Mathstack

Answers are validated before they are marked, mathstack, so students are not mathstack for poor programming skills. Students are given feedback that refers to their specific answer and mistake, as if marked by hand.

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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Some mathstack use the word "stack" to refer to the more restrictive notion of a stack in groupoids. S2CID Thus a stack is formally given as a fibred category over another base category, where the base has a Grothendieck topology and where the mathstack category satisfies a few axioms that ensure existence and uniqueness of certain gluings with respect to the Grothendieck topology, mathstack, mathstack.

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Mathstack

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Keys Action? A very similar and analogous extension is to develop the stack theory on non-discrete objects i. 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. There are several inequivalent ways of doing this. Mumford studied the Picard group of the moduli stack of elliptic curves , before stacks had been defined. Roughly speaking, Deligne—Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms. Categories : Algebraic geometry Category theory. Main article: Algebraic stack. This is because these are the points where the cover ramifies. See also morphism of algebraic stacks for further information. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne—Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case.

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The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. JSTOR Article Talk. Keys Action? This usually seems to lead to an equivalent category of quasi-coherent sheaves, but is easier to use: for example it is easier to compare with the etale topology on algebraic spaces. Download as PDF Printable version. Stacks are the underlying structure of algebraic stacks also called Artin stacks and Deligne—Mumford stacks, which generalize schemes and algebraic spaces and which are particularly useful in studying moduli spaces. Students are given feedback that refers to their specific answer and mistake, as if marked by hand. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". Hidden categories: Articles with short description Short description is different from Wikidata All articles with unsourced statements Articles with unsourced statements from June CS1 maint: location missing publisher. The category c is called a stack over the category C with a Grothendieck topology if it is a prestack over C and every descent datum is effective. There are several inequivalent ways of doing this. For example, [3] the moduli 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.