Essentially surjective functor information

mathematics, specifically in category theory, a functor F : C → D {\displaystyle F:C\to D} is essentially surjective if each object d {\displaystyle d} of D {\displaystyle...

anafunctor. For example, the statement "every fully faithful and essentially surjective functor is an equivalence of categories" is equivalent to the axiom...

morphisms between them. Thus any functor from C {\displaystyle C} to E {\displaystyle E} will not be essentially surjective. Consider a category C {\displaystyle...

are functions preserving this structure. There is a natural forgetful functor U : Top → Set to the category of sets which assigns to each topological...

if there is an equivalence between them. essentially surjective A functor F is called essentially surjective (or isomorphism-dense) if for every object...

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

{B}}\cong {\mathcal {C}}} if and only if there exists an exact and essentially surjective functor F : A → C {\displaystyle F\colon {\mathcal {A}}\to {\mathcal...

pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar...

which is an abelian category equipped with an exact functor from A to A/B that is essentially surjective and has kernel B. This quotient category can be constructed...

representation canonical bifunctor; but as (single) functor, of type [X, -], it appears as an adjoint functor to a functor of type (-×X) on objects; In functional...

to C) is all of C; that is, if and only if that map is an epimorphism (surjective, or onto). Therefore, the sequence 0 → X → Y → 0 is exact if and only...

groups 1 → A → H → G → 1 {\displaystyle 1\to A\to H\to G\to 1\!} the surjective homomorphism d : H → G {\displaystyle d\colon H\to G\!} together with...

is isomorphic to the quotient group G / ker(f). In particular, if f is surjective then H is isomorphic to G / ker(f). This theorem is usually called the...

A˘ • A ≤ I Q1: B˘ • B ≤ I Q2: A˘ • B = 1 Essentially these axioms imply that the universe has a (non-surjective) pairing relation whose projections are...

(this definition is motivated by a descent theorem of Grothendieck for surjective étale maps of affine schemes). With these definitions, the algebraic spaces...

is extra structure. Essentially, these are groupoids G 1 , G 0 {\displaystyle {\mathcal {G}}_{1},{\mathcal {G}}_{0}} with functors s , t : G 1 → G 0 {\displaystyle...

that monotone Galois connections are special cases of pairs of adjoint functors in category theory as discussed further below. Other terminology encountered...

or surjective, then the same is true for all above defined linear maps. In particular, the tensor product with a vector space is an exact functor; this...

sheaves, and a surjective map corresponds to an injection of sheaves. An alternative approach to the dual point of view is to use the functor of points. If...

example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below)...

continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd adjoint functor theorem)...