f_\beta:c_\beta \to c$.

John Baez (Sep 29 2024 at 16:49):

@David Egolf wrote:

I don't quite follow the parenthetical mention of pullbacks, so let me see if I can figure that out.

You actually talked about this earlier, so I figured you'd get it:

David Egolf said:

Chris Grossack (they/them) said:

To talk about "intersections" of elements of a cover, it's nice for your category to have pullbacks (do you see why a pullback is something that you might reasonably call an intersection?) but I think there's more technical versions that relax this condition.

If we work in the category $\mathcal{O}(X)$ where objects are open sets of a space $X$ and morphisms are inclusions, then I suspect that the pullback of $U \subseteq X$ and $V \subseteq X$ is $U \cap V \subseteq X$.

If that's true, then that would explain why we might want to call a pullback an intersection, perhaps especially in the case where the interesting morphisms in our original diagram (the diagram which our pullback is the limit of) are monomorphisms.

John Baez (Sep 29 2024 at 16:51):

So the idea is that pullbacks are the correct generalization of intersections when we generalize from the poset of open subsets of a topological space to an arbitrary category.

John Baez (Sep 29 2024 at 16:52):

David Egolf said:

Pullbacks were mentioned above as a way to generalize intersections of open sets. So if we think of $f_\alpha:c_\alpha \to c$ and $f_\beta:c_\beta \to c$ as "generalized open subsets of $c$", then we might try using this pullback square to talk about the "intersection" of those generalized open subsets:
pullback square

Right!

David Egolf (Sep 29 2024 at 16:52):

Applying $F:C^{\mathrm{op}} \to \mathsf{Set}$ to $p_\beta$ and $p_\alpha$ we get $F(p_\beta):F(c_\beta) \to F(c_\alpha \times_c c_\beta)$ and $F(p_\alpha):F(c_\alpha) \to F(c_\alpha \times_c c_\beta)$. I am guessing that these functions can act as our "restriction" functions.

John Baez (Sep 29 2024 at 16:53):

Right. Sorry to interrupt your flow.

David Egolf (Sep 29 2024 at 16:54):

Then we can say that $s_\alpha \in F(c_\alpha)$ and $s_\beta \in F(c_\beta)$ "agree on their overlap" if $F(p_\alpha)(s_\alpha) = F(p_\beta)(s_\beta)$.

David Egolf (Sep 29 2024 at 16:57):

I think that makes sense! So we can now talk about when two pieces of data $s_\alpha \in F(c_\alpha)$ and $s_\beta \in F(c_\beta)$ "agree on their overlap".

David Egolf (Sep 29 2024 at 16:58):

Hmm, there might be a bit more to this, actually.

We could have multiple different morphisms in $C$ from a fixed $c_\alpha$ to $c$. So then, we might need to consider a whole bunch of pullback squares associated to $c_\alpha$ and $c_\beta$ if we wish to assess if $s_\alpha \in F(c_\alpha)$ and $s_\beta \in F(c_\beta)$ agree "on their overlap".

David Egolf (Sep 29 2024 at 17:00):

We didn't have to think about this in the case of sheaves or presheaves on a topological spaces, because the open sets of a topological space form a thin category. That meant that there was only one pullback square associated to any two open subsets of our space.

Chris Grossack (they/them) (Sep 29 2024 at 17:06):

Remember that really the arrows are the things that cover, not the objects! So you don't need to worry about like, pulling back along all possible arrows from $c_\alpha \to c$ or something. The cover tells you which arrows are relevant

Chris Grossack (they/them) (Sep 29 2024 at 17:10):

To keep showing that the arrows are the important things: There might be two different arrows $f,g : D \to C$ in your cover. In that case you still have to consider the intersection $f \times_C g : D \times_C D \to C$. Despite the usual notation being only $D \times_C D$ (which is the less important part! Really the arrows are what matter!), this isn't really a self intersection since $f$ and $g$ might differ

David Egolf (Sep 29 2024 at 17:11):

That makes sense! This is reminding me a lot of how we can view a subgroup as a monomorphism (an arrow), which tells us how a given group gets "fit inside" another group. In general, there can be multiple ways to do this, and the particular choice of monomorphism tells us which way was chosen.

