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Stream: deprecated: topos theory

Topic: Grothendieck Topos and Giraud's Axioms

ADITTYA CHAUDHURI (Jul 08 2020 at 16:12):

I am a beginner in the subject . So I am seeking prior apology if I sound stupid.
Let $S$ be the category of Sets . A Grothendieck Topos is a Category of $S$-valued Sheaves over a site internal to $S$. According to https://ncatlab.org/nlab/show/Grothendieck+topos#Giraud a Grothendieck Topos can also be defined as a locally small category satisfying some properties called Giraud's Axioms. I do not know the Proof of the Giraud's theorem yet which characterises Grothendieck Topos in an Axiomatic way. Hopefully I will learn it soon in near future. But first I want to understand the statement that is "what Giraud's Axiom actually tells us".
In particular according to Wiki article https://en.wikipedia.org/wiki/Topos#Grothendieck_topoi_(topoi_in_geometry) it is mentioned " The main utility of this notion of Topos is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the heuristic". I can understand or technically feel that the notion of Topos expressed as Category of sheaves over a site expresses this fact . (In fact in the paper Convenient Categories of Smooth Spaces by Baez and Hoffnung https://arxiv.org/pdf/0807.1704.pdf the category of generalized smooth spaces are described as the category of concrete sheaves on a concrete site).
My question is the following:
How a Grothendieck Topos defined using Giraud's Axioms can be seen in a geometric way(The statement of Wikipedia I mentioned previously)?
Thank You

Jens Hemelaer (Jul 08 2020 at 20:00):

You can see a Grothendieck topos as a the category of sets parametrized over some "generalized space". For example, sheaves over a topological space are, informally, sets parametrized over this topological space.
If you want to make this precise, then you need to replace "sets parametrized over a space" by the notion of local homeomorphism.

Giraud's axioms hold in the category of sets, and then the idea is that they also hold for "sets parametrized over some generalized space".
They are concerned with the more "geometrical" features of working with sets. For example, the axiom about equivalence relations explains very accurately how the "gluing" of sets works.

ADITTYA CHAUDHURI (Jul 08 2020 at 21:20):

@Jens Hemelaer Thank you very much! Just to make sure.. By "If you want to make this precise, then you need to replace "sets parametrized over a space" by the notion of local homeomorphism." do you mean Etale spaces over the space?
Also what is meant by this here?
So as you suggested if we identify Set valued Sheaves on a space $X$ with Etale spaces over $X$ then can you please explain in little details how will it be convenient to see the geometric interpretation of Giraud's Axioms?

Morgan Rogers (he/him) (Jul 08 2020 at 21:36):

The advantage is that one can treat the objects of Grothendieck toposes as if they are spaces over some base and perform "geometric" operations on these spaces, and the resulting constructions will behave (from a categorical perspective) like they do over topological spaces.

Morgan Rogers (he/him) (Jul 08 2020 at 21:37):

Even when this is not true in a precise sense, the structural similarity typically allows geometric intuition to be extended to a form that makes sense in any (Grothendieck) topos

ADITTYA CHAUDHURI (Jul 08 2020 at 22:09):

@[Mod] Morgan Rogers Thanks! I got your point. But do identifying sheaves over a space as Etale spaces over the space will also help in understanding the Geometric Interpretation of Giraud's Axioms? (Note: I did not read the proof of Giraud's Theorem yet.. So I may sound very stupid)

Jens Hemelaer (Jul 09 2020 at 08:21):

ADITTYA CHAUDHURI said:

Jens Hemelaer Thank you very much! Just to make sure.. By "If you want to make this precise, then you need to replace "sets parametrized over a space" by the notion of local homeomorphism." do you mean Etale spaces over the space?
Also what is meant by this here?

Yes, I use local homeomorphism as a synonym for étale space here. And by 'this' I meant the concept of seeing objects of the topos as "sets parametrized over a space".

Jens Hemelaer (Jul 09 2020 at 08:50):

For me, one direction of Giraud's theorem is easier than the other one. The easier part is proving that the category of sheaves on a Grothendieck site satisfies Giraud's axioms (first prove it for categories of presheaves, then use that the sheafification functor has very good properties).

