Non-trivial automorphisms and descent

Question

In this expository paper by Low it says:

Roughly speaking, a topos in the sense of Grothendieck is the category of sheaves on a kind of generalised space whose “points” may have non-trivial automorphisms.

Question 1: What does that mean?

In a footnote it says:

More precisely, every Grothendieck topos is equivalent to the topos of equivariant sheaves on a localic groupoid; see [Joyal and Tierney, 1984].

Still, I wonder how this is related to points having non-trivial automorphisms.

The MathSciNet review of the paper by Johnstone says:

Basically, it is about descent theory: the question of whether, given a morphism of toposes $f: \mathcal F → \mathcal E$, it is possible to reconstruct objects $X$ of $\mathcal E$ from objects $f^* X$ of $\mathcal F$ equipped with “descent data”.

Question 2: In simple terms, what is the motivation for that kind of problem, and what is descent data? Also, why does it say "objects $f^* X$" - isn't $f^* X$ a single object?

Further in the review:

The main theorem (VIII 2.1) asserts that every open surjection is an effective descent morphism.

Question 3: What is the idea of an effective descent morphism?

Coupling this with the result, essentially due to R. Diaconescu[Comm. Algebra 4 (1976), no. 8, 723–729; MR0414658], that every Grothendieck topos $\mathcal E$ admits an open surjection $\mathcal F → \mathcal E$ where $\mathcal F$ is localic, yields the authors’ second main theorem (VIII 3.2) which is a representation theorem asserting that an arbitrary Grothendieck topos is equivalent to the topos of “equivariant sheaves” on a groupoid in the category of locales.

Question 4: Why does VIII 2.1 together with the theorem of Diaconescu imply the second main theorem (the representation theorem)? (I'm only interested in getting an idea how these statements fit together, i.e., a very rough proof sketch suffices.)

In the introduction to [Joyal and Tierney, 1984] the authors write:

In fact, our first descent theorem for modules is completely analogous to the usual descent theorems of commutative algebra.

Question 5: What are the usual descent theorems in commutative algebra? I did not find anything comprehensible googling "descent theorems in commutative algebra".

Caveat: I expect that some people will react to this question saying "just read the books and papers yourself", but I find that it is quite hard to go through tons of technical definitions and lemmas without knowing the main idea in advance. This is why I am asking these questions: in order that I am able to read the papers and books myself. :-)

I also tried this MO thread but I can't make head nor tail of it. There it says the main questions of descent theory are:

• When an object $G$ in $C_Y$ is in the image via $f^*$ of some object in $C_X$ ?
• Classify all forms of object $G\in C_Y$, that is find all $E\in C_X$ for which $f^*(E)\cong G$.

This setting seems to be different from the setting in the third quote above, because here we consider $G\in C_Y$ which is in the codomain of $f^*$ whereas above we are given $X\in\mathcal E$, and $\mathcal E$ is the domain of $f^*$. This increases my confusion.

Answer 1

Briefly, descent is an analogue of taking quotients.

In the category of sets, we have the following familiar facts:

1. an equivalence relations on a set is a relation $R \subseteq A \times A$ satisfying certain properties;

2. a quotient for an eq. rel. $R$ is a map $[-] : A \to Q$ that is universal among maps sending $R$ to equality;

3. we can construct a quotient for any equivalence relation (using equivalence classes);

4. for $x, y \in A$, we have $[x] = [y]$ if and only if $(x,y) \in R$;

5. every surjection $e : A \twoheadrightarrow A'$ is a quotient of $A$ by the relation “$e(x)=e(y)$”.

We can generalise these to arbitrary categories, ending up with the ingredients of a regular or exact category. The possibly-unexpected bit is the importance of property (4) — that elements become related in a quotient only if they were related by the given relation; this is not automatic from being a quotient, and is called an effective quotient. Lots of the ways we use quotients rely on effectivity (and not on anything else about the specific construction of the quotient).

The practical takehome is: If you can present an object as an effective quotient of a well-understood object, that can be very useful for understanding it. So it’s useful to have theorems for giving/identifying effective quotients, like fact (5) above.

Now since toposes are categorical objects, we have to generalise this to a 2-categorical version. When we “take a quotient” of an object of a 2-category, we want to “identify its elements/points/objects”, i.e. add new “isomorphisms”. But as ever, objects can be isomorphic via multiple different isomorphisms — equivalently, they can have non-trivial automorphisms. So when we “identify” them, we have to keep track of these extra automorphisms. So we want not just a “relation”, but extra data explaining what identifications should happen.

This is the idea of the truncated simplicial object that appears in the descent construction: it’s a generalisation of an equivalence relation. Effective descent morphisms of toposes are the analogue of effective quotients (and the category of descent data is a particular construction of such a quotient). And the motivation of the theorem is to give a tractable concrete presentation for an arbitrary topos, by expressing it as an “effective quotient” of a particularly nice and familiar kind of topos.

This tackles your questions 1–3. Questions 4 and 5 are a bit separate from these, and would be much better asked as separate questions.

Answer 2

Roughly speaking, a topos in the sense of Grothendieck is the category of sheaves on a kind of generalised space whose “points” may have non-trivial automorphisms.

Question 1: What does that mean?

An elementary answer to this question can be given in terms of sites: any Grothendieck topos is equivalent to the topos of sheaves of sets on a site, and an object in a site can have nontrivial automorphisms. These automorphisms survive when we pass to points of the site. For example, for the site of smooth manifolds and smooth maps, there is a single point for every $n≥0$ given by the germ of a point in $\def\R{{\bf R}}\R^n$. Such germs have nontrivial automorphisms.

Question 2: In simple terms, what is the motivation for that kind of problem, and what is descent data? Also, why does it say "objects f∗X" - isn't f∗X a single object?

An elementary example is given by cocycle data for objects like principal bundles, vector bundles, or just coverings. Here $\def\cF{{\cal F}}\def\cE{{\cal E}} f\colon \cF→\cE$ is an open cover of $\cE$ with $\cF=∐_k U_k$ being the disjoint union of opens in the cover.

Now a bundle $X$ over $\cE$ can be specified as a bundle $Y$ over $\cF$ (i.e., a bundle over every $U_i$) together with the descent data, which in this case boils down to the transition functions that satisfy a cocycle condition.

Question 3: What is the idea of an effective descent morphism?

In elementary terms, an effective descent morphism is a morphism $f\colon \cF→\cE$ for which the above gluing procedure works perfectly: there is an equivalence of categories between objects on $\cE$ and objects on $\cF$ equipped with a descent data.

Question 4: Why does VIII 2.1 together with the theorem of Diaconescu imply the second main theorem (the representation theorem)? (I'm only interested in getting an idea how these statements fit together, i.e., a very rough proof sketch suffices.)

The relevant construction for toposes can be seen as a glorified version of a traditional construction from topology: given a topological space $X$ with an open cover $\{U_i\}_{i∈I}$, how can we reconstruct $X$ from $U$? The answer is that we need to know the pairwise intersections $U_i∩U_j$ together with their embeddings into $U_i$ and $U_j$.

Question 5: What are the usual descent theorems in commutative algebra? I did not find anything comprehensible googling "descent theorems in commutative algebra".

For example, faithfully flat descent for modules. Roughly, this is an analogue of the gluing construction for vector bundles in the context of algebraic geometry.