Chris Grossack (they/them) (Sep 29 2024 at 17:11):

Exactly! And in both cases pullback gives intersection

Chris Grossack (they/them) (Sep 29 2024 at 17:13):

AND your subgroup examples shows that you can have $H \leq G$ in multiple ways, and the two different copies of $H$ might intersect in an interesting way in $G$, just like in my last message

John Baez (Sep 29 2024 at 20:26):

Do people routinely use $\le$ to mean "$H$ is a subgroup of $G$"?

Alex Kreitzberg (Sep 29 2024 at 20:51):

I've seen $\trianglelefteq$ mean "normal subgroup of"

Chris Grossack (they/them) (Sep 29 2024 at 20:51):

I see $\leq$ to mean subgroup pretty often, yeah

Chris Grossack (they/them) (Sep 29 2024 at 20:52):

Screenshot 2024-09-29 at 1.51.57 PM.png
here's a screenshot from Dummit and Foote

Alex Kreitzberg (Sep 29 2024 at 20:54):

Because I've seen the triangle normal subgroup notation, the inequality $\leq$ reads to me as "not even a normal subgroup :pensive:"

John Baez (Sep 29 2024 at 21:25):

I knew what Chris meant, so I'm not trying to roast them over the coals... well, not much :smiling_imp:... but according to the Dummit and Foote definition the phrase "$H \le G$ in multiple ways" doesn't literally make sense. So I was thinking maybe people use $H \le G$ in some subtly different way, which allows this usage. Anyway, never mind! I've never used any of those symbols $H \le G$, $H \triangleleft G$, etc. that some people use, much less $H \trianglelefteq G$, so I don't know all their nuances, and in fact I'm probably too old to start caring now.

Alex Kreitzberg (Sep 29 2024 at 21:40):

I may have misremembered $H \triangleleft G$ with an under bar, but LaTeX let me do it so I thought I was safe XD

Alex Kreitzberg (Sep 29 2024 at 21:47):

I took the pushout of $\leq$ and $\triangleleft$ :relieved:

David Egolf (Sep 30 2024 at 16:36):

John Baez said:

A Grothendieck topology is simply a rule that tells us when a bunch of morphisms

$f_\alpha : c_\alpha \to c$

can be said to 'cover' an object $c$. If they do, and we have a bunch of elements

$s_\alpha \in F(C_\alpha)$

that 'agree on the intersections' (i.e. pullbacks), then there exists a unique element

$s \in F(c)$

that 'restricts' to give each of these $s_\alpha$. The quoted phrase is a sloppy way to say

$F(f_\alpha)(s) = c_\alpha$

Last time, we discussed what it means for the various $s_\alpha$ to 'agree on the intersections'. Next, I would like to spell out what it means for $s \in F(c)$ to 'restrict' to some $s_\alpha$.

David Egolf (Sep 30 2024 at 16:36):

My first thought is to use this pullback square:
pullback square

I guess that we want $F(p_c)(s) = F(p_{c_\alpha})(s_\alpha)$, if $s$ is to restrict to $s_\alpha$ with respect to $f_\alpha:c_\alpha \to c$.

David Egolf (Sep 30 2024 at 16:36):

However, that condition doesn't immediately match with the following one, mentioned by @John Baez above: $F(f_\alpha)(s) = c_\alpha$. I'm guessing there was a typo, and that this condition should be $F(f_\alpha)(s) = s_\alpha$. But that still seems rather different from the condition I came up with!

David Egolf (Sep 30 2024 at 16:37):

My current guess is that the above pullback square simplifies quite a bit. If I can figure out how to simplify it, then hopefully these two conditions will end being secretly the same!

David Egolf (Sep 30 2024 at 16:42):

We can rewrite the pullback square like this:
rewritten pullback square

This makes me start to wonder if, in this special case, the pullback is the limit of the diagram formed by $f_\alpha$. [This seems plausible, because a cone over this new diagram is basically the same thing as a cone over the diagram involved with our pullback.]

David Egolf (Sep 30 2024 at 16:46):

Making that assumption for moment, I think we can figure out what $c \times_c c_\alpha$, and $p_{c_\alpha}$ and $p_c$ are:
limit

It seems that setting $c \times_c c_\alpha = c_\alpha$, and $p_{c_\alpha}=1_{c_\alpha}$, and $p_c = f_\alpha$ gives us a limit cone over the diagram formed by $f_\alpha$.

