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...

{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...

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...

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

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...

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...

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...

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...

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

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

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...

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...

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

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

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...

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...

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...

for any representable functor Hom(−, X) and any morphism Hom(−, X) → F, the fibered product G×FHom(−, X) is a representable functor Hom(−, Y) and the morphism...

family of functions. The notion of topological functors generalizes (and strengthens) that of fibered categories, for which one considers a single morphism...

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...

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

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

that of treating an object, such as S, as 'as good as' the representable functor it sets up. The Grothendieck–Riemann–Roch theorem from about 1956 is usually...

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

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...

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...

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