This is the generalization of the notion of ends from category theory to (∞,1)-category theory.


End of a space-valued functor

Given an (∞,1)-functor F:J op×JGpdF \colon J^{op} \times J \to \infty Gpd to ∞Groupoids from the product of a locally small (∞,1)-category JJ with its opposite (∞,1)-category, we define its end to be the (∞,1)-categorical hom-space in the (∞,1)-category of (∞,1)-functors from the hom-functor of JJ to FF:

JFGpd J op×J(J(,),F(,)) \int_J F \coloneqq \infty Gpd^{J^{op} \times J} \big( J(-,-), F(-,-) \big)

Ends and coends as a weighted (co)limit

More generally, given an (∞,1)-functor F:J op×JCF \colon J^{op} \times J \to C to ∞Groupoids to any locally small (∞,1)-category CC, we define its (co)end, when it exists, as the (co)representing object of

C(c, JF) JC(c,F)Gpd J op×J(J(,),C(c,F(,))) C\left(c, \int_J F \right) \simeq \int_J C(c, F) \simeq \infty Gpd^{J^{op} \times J} \big( J(-,-), C(c, F(-,-)) \big)
C( JF,c) J opC(F,c)Gpd J×J op(J op(,),C(F(,),c)) C\left(\int^J F, c \right) \simeq \int_{J^{op}} C(F, c) \simeq \infty Gpd^{J \times J^{op}}( J^{op}(-,-), C(F(-,-), c) )

When C=GpdC = \infty Gpd, this definition agrees with the previous.

As (co)limits over the twisted arrow category

Let F:J op×JCF : J^{op} \times J \to C be a functor between (∞,1)-categories. We can express the (co)end as (co)limits over the twisted arrow (∞,1)-category by

JFlim(Tw(J)(src,tgt)J op×JFC) \int_J F \simeq \lim \left( Tw(J) \xrightarrow{(src, tgt)} J^{op} \times J \xrightarrow{F} C \right)
JFcolim(Tw(J op) op(src,tgt)J op×JFC) \int^J F \simeq \colim \left( Tw(J^{op})^{op} \xrightarrow{(src, tgt)} J^{op} \times J \xrightarrow{F} C \right)

The second of these, for example, follows from the fact the projection p:Tw(J op)J×J opp : Tw(J^{op}) \to J \times J^{op} is the left fibration classified by J op(,)J^{op}(-,-), and so we can write the hom-functor as the left Kan extension p !(1)=J op(,)p_!(1) = J^{op}(-,-) and take an adjoint transpose.


Proposition (Fubini)

Given a functor F:J op×J×K op×KCF : J^\op \times J \times K^\op \times K \to C, if the relevant ends/coends exist, we have natural equivalences

J KF J×KF J KF J×KF \int_J \int_K F \simeq \int_{J \times K} F \qquad \qquad \int^J \int^K F \simeq \int^{J \times K} F

For ends, using the fact TwTw preserves products we can compute

J KF lim jjTw(J)lim kkTw(K)F(j,j,k,k) lim (j,k)(j,k)Tw(J×K)F(j,j,k,k) J×KF \begin{aligned} \int_J \int_K F &\simeq \lim_{j \to j' \in Tw(J)} \lim_{k \to k' \in Tw(K)} F(j,j',k,k') \\&\simeq \lim_{(j,k) \to (j',k') \in \Tw(J \times K)} F(j,j',k,k') \\&\simeq \int_{J \times K} F \end{aligned}

The same argument applies for coends.


Given functors F,G:XYF,G : X \to Y between two (∞,1)-categories, the space of natural transformations between them is given by the end

Y X(F,G) xXY(F(x),G(x)) Y^X(F, G) \simeq \int_{x \in X} Y(F(x), G(x))

This is proposition 5.1 of Gepner-Haugseng-Nikolaus 15.


Last revised on May 6, 2021 at 01:19:27. See the history of this page for a list of all contributions to it.