nLab (∞,1)-end

Idea

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

Definition

End of a space-valued functor

Given an (∞,1)-functor $F \colon J^{op} \times J \to \infty Gpd$ to ∞Groupoids from the product of a locally small (∞,1)-category $J$ 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 $J$ to $F$:

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

Ends and coends as a weighted (co)limit

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

$$C\left(c, \int_J F \right) \simeq \int_J C(c, F) \simeq \infty Gpd^{J^{op} \times J} \big( J(-,-), C(c, F(-,-)) \big)$$

$$C\left(\int^J F, c \right) \simeq \int_{J^{op}} C(F, c) \simeq \infty Gpd^{J \times J^{op}}( J^{op}(-,-), C(F(-,-), c) )$$

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

As (co)limits over the twisted arrow category

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

$$\int_J F \simeq \lim \left( Tw(J) \xrightarrow{(src, tgt)} J^{op} \times J \xrightarrow{F} C \right)$$

$$\int^J F \simeq \colim \left( Tw(J^{op})^{op} \xrightarrow{(src, tgt)} J^{op} \times J \xrightarrow{F} C \right)$$

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

Properties

Proposition (Fubini)

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

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

Proof

For ends, using the fact $Tw$ preserves products we can compute

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

The same argument applies for coends.

Proposition

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

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

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

Related concepts

• twisted arrow (∞,1)-category
• lax (∞,1)-colimit

References

• David Gepner, Rune Haugseng, Thomas Nikolaus, Lax colimits and free fibrations in $\infty$-categories (arXiv:1501.02161)