David Egolf (Sep 30 2024 at 16:48):

So, our condition $F(p_c)(s) = F(p_{c_\alpha})(s_\alpha)$ then becomes $F(f_\alpha)(s) = F(1_{c_\alpha})(s_\alpha) = s_\alpha$. And this I think matches the condition given by John Baez above!

David Egolf (Sep 30 2024 at 16:51):

Intuitively, it also makes sense that if $f_\alpha:c_\alpha \to c$ is a sort of "generalized open subset inclusion into $c$", then $F(f_\alpha)$ is a sort of "generalized restriction" from $F(c)$ to $F(c_\alpha)$.

John Baez (Sep 30 2024 at 18:54):

Right!

John Baez (Sep 30 2024 at 18:54):

David Egolf said:

However, that condition doesn't immediately match with the following one, mentioned by John Baez above: $F(f_\alpha)(s) = c_\alpha$. I'm guessing there was a typo, and that this condition should be $F(f_\alpha)(s) = s_\alpha$.

Yes, that was a typo.

Peva Blanchard (Sep 30 2024 at 21:14):

I tend to see a cover as a decomposition of a big piece into smaller pieces. Hence, a Grothendieck topology specifies all the valid decompositions of any object.

Maybe it is a false intuition, but it feels "algebraic" (or "coalgebraic"?). A specific cover, or decomposition, is like an operation $c \to \prod_i c_i$. With a presheaf, I can move data in one direction (restriction), and, if is a sheaf, I can also move data in the other direction (gluing).

Does it make sense to see a Grothendieck topology as some kind of Lawvere theory?

Kevin Carlson (Sep 30 2024 at 21:24):

It's a natural thought, but it doesn't quite work. A cover of $c$ makes $c$ into a certain limit of the $c_i$ in the category of sheaves, but that limit isn't necessarily a product; a Lawvere theory can only encode restrictions that sends certain objects to a product at the values of another object.

Mike Shulman (Oct 01 2024 at 01:15):

However, there is a generalization of a Lawvere theory called a [[sketch]], and a Grothendieck topology can be seen as a certain kind of sketch.

John Onstead (Oct 01 2024 at 11:26):

I like the product perspective and I really want to learn more about it! But when I think of a covering, I think of a coproduct. For example, in Top, let's say we have a covering ci: Ui -> X where Ui are open subsets of X. We can take the coproduct of all Ui to get a single object U, in which case we have a resulting morphism c: U -> X such that ci is the composition from the projection Ui -> U with c: U -> X. The interesting thing about this morphism is that it's an epimorphism, and in a sense that's why it "covers" X.

What truly blows my mind is that, in a Grothendieck topology, you can just declare a family of morphisms ci: Ui -> X to be a "covering" without any of those conditions holding. That means if you take the coproduct of all Ui and find the resulting morphism c: U -> X, it doesn't have to be an epimorphism. In fact, the coproduct of all Ui doesn't even have to exist in the first place! And yet, despite ci: Ui -> X not satisfying these properties we'd expect intuitively a notion of "cover" to satisfy, everything still works out in the end. That's probably got to be the coolest fact about general Grothendieck topologies and I'm still trying to wrap my head around how that's even possible!

Kevin Carlson (Oct 01 2024 at 16:22):

I think the short answer to why it's possible is that the presheaves which satisfy the sheaf condition for that cover are precisely the presheaves that "think" the cover "really is" a cover, in the sense that they think X really is the colimit of the U_i over their various intersections. Whether or not it's actually true that X is such a colimit, there are a whole lot of presheaves, so some of them will have that impression!

Mike Shulman (Oct 01 2024 at 23:02):

And this generalizes to sketches. A sketch is a category with a distinguished set of cones and cocones that may not actually be limiting or colimiting, while a "model" of the sketch is a Set-valued (or other-category-valued) functor that "thinks" they are limiting or colimiting, in the sense that it sends them to actual limits or colimits in Set.