Regarding the geometric interpretation of the axioms:
Colimits are very geometrical, because they are about gluing things. Now take for example the third axiom here: a Grothendieck topos has finite coproducts that are disjoint and pullback-stable.
This means that you can visualize a coproduct of two objects $A$ and $B$ as "just putting $A$ and $B$ together".
If you look at the copy of $A$ and the copy of $B$ inside $A \sqcup B$, then their intersection is empty.
Pullback-stability says for example that $(A \sqcup B) \times I \simeq (A \times I) \sqcup (B \times I)$, so if you visualize $I$ as a line or interval, then this is the behaviour that you would expect, geometrically.

When visualizing things, it helps to see how things work at the level of points (although not all Grothendieck toposes have enough points).
By definition, taking the "fiber" above a point in a topos preserves colimits and finite limits. So a colimit or a finite limit of étale spaces will still be a colimit or finite limit if you zoom in to the fiber above one point.

Jens Hemelaer (Jul 09 2020 at 08:57):

A fun consequence of pullback-stability of coproducts:
Take the initial object $0$, this is the empty coproduct. Pullback-stability says that $0 \times I \simeq 0$ for all objects $I$. This expresses that the initial object $0$ is "really empty".

A counter-example where this pullback-stability of coproducts does not hold, is the category of pointed sets. The initial object is the set with one element. However, if you take the product of this initial object with some pointed set $I \neq 0$,
you get $0 \times I = I \neq 0$. So the initial object was not "really empty", and as a result the category of pointed sets is not a Grothendieck topos.

ADITTYA CHAUDHURI (Jul 09 2020 at 12:52):

@Jens Hemelaer Thanks a lot for the explanation. I got the geometric motivation.

Can you please explain in little detail what did you mean by "By definition, taking the "fiber" above a point in a topos preserves colimits and finite limits. So a colimit or a finite limit of étale spaces will still be a colimit or finite limit if you zoom in to the fiber above one point."
For me fiber over point $x$ is something like this
taaaa.png

where "1" is a terminal object in the category. I did not get the line taking "fibre" above a point in a topos preserves colimits and finite limits .

I observed another geometric interpretation in the the 2nd Axiom https://ncatlab.org/nlab/show/Grothendieck+topos#Giraud that is the existence of finite limits and hence existence of terminal object which is ensuring that we can talk about fibre over a point where by a point $x$ in an object $M$ of the topos I actually mean a morphism $x:1 \rightarrow M$ . So I feel literally they behave like points!

Reid Barton (Jul 09 2020 at 13:07):

There are two different meanings of "point" here which arise from the blurred distinction between topos-as-category and topos-as-space. If $X$ is an object of a topos, then a "point of $X$" is a map $x : * \to X$ like you write above. On the other hand, a "point of the topos" itself is a point of the imaginary space on which the topos is the category of sheaves. On the category level, it corresponds to a certain kind of functor.

ADITTYA CHAUDHURI (Jul 09 2020 at 13:35):

@Reid Barton Thanks a lot for the clarification. Did you mean that since the category Sets is a terminal object in the 2-category of Grothendieck Topos so a geometric morphism $x: Sets \rightarrow E$ can be thought as a point in the Topos $E$?(Precisely due to the same reason I explained previously)

Jens Hemelaer (Jul 09 2020 at 14:11):

By a point, I meant a point of the topos $\mathcal{E}$ in the sense of a geometric morphism $x: \mathbf{Sets} \to \mathcal{E}$. Taking the fiber is then applying $x^*$, the left adjoint part of the geometric morphism. In order for $x$ to qualify as a geometric morphism, $x^*$ should preserve colimits and finite limits.

The diagram that you posted is good intuition. In fact, for each geometric morphism $X$ of $\mathcal{E}$, there is an associated topos $\mathcal{E}/X$, the slice topos over $X$. If you think of $\mathcal{E}$ as a generalized space, then $\mathcal{E}/X$ is the étale space over it, corresponding to the object $X$.

Now, if you have a geometric morphism $x : \mathbf{Sets} \to \mathcal{E}$, then there is the following pullback diagram in the 2-category of Grothendieck toposes:
Screenshot-2020-07-09-at-16.08.58.png

ADITTYA CHAUDHURI (Jul 09 2020 at 15:06):

@Jens Hemelaer Thanks a lot for the explanation!