End (category theory)

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

More explicitly, this is a pair [math]\displaystyle{ (e,\omega) }[/math], where e is an object of X and [math]\displaystyle{ \omega:e\ddot\to S }[/math] is an extranatural transformation such that for every extranatural transformation [math]\displaystyle{ \beta : x\ddot\to S }[/math] there exists a unique morphism [math]\displaystyle{ h:x\to e }[/math] of X with [math]\displaystyle{ \beta_a=\omega_a\circ h }[/math] for every object a of C.

By abuse of language the object e is often called the end of the functor S (forgetting [math]\displaystyle{ \omega }[/math]) and is written

[math]\displaystyle{ e=\int_c^{} S(c,c)\text{ or just }\int_\mathbf{C}^{} S. }[/math]

Characterization as limit: If X is complete and C is small, the end can be described as the equalizer in the diagram

[math]\displaystyle{ \int_c S(c, c) \to \prod_{c \in C} S(c, c) \rightrightarrows \prod_{c \to c'} S(c, c'), }[/math]

where the first morphism being equalized is induced by [math]\displaystyle{ S(c, c) \to S(c, c') }[/math] and the second is induced by [math]\displaystyle{ S(c', c') \to S(c, c') }[/math].

Coend

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

Thus, a coend of S consists of a pair [math]\displaystyle{ (d,\zeta) }[/math], where d is an object of X and [math]\displaystyle{ \zeta:S\ddot\to d }[/math] is an extranatural transformation, such that for every extranatural transformation [math]\displaystyle{ \gamma:S\ddot\to x }[/math] there exists a unique morphism [math]\displaystyle{ g:d\to x }[/math] of X with [math]\displaystyle{ \gamma_a=g\circ\zeta_a }[/math] for every object a of C.

The coend d of the functor S is written

[math]\displaystyle{ d=\int_{}^c S(c,c)\text{ or }\int_{}^\mathbf{C} S. }[/math]

Characterization as colimit: Dually, if X is cocomplete and C is small, then the coend can be described as the coequalizer in the diagram

[math]\displaystyle{ \int^c S(c, c) \leftarrow \coprod_{c \in C} S(c, c) \leftleftarrows \coprod_{c \to c'} S(c', c). }[/math]

Examples

• Natural transformations:

  Suppose we have functors [math]\displaystyle{ F, G : \mathbf{C} \to \mathbf{X} }[/math] then

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

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

  [math]\displaystyle{ \int_c \mathrm{Hom}_{\mathbf{X}}(F(c), G(c)) = \mathrm{Nat}(F, G) }[/math]

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

• Geometric realizations:

  Let [math]\displaystyle{ T }[/math] be a simplicial set. That is, [math]\displaystyle{ T }[/math] is a functor [math]\displaystyle{ \Delta^{\mathrm{op}} \to \mathbf{Set} }[/math]. The discrete topology gives a functor [math]\displaystyle{ d:\mathbf{Set} \to \mathbf{Top} }[/math], where [math]\displaystyle{ \mathbf{Top} }[/math] is the category of topological spaces. Moreover, there is a map [math]\displaystyle{ \gamma:\Delta \to \mathbf{Top} }[/math] sending the object [math]\displaystyle{ [n] }[/math] of [math]\displaystyle{ \Delta }[/math] to the standard [math]\displaystyle{ n }[/math]-simplex inside [math]\displaystyle{ \mathbb{R}^{n+1} }[/math]. Finally there is a functor [math]\displaystyle{ \mathbf{Top} \times \mathbf{Top} \to \mathbf{Top} }[/math] that takes the product of two topological spaces.

  Define [math]\displaystyle{ S }[/math] to be the composition of this product functor with [math]\displaystyle{ dT \times \gamma }[/math]. The coend of [math]\displaystyle{ S }[/math] is the geometric realization of [math]\displaystyle{ T }[/math].

References

• Mac Lane, Saunders (2013). Categories For the Working Mathematician. Springer Science & Business Media. pp. 222–226. 
• Loregian, Fosco (2015). "This is the (co)end, my only (co)friend". arXiv:1501.02503 [math.CT].

External links

• end in nLab