Full and faithful functors information

category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties...

embedding to be a full and faithful functor that is injective on objects. Other authors define a functor to be an embedding if it is faithful and injective on...

field of topology; see Full set A property of functors in the mathematical field of category theory; see Full and faithful functors Satiety, the absence...

addition to those functors that delete some of the operations, there are functors that forget some of the axioms. There is a functor from the category...

C} . Any functor that is part of an equivalence of categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially...

between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category...

{\displaystyle F\dashv G} and both F and G are full and faithful. When adjoint functors F ⊣ G {\displaystyle F\dashv G} are not both full and faithful, then we may...

object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined...

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...

, h ∈ G {\displaystyle g,h\in \mathbb {G} } and S g {\displaystyle S_{g}} is a full and faithful functor for every g ∈ G {\displaystyle g\in \mathbb {G}...

equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below). This functor makes it possible...

(small, thin, and skeletal) category such that each homset has at most one element. An order embedding A → B is a full and faithful functor from A to B...

both an epimorphism and a monomorphism. Bousfield localization See Bousfield localization. calculus of functors The calculus of functors is a technique of...

epimorphism and it induces a full and faithful functor on derived categories: D(f) : D(B) → D(A). A morphism that is both a monomorphism and an epimorphism...

categories Subcategory Faithful functor Full functor Forgetful functor Yoneda lemma Representable functor Functor category Adjoint functors Galois connection...

topological spaces and their continuous functions embeds in Chu(Set, 2) in the sense that there exists a full and faithful functor F : Top → Chu(Set, 2)...

there are forgetful functors A : Ring → Ab M : Ring → Mon which "forget" multiplication and addition, respectively. Both of these functors have left adjoints...

the higher cohomology groups are the derived functors of the functor of G-invariants. Given g in G and x in X with g⋅x = x, it is said that "x is a fixed...

{\displaystyle \mathbb {N} } and rejected equality maps to { } {\displaystyle \{\}} . This gives rise to a full and faithful functor ∇ : S e t s → E f f {\displaystyle...

also fibred, and the inverse image functors are the ordinary pull-back functors for vector bundles. These fibred categories are (non-full) subcategories...

semigroup action on that set. Such actions are characterized by being faithful, i.e., if two elements of the semigroup have the same action, then they...

{\displaystyle \to } S-Mod and G:S-Mod → {\displaystyle \to } R-Mod are additive (covariant) functors, then F and G are an equivalence if and only if there is a...

model structures and contains as a full subcategory the category of spaces and proper maps; that is, there is full and faithful functor P→E which carries...

stems from the fact that kernels of exact functors between abelian categories are Serre subcategories, and that one can build (for locally small A {\displaystyle...

isomorphism in the homotopy category if and only if it is a weak homotopy equivalence. There are obvious functors from the category of topological spaces...

using the Tor functors, the left derived functors of the tensor product. A left R {\displaystyle R} -module M {\displaystyle M} is flat if and only if Tor...