Mathstack

Answers are validated before they mathstack marked, so students are not penalised for poor programming skills.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory , and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic , compatible geometrical objects such as vector bundles on topological spaces can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks ; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work".

Mathstack

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In the same way, moduli spaces of mathstack, vector bundles, mathstack, or other geometric objects are often best defined as stacks instead of schemes.

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Mathstack

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In a more general set-up the restrictions are replaced with pullbacks ; fibred categories then make a good framework to discuss the possibility of such gluing. Main article: Quasi-coherent sheaf on an algebraic stack. Archived PDF from the original on Toggle limited content width. For example, the big fppf topology leads to essentially the same category of quasi-coherent sheaves as the Lis-Et topology, but has a subtle problem: the natural embedding of quasi-coherent sheaves into O X modules in this topology is not exact it does not preserve kernels in general. This is because these are the points where the cover ramifies. If A is a quasi-coherent sheaf of algebras in an algebraic stack X over a scheme S , then there is a stack Spec A generalizing the construction of the spectrum Spec A of a commutative ring A. For example, if a few points have non-trivial stabilisers, then the categorical quotient will not exist among schemes, but it will exist as a stack. A stack is called a stack in groupoids or a 2,1 -sheaf if it is also fibered in groupoids, meaning that its fibers the inverse images of objects of C are groupoids. Differentiable stacks and topological stacks are defined in a way similar to algebraic stacks, except that the underlying category of affine schemes is replaced by the category of smooth manifolds or topological spaces. Main article: Algebraic stack.

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ISSN X. In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Main article: Quasi-coherent sheaf on an algebraic stack. S2CID Keys Action? These moduli spaces are denoted [3]. Stacky curves , or orbicurves, can be constructed by taking the stack quotient of a morphism of curves by the monodromy group of the cover over the generic points. In this paper they also introduced Deligne—Mumford stacks , which they called algebraic stacks, though the term "algebraic stack" now usually refers to the more general Artin stacks introduced by Artin For example, [3] the moduli stack. The Lis-Et topology has a subtle technical problem: a morphism between stacks does not in general give a morphism between the corresponding topoi. For example, the big fppf topology leads to essentially the same category of quasi-coherent sheaves as the Lis-Et topology, but has a subtle problem: the natural embedding of quasi-coherent sheaves into O X modules in this topology is not exact it does not preserve kernels in general. More generally one can define the notion of an n -sheaf or n —1 stack, which is roughly a sort of sheaf taking values in n —1 categories. Students are given feedback that refers to their specific answer and mistake, as if marked by hand. 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.