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- Grothendieck fibration in nLab
A Grothendieck fibration (named after Alexander Grothendieck, also called a fibered category or just a fibration) is a functor p: E → B such that the fibers E b = p 1 (b) depend (contravariantly) pseudofunctorially on b ∈ B
- CATEGORICAL NOTIONS OF FIBRATION - arXiv. org
Fibrations over a category B, introduced to category the-ory by Grothendieck, encode pseudo-functors Bop Cat, while the special case of discrete fibrations encode presheaves Bop → Set A two-sided discrete variation encodes functors Bop × A → Set, which are also known as profunctors from A to B
- math. uchicago. edu
There is a dual notion of cocartesian functor between Grothendieck opfibrations over $B$ and a notion of bicartesian functor between Grothendieck bifibrations
- About Grothendieck Fibrations | Springer Nature Link (formerly . . .
Fibrations are a very sophisticated tool, very flexible and pervasive After reviewing some elementary, known properties of Grothendieck fibrations, we shall address their applications in logic, with the use of several examples
- Grothendieck fibrations or When aestethics drives mathematics
Fibrations are categories varying over a category Definition Afibrationis a functorBopP Cat i e data as follows for every objectbinBa categoryP(b) for everyf:b0 binBa functorP(f):P(b) P(b0) for every objectbinBa commutative diagramP(b) IdP(b) P(idb) P(b) for every composable pairg:b00 b0,f:b0 binBa commutative diagramP(b0)P(g)
- A model structure for Grothendieck fibrations
Given a functor F: C op → Cat, its Grothendieck construction yields a Grothendieck fibration over C; that is, a functor P: P → C with the property that, for every morphism f: c → P a in C, there is a P -cartesian morphism g: a → a ′ in P such that P g = f
- Grothendieck construction in nLab
A functor p: E → C is a Grothendieck fibration if for every object e ∈ E and every morphism f: c → p (e) in C there is a morphism f ^: c ^ → e in E that lifts f in that p (f ^) = f and which is a Cartesian morphism
- Fibrations and Grothendieck topologies - Cambridge University Press . . .
We define a notion of fibration for morphisms of presheaves that is well behaved with respect to composition, base change and exponentiation, and trivializes on the topos S
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