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End (category theory)

Mathematical concept

Note

the type of transformation

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

More explicitly, this is a pair (e,\omega), where e is an object of \mathbf X and \omega\colon e\ddot\to S is an extranatural transformation such that for every extranatural transformation \beta\colon x\ddot\to S there exists a unique morphism h\colon x\to e of \mathbf X with \beta_a=\omega_a\circ h for every object a of \mathbf 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.

Ends can also be described using limits. If \mathbf X is complete and \mathbf C is small, the end can be described as the equalizer 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 being equalized is induced by S(c, c) \to S(c, c') and the second is induced by S(c', c') \to S(c, c').

Coend

The definition of the coend of a functor S\colon \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 \mathbf X and \zeta\colon S\ddot\to d is an extranatural transformation, such that for every extranatural transformation \gamma\colon S\ddot\to x there exists a unique morphism g\colon d\to x of \mathbf X with \gamma_a=g\circ\zeta_a for every object a of \mathbf C.

The coend d of the functor S is written

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

Coends have a characterization using limits dual to the characterization of ends. If \mathbf X is cocomplete and \mathbf C is small, 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

Natural transformations:

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.

Geometric realizations:

Let T be a simplicial set. That is, T is a functor \Delta^{\mathrm{op}} \to \mathbf{Set}. The discrete topology gives a functor d:\mathbf{Set} \to \mathbf{Top}, where \mathbf{Top} is the category of topological spaces. Moreover, there is a map \gamma:\Delta \to \mathbf{Top} sending 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} that takes the product of two topological spaces.

Define S to be the composition of this product functor with dT \times \gamma. The coend of S is the geometric realization of T.

Notes

References

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