equivalences in/of $(\infty,1)$-categories
Ends and coends are special sorts of limit and colimit, respectively, and have corresponding sorts of homotopy limits and colimits – homotopy ends and coends. Since a derivator is a formal structure for computing homotopy limits and colimits, there are corresponding notions of ends and coends in a derivator.
Let $D$ be a derivator indexed on a 2-category $Dia$ of diagram shapes, let $A\in Dia$, and let $H\in D(A^{op}\times A)$; we wish to define the coend of $H$ (obvious dualizations will yield its end). We will give four equivalent definitions, each of which generalizes a classical construction of ordinary coends in terms of colimits.
One classical construction of a coend as a colimit proceeds by constructing an auxiliary category $A^{tw}$, whose objects are the objects and arrows of $A$, with morphisms from each arrow of $A$ (regarded as an object of $A^{tw}$) to its domain and codomain. There is a functor $p\colon A^{tw}\to A^{op}\times A$ which sends each object $x$ to $(x,x)$ and each arrow $f\colon x\to y$ to $(y,x)$, and the coend of $H\colon A^{op}\times A\to C$ can be constructed as the colimit of $p^* H\colon A^{tw}\to C$.
We can mimic this in a derivator, except that we need to “homotopify” $A^{tw}$ by including higher information as well. Thus, let $(\Delta / A)^{op}$ denote the opposite of the category of simplices of $A$. Thus its objects are functors $x\colon [n]\to A$, and its morphisms from $x\colon [n]\to A$ to $y\colon [m]\to A$ are functors $[m]\to [n]$ making the evident triangle commute. There is a functor $p\colon (\Delta / A)^{op} \to A^{op}\times A$ which sends $x\colon [n]\to A$ to $(x(n), x(0))$.
Note that $A^{tw}$, as constructed above, is the full subcategory of $(\Delta / A)^{op}$ containing only the 0-simplices and 1-simplicies. The inclusion of this subcategory is final, but not homotopy final. Thus, for ordinary colimits it suffices to consider $A^{tw}$, but for homotopy colimits we need all of $(\Delta / A)^{op}$.
Hence, we define the (homotopy) coend of $H\in D(A^{op}\times A)$ to be the (homotopy) colimit of $p^* H \in D((\Delta / A)^{op})$.
Another classical construction of a coend as a colimit involves a different auxiliary category, the opposite twisted arrow category $Tw(A)^{op}$. The objects of $Tw(A)^{op}$ are arrows $f\colon x\to y$ in $A$, and its morphisms from $f\colon x\to y$ to $g\colon z\to w$ are commutative squares
in $A$. There is a projection
sending $f\colon x\to y$ to $(y,x)$, and the coend of $H\colon A^{op}\times A\to C$ can be constructed as the colimit of $r^* H\colon Tw(A)^{op} \to C$.
Amazingly, this version needs no modification to become homotopical. Given $H\in D(A^{op}\times A)$ in a derivator, we can simply restrict along $r$ to $Tw(A)^{op}$, then take the (homotopy) colimit. To see that this agrees with the previous definition, it suffices to factor $q$ through $r$ via a homotopy final functor $s$:
The definition of $s$ is simple: we regard an $n$-simplex as a string of $n$ composable arrows in $A$ and take its composite. The morphisms in the two categories match nicely. To show that $s$ is homotopy final, we must show that
is a homotopy exact square. For this, it suffices to show that it becomes homotopy exact when pasted with any comma square
The objects of the category $Q_{f}$ are strings of composable $n\ge 2$ arrows whose composite is $f$. Its morphisms are like those of $(\Delta / A)^{op}$, but the first and last face maps are also given by composition instead of forgetting. In fact, it is precisely the category of simplices of the fiber of the two-sided bar construction $B(A(-,b),A,A(a,-))$ over $f$.
However, the simplicial map $B(A(-,b),A,A(a,-)) \to A(a,b)$, with $A(a,b)$ a discrete simplicial set, is well-known to be a simplicial homotopy equivalence? and thus a weak equivalence of simplicial sets. Thus, each of its fibers is simplicially contractible, and hence each $Q_f$ has contractible nerve. This implies that
is homotopy exact, and thus $s$ is homotopy final.
Another classical construction of a coend as a colimit is as the coequalizer of a parallel pair
This can be obtained in a straightforward way from the previous construction. If $Ppr$ denotes the walking parallel pair $(1 \rightrightarrows 0)$, then there is a functor $q\colon A^{tw}\to Ppr$ sending each object of $A$ to 0, each arrow of $A$ to 1, and sorting the morphisms by whether they map an arrow to its domain or to its codomain.
Then the above parallel pair is the (pointwise) left Kan extension of $p^* H \colon A^{tw} \to C$ along $q$. Because $q$ is a discrete opfibration, left Kan extension along it can be computed with colimits over its fibers – since each fiber is discrete, we obtain the coproducts above. And since the colimit of $p^*H$ is equivalently its left Kan extension to the point, the functoriality of Kan extensions means that $\colim^{A^{tw}} p^* H$ is isomorphic to $\colim^{Ppr} Lan_q p^* H$, the latter being precisely the above coequalizer.
We can homotopify this in a straightforward way as well. Let $p\colon (\Delta / A)^{op} \to A^{op}\times A$ be as above, and let $q\colon (\Delta / A)^{op} \to \Delta^{op}$ be the obvious forgetful functor (whose target is the opposite of the simplex category). Note that as before, $Ppr$ is a final, but not homotopy-final, subcategory of $\Delta^{op}$. The functoriality of homotopy Kan extensions in a derivator means that the homotopy colimit of $p^* H\in D((\Delta / A)^{op})$ can equivalently be calculated as the homotopy colimit of $q_! p^* H \in D(\Delta^{op})$.
Note that since an object $D(\Delta^{op})$ is a simplicial object of $D$, it makes sense to call its colimit geometric realization. Moreover, the homotopy version of $q$ is also a discrete opfibration, and since pullbacks of fibrations are homotopy exact, homotopy Kan extensions along $q$ are also computed as colimits over its fibers. These fibers are also discrete, so we obtain a simplicial diagram of the following sort:
This is a derivator version of the bar construction of $H$. (A bar construction is perhaps the most classical construction of homotopy coends.)
A last classical construction of a coend is as a weighted colimit of $H\colon A^{op}\times A\to C$, weighted by the hom-functor $hom\colon A^{op}\times A \to Set$.
In order to homotopify this, recall that weighted colimits can be constructed in terms of Kan extensions by first Kan extending to the collage of the weighting (pro)functor, then restricting to the target object. Thus, let $B$ denote the collage of $hom\colon A^{op}\times A \to Set$ regarded as a profunctor $1 ⇸ A^{op}\times A$, with inclusions $u\colon 1\to B$ and $v\colon A^{op}\times A \to B$. We can therefore define the homotopy coend of $H\in D(A^{op}\times A)$ to be $u^* v_! H$.
To show that this is the same as the previous definitions, we simply observe that there is a comma square
Thus, by one of the axioms of a derivator, $u^* v_! H \cong \colim (r^* H)$.