Questions tagged [algebraic-stacks]

Use this tag for questions related to algebraic stacks, which are a stacks in groupoids X over the etale site such that the diagonal map of X is representable, and there exists a smooth surjection from a scheme to X.

An algebraic stack or Artin stack is a stack in groupoids $X$ over the etale site such that the diagonal map of $X$ is representable, and there exists a smooth surjection from (the stack associated to) a scheme to $X$. A morphism $Y \rightarrow X$ of stacks is representable if for every morphism $S \rightarrow X$ from (the stack associated to) a scheme to $X$, the fiber product $Y \times_X S$ is isomorphic to (the stack associated to) an algebraic space. The fiber product of stacks is defined using the usual universal property and changing the requirement that diagrams commute to the requirement that they 2-commute.

The motivation behind the representability of the diagonal is that the diagonal morphism $\Delta : \mathfrak X \to \mathfrak X \times \mathfrak X$ is representable if and only if for any pair of morphisms of algebraic spaces $X,$ $Y \to \mathfrak X$, their fiber product $X \times_{\mathfrak X} Y$ is representable.

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Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a topological space $X$, which leads one to the notion of…
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Good Introduction to Hopf Algebras with Examples

I want to learn more about hopf algebras but I am having trouble finding a down to earth introduction to the subject with lots of motivation and examples. My algebra knowledge ranges from Dummit and Foote, Atiyah-Macdonald commutative algebra, and…
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Is an étale morphism of algebraic stacks locally quasi-finite?

An étale morphism of schemes is unramified, and an unramified morphism is locally quasi-finite. Does the same hold for étale morphisms of algebraic stacks? Let us recall the definitions, following the Stacks Project[1]. Suppose $P$ is a property of…
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Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to consider sheaves that have values in…
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Three meanings of étale sheaf on X

When I am studying stacky stuffs, I am always confused by the notion of étale abelian sheaves on $X$, because conceivably there might be three different meanings of that: Take the global étale site (the category of scheme with topology defined by…
waikit
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Homology of stack points

This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the classifying stack $\mathcal{B}G$. This has a (nonrepresentable) morphism to a point:…
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Why is this called the cocycle condition?

My question is about why the condition below is called the cocycle condition. Surely it is named after some interpretation of it in cohomology. Let $C$ be a site, and let $F$ be a fibered category over $C$; for $U \in \text{Obj}(C)$ in we write…
Kind Bubble
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Corollary (1.6.2)(b) in "Champs algébriques"

I believe this hasn't been asked on this platform, so here goes. Part (b) of Corollary (1.6.2) in Gérard Laumon and Laurent Moret-Bailly's text `Champs algébriques' states the following: Let $S$ be a scheme. Let $X \xrightarrow{f} Y \xrightarrow{g}…
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Stacks and Grothendieck topology.

I was reading https://math.dartmouth.edu/~jvoight/notes/moduli-red-harvard.pdf, introduction of some notes on algebraic geometry (stacks, I do not know what it is) and came across following statement : In beginning of algebraic geometry, one starts…
user87543
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Moduli functor $\mathcal{M}_g$ of smooth curves of genus $g$ not representable

Let $S$ be a scheme. By a smooth curve of genus $g$ over $S$ we mean a proper, flat, family $C \to S$ whose geometric fibers are smooth, connected $1$-dimensional schemes of genus $g$. The moduli functor $\mathcal{M}_g$ of smooth curves of genus $g$…
user526728
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How is a Stack the generalisation of a sheaf from a 2-category point of view?

A stack is usually given in terms of: -A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf -The descent data are effective. There is an equivalent definition, using the Grothendieck…
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Is my algebraic space a scheme?

Consider $\mathcal{M}_{1,1}$ over $\bar{\mathbb{Q}}$. I have an algebraic stack $\mathcal{M}$ finite etale over $\mathcal{M}_{1,1}$ I can prove that it is an algebraic space (essentially because all its "hidden fundamental groups" are trivial - ie,…
oxeimon
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What are Mumford's 'moduli topologies'?

I've been reading Mumford's Paper 'Picard Groups of Moduli Problems'. Stated in modern language, the most famous result from the paper is that the moduli stack of elliptic curves has Picard group $\mathbb Z/12$. In the original paper, however,…
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What's the explicit description of the atlas of quotient stack?

Let $U\to[U/G]$ be a quotient stack. How does one associate a $G$-torsor $P\to S$ and a equivariant map $P\to U$ to each $S\to U$?
Sayako Hoshimiya
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Is there any known categorification of the completion of the rational numbers to the real numbers?

Classically we start with natural numbers and integers, then invert to obtain rational numbers. We can then complete the rational numbers to obtain the real numbers. One categorical level up, natural numbers can be realised as the cardinalities of…
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