25
votes

Accepted

### What is Barr-Beck?

It is well-attested in the category theory literature (e.g. in Mac Lane's Categories for the Working Mathematician, Chapter VI) that the well-known theorem giving necessary and sufficient conditions ...

24
votes

Accepted

### How do $\infty$-categories allow us to do descent on the derived level?

Let $X$ be a topological space covered by open sets $U$ and $V$.
Let $\mathscr{F}$ and $\mathscr{G}$ be complexes of sheaves defined on $U$ and $V$, respectively. Suppose you are given an isomorphism $...

17
votes

Accepted

### Difficulties with descent data as homotopy limit of image of Čech nerve

To answer your question I'll need to do a fairly long digression on homotopy limits and colimits. Before I delve deep into the topic let me say that there's more than one way to describe this topic, ...

11
votes

Accepted

### The derived category does not satisfy descent - example

Consider the canonical functor $H$ from the homotopy category of homotopy coherent descent data in the ∞-category of coherent sheaves
to the category of descent data in the derived category of ...

10
votes

Accepted

### Which simplicial objects are Čech nerves?

The analogous 1-categorical version of your first question would be "which parallel pairs are kernel pairs?" As far as I know this does not have a non-tautological answer in an arbitrary category, ...

9
votes

### Descent a representation over finite field

As stated, there is a non-semisimple counterexample: take $G$ to be the additive group $\mathbb{F}_q$ with a unipotent representation $\rho(a)=\left(\begin{matrix}1 & a\\ 0 & 1\end{matrix}\...

9
votes

### What is Barr-Beck?

It looks like Alexander's answer hits the nail on the head, but I thought I might add what Barr himself says. Barr and Wells write in Toposes, Triples, and Theories (on page 121 in the "...

8
votes

### English Reference for the Bénabou-Roubaud theorem

The following is a reasonably literal translation (made by me from the French original) of the published article: Jean Bénabou, Jacques Roubaud. Monades et descente. C. R. Acad. Sc. Paris, t. 270 (12 ...

8
votes

Accepted

### Basic example of derived descent

$\newcommand{\Z}{\mathbf{Z}}\newcommand{\FF}{\mathbf{F}}\newcommand{\H}{\mathrm{H}}$Let me do the universal case of your situation, which is understanding descent along the map $\Z[t] \to \Z$ sending $...

7
votes

Accepted

### Descent for $K(1)$-local spectra

It's not quite true: need to require a $p$-adic continuity condition for the $\Psi^g$-semilinear automorphism of the $K(1)$-local $K$-module. You can see https://arxiv.org/pdf/2001.11622.pdf ...

7
votes

### What is Barr-Beck?

M. Barr and J. Beck, Acyclic models and triples, Proc. Conference Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 336-343.
The "triple" is an older name for a ...

6
votes

Accepted

### Geometric intuition for the condition of Galois descent

Let $G \to E \to B$ be a principal bundle. It's classified by a map $f : B \to BG$ in the sense that $E$ is the homotopy fiber of this map. This means that $E$ has a certain universal property: namely,...

6
votes

### English Reference for the Bénabou-Roubaud theorem

This is incoporated into an existing answer for several reasons, e.g. to avoid moving it to the front page. I only noticed this reference now, when I continued to work on this. While the 'translation' ...

6
votes

Accepted

### Descent of Higher categorical structures along geometric morphisms

I have been able to gather all the elements for the case of $2$-categorical descent along open surjections, so I will write it as an answer.
It is worth noting that the proof below follows the exact ...

6
votes

### Non-trivial automorphisms and descent

Briefly, descent is an analogue of taking quotients.
In the category of sets, we have the following familiar facts:
an equivalence relations on a set is a relation $R \subseteq A \times A$ satisfying ...

5
votes

Accepted

### Fpqc-locally constant if and only if étale-locally constant?

The answer is Yes, but it fails for some slight variants, so let me first discuss an analogue for sheaves of sets. In that case, this is closely related to the discussion of pro-etale fundamental ...

5
votes

Accepted

### When can a scheme be recovered from its descent groupoid?

In any topos, if $Y \rightarrow X$ is an epimorphism then:
$$Y \times_X Y \rightrightarrows Y \rightarrow X $$
is indeed a colimit diagram.
If you have a site $S$ and a cover $Y \rightarrow X$ ...

5
votes

Accepted

### Infinite Galois descent for finitely generated commutative algebras over a field

As an alternative to my first answer, here is an easier (but less powerful) approach that uses the finite generation in a more essential way:
Let $k \to \ell$ be a Galois field extension with Galois ...

5
votes

### Non-trivial automorphisms and descent

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?
...

5
votes

Accepted

### Gluing data for modules over a ring with idempotents

If $m=2$, your data gives no information about the relationship between $e_1M$ as an $e_1Ae_1$-module and $e_2M$ as an $e_2Ae_2$-module. So this is false. For example, take a quiver with two vertices, ...

4
votes

### English Reference for the Bénabou-Roubaud theorem

Here is the copy of The Bénabou-Roubaud monadic descent theorem via string diagrams paper:
https://www.dropbox.com/s/236deicmr3636d5/draft.pdf?dl=0
Disclaimer: the papers was, and still is, just a ...

4
votes

Accepted

### Projective after fpqc base change

Hironaka's example is a locally projective birational morphism $f \colon \tilde X \to X = \mathbf P^3$ where $\tilde X$ is not projective; in particular $f$ is not projective. This already shows that ...

4
votes

### English Reference for the Bénabou-Roubaud theorem

I cooked up a detailed proof in this file, even if it is not original, because I wanted to understand everything.
About Zoran Škoda's blog: it is factually false. Fibred categories appear right at the ...

3
votes

Accepted

### Descending central extensions to homogeneous spaces

First up, $\sigma\colon G\times H \to G\times_f G$ sending $(g,h)\mapsto (g,gh)$ is a diffeomorphism. For this, all you need is that $G\to G/H$ is a locally trivial $H$-bundle (so, with care, this ...

3
votes

Accepted

### Pushouts of commutative pseudomonoids

To summarize some of the comments:
I don't know a short answer for why a bicategorical coequalizer doesn't work. If you try to give the bicategorical coequalizer the structure and universal property,...

3
votes

### Infinite Galois descent for finitely generated commutative algebras over a field

Let me reduce the statement to one about fpqc descent. There should probably be an easier way to do this, but this is arguably the more 'correct' argument. The reduction of Galois descent for a finite ...

3
votes

Accepted

### Faithfully flat descent in complex analytic geometry

There will not be a way to directly translate faithfully flat descent into gluing over analytic opens; the former is genuinely stronger than the latter in the analytic category. However basically any ...

2
votes

Accepted

### Stack descent to sheaf descent via Grothendieck construction?

Here's a variation which is true, when interpreted in a suitably non-strict / higher categorical sense (for example, "functor" means "pseudofunctor" below). I'm not sure on which side of the ...

2
votes

### What is etale descent?

When a functor happens to be a sheaf in étale topology you say it satisfies étale descent. So for example algebraic K-theory does not satisfy étale descent but Bott periodic algebraic K-theory modulo ...

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