You can think of a sketch as a "presentation" of a category with limits/colimits, and the specified cones/cocones as "relations" -- compare how in a group presentation, the "equations" specified by the relations don't actually "hold" as equations between words in the generators, but they are "forced" to hold in the actual group that's presented. Similarly, the cones and cocones in a sketch are "relations" that are forced to hold in the category of models it presents, and as a particular case the covering families in a Grothendieck topology are "forced to become covers" in the topos of sheaves, whether or not they originally were.

Mike Shulman (Oct 01 2024 at 23:03):

A site in which the covering families are "actually" covers is called a [[subcanonical site]].

John Baez (Oct 01 2024 at 23:14):

@Mike Shulman wrote:

And this generalizes to sketches. A sketch is a category with a distinguished set of cones and cocones that may not actually be limiting or colimiting, while a "model" of the sketch is a Set-valued (or other-category-valued) functor that "thinks" they are limiting or colimiting, in the sense that it sends them to actual limits or colimits in Set.

James Dolan has developed a method that builds on this notion of "thinking": he calls it the "belief method", because it involves objects that "believe" certain axioms. He calls these objects the "believers". This is even more amusing because he applies it to doctrines.

Peva Blanchard (Oct 02 2024 at 04:42):

This looks very interesting. Could you share a reference where the belief method is defined?

I couldn’t find it on nlab.

John Baez (Oct 02 2024 at 04:45):

James Dolan doesn't publish anything, so the only place to find out about the belief method is to talk to him or watch four videos where he explains it to me. (Search under "belief method".)

David Egolf (Oct 06 2024 at 22:31):

Chris Grossack (they/them) said:

So this means if you have a lattice of opens of a space, there's three different topologies you can do computations with to gain intuition. The trivial topology (only isos cover), the atomic topology (every arrow covers), and the "canonical topology" (open covers in the usual sense)

Chris Grossack (they/them) said:

For a simple topological space, it might be fun to look at sheaves for all three of these topologies and see how they compare!

This sounds like an interesting thing to do! The first topological space that comes to mind for me is $\mathbb{R}$ together with its usual topology. This topology gives us a bunch of open sets, which assemble to form a poset (where $U \leq V$ iff $U \subseteq V$), which can be thought of as a category. If I understand correctly, the idea now is to consider different Grothendieck topologies on this category.

David Egolf (Oct 06 2024 at 22:34):

I want to start by contemplating the "trivial" Grothendieck topology on this category. To do this, I think I need to understand, for each object $U$ in this category, which collections of morphisms (with target $U$) are said to "cover" $U$.

David Egolf (Oct 06 2024 at 22:37):

The only isomorphisms in this category are the identity morphisms. So, in this case I think there is only one "covering" of $U$, for any object $U$. And this covering is given by the identity morphism $1_U: U \to U$.

David Egolf (Oct 06 2024 at 22:42):

Given this, I think we can now try to work out what a sheaf is on our poset $C$ of open subsets of $\mathbb{R}$. It should be a presheaf to start out with, so a functor $F:C^{\mathrm{op}} \to \mathsf{Set}$. However, since we have a Grothendieck topology, we can also ask if data attached to "parts" of an object that "agree on overlaps" can be "glued together" to form some data attached to the entire object.

David Egolf (Oct 06 2024 at 22:44):

However, I am guessing that this condition will be rather trivial in this case. That's because any object $U$ only has a "part" given by itself, or rather given by $1_U:U \to U$.

John Baez (Oct 06 2024 at 22:48):

And so....?

David Egolf (Oct 06 2024 at 22:52):

And so I expect that every presheaf on $C$ will in fact be a sheaf, given this choice of Grothendieck topology.

David Egolf (Oct 06 2024 at 22:53):

(I'm running out of steam for today, but hopefully (1) that's correct and (2) next time I can start to think about the "atomic topology" case.)

John Baez (Oct 06 2024 at 22:58):

Yes, you're right! And you used nothing about $\mathbb{R}$, so this is a general fact: we can always give the poset of open subsets of a topological space a 'trivial' Grothendieck topology, so that sheaves on it are just the same as presheaves!

John Baez (Oct 06 2024 at 23:00):

And it's actually a super-general fact: every category has a 'trivial' Grothendieck topology, for which the sheaves on it are the same as presheaves.

