End (category theory)

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Not to be confused with the use of End to represent (categories of) endomorphisms.

In category theory, an end of a functor S:\mathbf{C}^{\mathrm{op}}\times\mathbf{C}\to \mathbf{X} is a universal dinatural transformation from an object e of X to S.

More explicitly, this is a pair (e,\omega), where e is an object of X and

\omega:e\ddot\to S

is a dinatural transformation from the constant functor whose value is e on every object and 1_e on every morphism, such that for every dinatural transformation

\beta : x\ddot\to S

there exists a unique morphism

h:x\to e

of X with

\beta_a=\omega_a\circ h

for every object a of C.

By abuse of language the object e is often called the end of the functor S (forgetting \omega) and is written

e=\int_c^{} S(c,c)\text{ or just }\int_\mathbf{C}^{} S.

If X is complete, the end can be described as the equaliser in the diagram

\int_c S(c, c) \to \prod_{c \in C} S(c, c) \rightrightarrows \prod_{c \to c'} S(c, c'),

where the first morphism is induced by S(c, c) \to S(c, c') and the second morphism is induced by S(c', c') \to S(c, c').

Coend[edit]

The definition of the coend of a functor S:\mathbf{C}^{\mathrm{op}}\times\mathbf{C}\to\mathbf{X} is the dual of the definition of an end.

Thus, a coend of S consists of a pair (d,\zeta), where d is an object of X and

\zeta:S\ddot\to d

is a dinatural transformation, such that for every dinatural transformation

\gamma:S\ddot\to x

there exists a unique morphism

g:d\to x

of X with

\gamma_a=g\circ\zeta_a

for every object a of C.

The coend d of the functor S is written

d=\int_{}^c S(c,c)\text{ or }\int_{}^\mathbf{C} S.

Dually, if X is cocomplete, then the coend can be described as the coequalizer in the diagram

\int^c S(c, c) \leftarrow \coprod_{c \in C} S(c, c) \leftleftarrows \coprod_{c \to c'} S(c', c).

Examples[edit]

Suppose we have functors F, G : \mathbf{C} \to \mathbf{X} then

\mathrm{Hom}_{\mathbf{X}}(F(-), G(-)) : \mathbf{C}^{op} \times \mathbf{C} \to \mathbf{Set}.

In this case, the category of sets is complete, so we need only form the equalizer and in this case

\int_c \mathrm{Hom}_{\mathbf{X}}(F(c), G(c)) = \mathrm{Nat}(F, G)

the natural transformations from F to G. Intuitively, a natural transformation from F to G is a morphism from F(c) to G(c) for every c in the category with compatibility conditions. Looking at the equalizer diagram defining the end makes the equivalence clear.

Let T be a simplicial set. That is, T is a functor \Delta^{\mathrm{op}} \to \mathbf{Set}. The discrete topology gives a functor \mathbf{Set} \to \mathbf{Top}, where \mathbf{Top} is the category of topological spaces. Moreover, there is a map \gamma:\Delta \to \mathbf{Top} which sends the object [n] of \Delta to the standard n simplex inside \mathbb{R}^{n+1}. Finally there is a functor \mathbf{Top} \times \mathbf{Top} \to \mathbf{Top} which takes the product of two topological spaces. Define S to be the composition of this product functor with T \times \gamma. The coend of S is the geometric realization of T.

References[edit]