26
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 $...
- 17.8k
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 ...
- 5,539
12
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 ...
- 37.8k
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 "...
- 28.1k
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}\...
- 7,377
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 ...
- 6,051
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 $...
- 5,760
8
votes
Accepted
What are the intermediate semisimple groups of type A?
$\DeclareMathOperator\SL{SL}\SL_n/\mu_d (k)$ is the group consisting of pairs of an $n\times n$ matrix $M$ over $k$ and an element $a$ of $k^\times$ such that $\det(M) = a^{n/d}$, modulo the subgroup $...
- 148k
8
votes
Accepted
effective descent of coherent sheaves
The answer depends on the action:
If you take an action which is free, then the morphism $\pi$ is étale and, by Etale descent (cf. Bosch-Lutkebohmer-Raynaud "Neron Models page. 139) and you have ...
- 368
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 ...
- 9,273
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 ...
- 188k
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' ...
- 6,051
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 ...
- 21.3k
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?
...
- 37.8k
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 ...
- 21.3k
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 $Y \rightarrow X$ ...
- 42.4k
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 ...
- 28.7k
5
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 ...
- 532
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, ...
- 16.2k
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 ...
- 41
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 ...
- 28.7k
4
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 ...
- 91
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 ...
- 35.4k
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 ...
- 28.7k
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,...
- 66.7k
3
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 ...
- 31
2
votes
English Reference for the Bénabou-Roubaud theorem
There is an excellent translation of the French article presenting the
Benabou–Roubaud Theorem to English by Zoran Škoda which he certainly will provide on request.
On p.101 of Fibered Categories (à ...
- 121
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 ...
- 63.9k
2
votes
Accepted
Number Rings and (Galois) Descent
The categorical Galois theory of Borceux and Janelidze given in chapter 4 for commutative rings applies to your situation.
In particular, it applies to any 'effective Galois descent morphism' defined ...
- 10.1k
Only top scored, non community-wiki answers of a minimum length are eligible