This is a very useful result, because it allows you to say you're doing sheaf theory when you're messing around with presheaves. :upside_down:

David Egolf (Oct 07 2024 at 16:30):

Before moving on, I want to think a bit more about this "trivial" Grothendieck topology. The definition of a Grothendieck topology on a category $C$ as given in "Sheaves and Geometry in Logic" involves assigning to each object $c$ of $C$ a collection of "sieves". A sieve $S$ on $c$ is a set of morphisms with target $c$, such that if $f \in S$ then $f \circ h \in S$, for any morphism $h$ in $C$ such that $f \circ h$ is defined.

David Egolf (Oct 07 2024 at 16:31):

At first I had thought that a sieve on $c$ would correspond to a collection of morphisms that "cover" $c$, as discussed above.

David Egolf (Oct 07 2024 at 16:32):

However, when working in the poset of open sets of $\mathbb{R}$ (equipped with its usual topology), if we have $1_U: U \to U$ in a sieve on $U$, then $1_U \circ i_V:V \to U \to U$ would also need to be in our sieve, for any open subset $V$ of $U$.

David Egolf (Oct 07 2024 at 16:35):

So, it seems that a sieve on $c$ is something different from a collection of morphisms that "cover" $c$ in the sense discussed above.

David Egolf (Oct 07 2024 at 16:37):

I am guessing that what we've been discussing instead corresponds to what "Sheaves in Geometry and Logic" calls a "basis" for a Grothendieck topology.

John Baez (Oct 07 2024 at 17:14):

That sounds right. Maybe @Chris Grossack (they/them) should say what they meant.

Chris Grossack (they/them) (Oct 07 2024 at 18:04):

That's right, it's common to abuse notation and identify a basis with the topology it generates. The topology itself is canonical which makes it really useful for proving theorems. Unfortunately, these sieves are big and kind of unwieldy, which makes them less useful for calculations. So instead of working with the sieves we work with a basis generating the topology instead, which is smaller and more manageable.

Chris Grossack (they/them) (Oct 07 2024 at 18:05):

This is a super common pattern in math, and moving fluently between the "abstract"/"canonical" definition (which is good for proving theorems) and the "concrete" definition relying on a sequence of arbitrary choices (which is good for calculating) is a really important skill to develop in basically any subject you learn

Chris Grossack (they/them) (Oct 07 2024 at 18:09):

To draw an analogy with pointset topology, there it's often the case that you can reduce a question about an arbitrary open to a question about a basic open, since the question you're asking is stable under union.

In the localic picture, where we think of topologies as frames (which are special lattices), the choice of a basis for your topology is literally the choice of a presentation of your frame. Then computing with "basic opens" amounts to computing with this presentation. You can push this so far that the [[formal topology]] people don't bother "completing" to the canonical object at all, and just work directly with the presentation (since this is [[predicative]], which some people care about)

Chris Grossack (they/them) (Oct 07 2024 at 18:18):

Similarly, for grothendieck topologies, it's often the case that you can reduce a question about an arbitrary sieve $R$ to a question about a generating family $\{f_\alpha\}$, since the question you're asking is stable under composition.

Similar to formal topology, there are people (Steve Vickers being the first to come to mind) who work with a definition of site that's geometric (and maybe even predicative? I don't remember). They choose sites with covering families as the basic concept, rather than completing to the whole topos, so that everything they do stays finitary

Chris Grossack (they/them) (Oct 07 2024 at 18:26):

To see the relationship between these "covering families" and the "sieves" you were expecting, it might be helpful to take a detour from Mac Lane & Moerdijk to look at Chapter C2.1 in The Elephant. This chapter is well written, and even in the first few pages, you'll see another take on what's going on here ^_^

David Egolf (Oct 07 2024 at 21:22):

Chris Grossack (they/them) said:

To see the relationship between these "covering families" and the "sieves" you were expecting, it might be helpful to take a detour from Mac Lane & Moerdijk to look at Chapter C2.1 in The Elephant. This chapter is well written, and even in the first few pages, you'll see another take on what's going on here ^_^

Thanks for elaborating on this point! I'll plan to take a look at Chapter C2.1!

David Egolf (Oct 08 2024 at 23:27):

I started reading C2.1, and I wanted to highlight a blog post which gives a nice explanation of the first definition which appears in C2.1.

