- (Fop)op=F{\displaystyle \left(F^{\mathrm {op} }\right)^{\mathrm {op} }=F}. A
bifunctor (also
known as a
binary functor) is a
functor whose domain is a product...
- B′. The
commutativity of the
above diagram implies that Hom(–, –) is a
bifunctor from C × C to Set
which is
contravariant in the
first argument and covariant...
-
Cartesian product of two sets.
Product categories are used to
define bifunctors and multifunctors. The
product category C × D has: as objects:
pairs of...
- or map object. It
appears in one way as the
representation canonical bifunctor; but as (single) functor, of type [X, -], it
appears as an
adjoint functor...
- category) is a
category C{\displaystyle \mathbf {C} }
equipped with a
bifunctor ⊗:C×C→C{\displaystyle \otimes :\mathbf {C} \times \mathbf {C} \to \mathbf...
- Let I be a
finite category and J be a
small filtered category. For any
bifunctor F:I×J→Set,{\displaystyle F:I\times J\to \mathbf {Set} ,}
there is a natural...
- does not fit. First,
consider the
binary product functor,
which is a
bifunctor. For f1:X1→Y1,f2:X2→Y2{\displaystyle f_{1}:X_{1}\to Y_{1},f_{2}:X_{2}\to...
- they are
given by
groupoids A {\displaystyle \mathbb {A} }
which have a
bifunctor + : A × A → A {\displaystyle +:\mathbb {A} \times \mathbb {A} \to \mathbb...
- In
terms of
category theory, this
means that the
tensor product is a
bifunctor from the
category of
vector spaces to itself. If f and g are both injective...
-
recognize homC(F–, –) and homD(–, G–) as functors. In fact, they are both
bifunctors from Dop × C to Set (the
category of sets). For details, see the article...