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...

G {\displaystyle g,h\in G} and S g {\displaystyle S_{g}} is a full and faithful functor for every g ∈ G {\displaystyle g\in G} we say that ( C , S ) {\displaystyle...

category that is equipped with a faithful functor to the category of sets (or sometimes to another category). This functor makes it possible to think of...

{\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...

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...

the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf...

of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their...

commutative) and a full, faithful and exact functor F: A → R-Mod (where the latter denotes the category of all left R-modules). The functor F yields an...

t-structure do not assume the existence of truncation functors, such functors can always be constructed and are essentially unique. Suppose that D {\displaystyle...

{\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...

(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...

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...

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)...

{\displaystyle u^{*}} is the left adjoint in a pair of adjoint functors and is a full and faithful functor. The category of presheaves over any Q-category is itself...

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...

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...

{\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...

resolutions and thereby the use of the tools of homological algebra in A {\displaystyle {\mathcal {A}}} , in order to define derived functors. (Note that...

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

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

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...

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...

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...

speak of an exact functor between exact categories exactly as in the case of exact functors of abelian categories: an exact functor F {\displaystyle F}...