Fiber functors in category theory, topology and algebraic geometry refer to several loosely related functors that generalise the functors taking a covering...

gist of the theory is that the fiber functor Φ of the Galois theory is replaced by an exact and faithful tensor functor F from C to the category of finite-dimensional...

Formally, a diagram of shape J {\displaystyle J} in C {\displaystyle C} is a functor from J {\displaystyle J} to C {\displaystyle C} : F : J → C . {\displaystyle...

geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each...

Presh(D) denotes the category of contravariant functors from D to the category of sets; such a contravariant functor is frequently called a presheaf. Giraud's...

category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit...

{F}}_{c}} , and a morphism d → c {\displaystyle d\to c} induces a functor from the fibered category structure. Namely, for an object x ∈ Ob ( F c ) {\displaystyle...

theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting...

In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental...

direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in...

section of the pullback (fiber-product) bundle f ∗ E {\displaystyle f^{*}E} over M . {\displaystyle M.} Inverse image functor – functor between categories of...

construction associates to a pseudofunctor a fibered category. Lax functor Prestack (an example of pseudofunctor) Fibered category C. Sorger, Lectures on moduli...

mixed-characteristic local fields. Section conjecture Class field theory Fiber functor Neukirch–Uchida theorem Belyi's theorem Frobenioid Inter-universal Teichmüller...

this is a part of the study of atomic toposes. Tannakian formalism Fiber functor Anabelian geometry Grothendieck, A.; et al. (1971). SGA1 Revêtements...

category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category...

an assembly map is a universal approximation of a homotopy invariant functor by a homology theory from the left. From the geometric viewpoint, assembly...

such that π {\displaystyle \pi } is injective as a function. fiber functor fiber functor. Frobenius reciprocity The Frobenius reciprocity states that...

continuous functor sends covering sieves to covering sieves. If J is the topology defined by a pretopology, and if u commutes with fibered products, then...

Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor Hom functor Product (category theory) Equaliser...

involves a functor, the nearby cycle functor, with a definition by means of the higher direct image and pullbacks. The vanishing cycle functor then sits...

indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top. Top is also fiber-complete meaning that the category of all topologies...

In the mathematical field of topology, a section (or cross section) of a fiber bundle E {\displaystyle E} is a continuous right inverse of the projection...

Dolbeault–Grothendieck lemma. The construction of a scheme structure on (representable functor version of) the Picard group, the Picard scheme, is an important step in...

In mathematics, the theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated...

Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful...

mathematics, restriction of scalars (also known as "Weil restriction") is a functor which, for any finite extension of fields L/k and any algebraic variety...

respect to some universe) and the morphisms functors. Fct(C, D), the functor category: the category of functors from a category C to a category D. Set, the...

or a colimit of presheaves on a category C is a limit or colimit in the functor category C ^ = F c t ( C op , S e t ) {\displaystyle {\widehat {C}}=\mathbf...

form a tannakian category RepG. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine...

overcounted. A category c {\displaystyle c} with a functor to a category C {\displaystyle C} is called a fibered category over C {\displaystyle C} if for any...