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Fibered Category

A fibered category F over a topological space X consists of

1. a category F(U) for each open subset U subset= X,

2. a functor i^*:F(U)->F(V) for each inclusion i:V↪U, and

3. a natural isomorphism

tau=tau_(i,j):(ij)^*->j^*i^*

for each pair of inclusions j:W↪V, i:V↪U.

FiberedCategoryDiagram

In addition, for any three composable inclusions k:N↪W, j:W↪V, and i:V↪U, there exists a natural commuting as shown above.

Sometimes, the pair (F,tau) is used to denote a fibered category with more precision while the shorthand a|V is sometimes used for i^*(a), i:V↪U, a in F(U).

See also

Category, Commutative Diagram, Composition, Fibered Category Morphism, Functor, Inclusion Map, Isomorphism, Morphism, Open Set, Topological Space

This entry contributed by Christopher Stover

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References

Moerdijk, I. "Introduction to the Language of Stacks and Gerbes." 2002. http://arxiv.org/pdf/math/0212266v1.pdf.

Cite this as:

Stover, Christopher. "Fibered Category." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FiberedCategory.html

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