The first definition in C2.1 is:

Let $C$ be a category. By a coverage on $C$ we mean a function assigning to each object $U$ of $C$ a collection $T(U)$ of families $(f_i:U_i \to U | i \in I)$ of morphisms with common codomain $U$ (called $T$-covering families), such that

(C) If $(f_i:U_i \to U | i \in I)$ is a $T$-covering family and $g:V \to U$ is any morphism with codomain $U$, there exists a $T$-covering family $(h_j:V_j \to V | j \in J)$ such that each $g h_j$ factors through some $f_i$.

David Egolf (Oct 08 2024 at 23:27):

The blog post I linked above has a very nice picture illustrating the condition in the definition:
picture part 1

picture part 2

David Egolf (Oct 08 2024 at 23:30):

I believe the idea is to build intuition from the case where $C$ is a poset of open sets of some topological space. (So, for some open sets $v$ and $u$ in our topological space, we have a morphism $:v \to u$ iff $v \subseteq u$). Then a $T$-covering family for a given object $u$ involves a collection of morphisms to $u$, which corresponds to a bunch of open subsets of $u$.

David Egolf (Oct 08 2024 at 23:31):

We want this collection of morphisms $(f_i:u_i \to u)$ to really "cover" $u$. If these morphisms really do cover $u$, then they should induce a cover of any open subset $v$ of $u$, given by taking intersections (as illustrated in the image I've labelled "picture part 2").

David Egolf (Oct 08 2024 at 23:34):

In diagram form, if we have a cover $(f_i:u_i \to u)$ for $u$ and we have some open subset $v \subseteq u$, then there should be a cover of $v$ formed from various open subsets $v_j \subseteq v$ such that there is, for any $v_j$, always some $u_i$ so that the dashed morphism exists:
diagram

David Egolf (Oct 08 2024 at 23:35):

In the pictured example above, each $v_j$ can be obtained by intersecting $u_j$ with $v$. So, $v_j$ is the part of $v \subseteq u$ induced by some part $u_i$ of $u$. This cover for $v$ is induced by the cover for $u$ that we started with, by taking intersections with $v$.

David Egolf (Oct 08 2024 at 23:42):

The definition of a coverage then ensures that any $T$-covering family for an object $U$ behaves in this sense like a cover of open sets. Intuitively, if we have a $T$-covering family for $U$ and a morphism $g:V \to U$, there is some $T$-covering family for $V$ where each "part" of $V$ in this $T$-covering family can be associated to some part of $U$ in our $T$-covering family for $U$.

David Egolf (Oct 09 2024 at 00:11):

(At first glance, this notion of "coverage" appears to be different than both the notion of "Grothendieck topology" and the notion of "basis for a Grothendieck topology". Hopefully how all these ideas relate will become clearer eventually!)

John Baez (Oct 09 2024 at 01:24):

According to the nLab article [[coverage]]:

The traditional name for a coverage, with the extra saturation conditions imposed, is a Grothendieck topology, and this is still widely used in mathematics. Following the Elephant, on this page we use coverage for a pullback-stable system of covering families and Grothendieck coverage if the extra saturation conditions are imposed. See Grothendieck topology for a discussion of the objections to that term.

A related notion is that of basis for a Grothendieck topology, which is similar to the notion of coverage, and similarly induces a Grothendieck topology, but assumes existence of pullbacks and closure of covering families under these pullbacks.

So basically a "coverage" and a "basis for a Grothendieck topology" are two ways to describe a Grothendieck topology, which is ultimately a way to describe a topos of sheaves.

John Baez (Oct 24 2024 at 17:09):

Hey @David Egolf - I hope you're doing okay. You might like this:

• Bartosz Milewski on sheaves.

I think you already read his previous post on 'coverages', which are a way of presenting Grothendieck topologies.

David Egolf (Oct 24 2024 at 19:56):

Hi @John Baez ! I'm doing alright. I've been less active here recently because I've started doing a little tutoring (in 100-level physics). Tutoring has been interesting, but it's been using up most of the energy I have at the moment.

I saw that post and it looks quite interesting, although I haven't read it yet. Thanks for pointing it out! I hope to get back to sheaves in some form sooner than later, energy allowing. Maybe I'll take a good look at that post, or maybe I'll try to start on Part 4 of the topos theory posts.