In algebraic geometry, a prestackF over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object.
Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks.
Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category C; for example, C can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology.
In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying...
of canonical maps or canonical isomorphisms; for a typical example, see prestack. For a discussion of the problem of defining a canonical map see Kevin...
the following are the most important: anisotropic parameter estimation, prestack depth anisotropy migration, and fracture characterization based on anisotropy...
X,i=1,...,m} . There is an analog of a Ran space for a scheme: the Ran prestack of a quasi-projective scheme X over a field k, denoted by Ran ( X ) {\displaystyle...
of groupoids. This way, each groupoid object determines a prestack in groupoids. This prestack is not a stack but it can be stackified to yield a stack...
in this area: (the prestack of quasi-coherent sheaves over a scheme S means that, for any S-scheme X, each X-point of the prestack is a quasi-coherent...
industry, he earned his PhD from Stanford in 1979. His dissertation on prestack partial migration was a major contribution to seismic processing. Dr. Yılmaz...
functor. Via p, Vect n {\displaystyle \operatorname {Vect} _{n}} is a prestack over C. That it is a stack over C is precisely the statement "vector bundles...
presentation or cover of the stack X {\displaystyle X} . Recall that a prestack (of groupoids) on a category C {\displaystyle {\mathcal {C}}} , also known...
cartesian morphisms. cartesian morphism 1. Given a functor π: C → D (e.g., a prestack over schemes), a morphism f: x → y in C is π-cartesian if, for each object...
stacks. Hopf algebroid - encodes the data of quasi-coherent sheaves on a prestack presentable as a groupoid internal to affine schemes (or projective schemes...
q\circ f=p} . More generally, one can also consider a morphism between prestacks; (a stackification would be an example.) One particular important example...
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