coend in a derivator


Homotopy theory

(,1)(\infty,1)-Category theory

Ends and coends in a derivator


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 DD be a derivator indexed on a 2-category DiaDia of diagram shapes, let ADiaA\in Dia, and let HD(A op×A)H\in D(A^{op}\times A); we wish to define the coend of HH (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.

As a colimit over simplices

One classical construction of a coend as a colimit proceeds by constructing an auxiliary category A twA^{tw}, whose objects are the objects and arrows of AA, with morphisms from each arrow of AA (regarded as an object of A twA^{tw}) to its domain and codomain. There is a functor p:A twA op×Ap\colon A^{tw}\to A^{op}\times A which sends each object xx to (x,x)(x,x) and each arrow f:xyf\colon x\to y to (y,x)(y,x), and the coend of H:A op×ACH\colon A^{op}\times A\to C can be constructed as the colimit of p *H:A twCp^* H\colon A^{tw}\to C.

We can mimic this in a derivator, except that we need to “homotopify” A twA^{tw} by including higher information as well. Thus, let (Δ/A) op(\Delta / A)^{op} denote the opposite of the category of simplices of AA. Thus its objects are functors x:[n]Ax\colon [n]\to A, and its morphisms from x:[n]Ax\colon [n]\to A to y:[m]Ay\colon [m]\to A are functors [m][n][m]\to [n] making the evident triangle commute. There is a functor p:(Δ/A) opA op×Ap\colon (\Delta / A)^{op} \to A^{op}\times A which sends x:[n]Ax\colon [n]\to A to (x(n),x(0))(x(n), x(0)).

Note that A twA^{tw}, as constructed above, is the full subcategory of (Δ/A) op(\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 twA^{tw}, but for homotopy colimits we need all of (Δ/A) op(\Delta / A)^{op}.

Hence, we define the (homotopy) coend of HD(A op×A)H\in D(A^{op}\times A) to be the (homotopy) colimit of p *HD((Δ/A) op)p^* H \in D((\Delta / A)^{op}).

As a colimit over arrows

Another classical construction of a coend as a colimit involves a different auxiliary category, the opposite twisted arrow category Tw(A) opTw(A)^{op}. The objects of Tw(A) opTw(A)^{op} are arrows f:xyf\colon x\to y in AA, and its morphisms from f:xyf\colon x\to y to g:zwg\colon z\to w are commutative squares

x z f g y w \array{ x & \to & z\\ ^f \downarrow & & \downarrow^g\\ y & \leftarrow & w }

in AA. There is a projection

r:Tw(A) opA op×A r \colon Tw(A)^{op} \to A^{op}\times A

sending f:xyf\colon x\to y to (y,x)(y,x), and the coend of H:A op×ACH\colon A^{op}\times A\to C can be constructed as the colimit of r *H:Tw(A) opCr^* H\colon Tw(A)^{op} \to C.

Amazingly, this version needs no modification to become homotopical. Given HD(A op×A)H\in D(A^{op}\times A) in a derivator, we can simply restrict along rr to Tw(A) opTw(A)^{op}, then take the (homotopy) colimit. To see that this agrees with the previous definition, it suffices to factor qq through rr via a homotopy final functor ss:

Tw(A) op s r (Δ/A) op q A op×A \array{ & & Tw(A)^{op} \\ & ^{s}\nearrow & & \searrow^{r}\\ (\Delta / A)^{op} & & \underset{q}{\to} & & A^{op}\times A }

The definition of ss is simple: we regard an nn-simplex as a string of nn composable arrows in AA and take its composite. The morphisms in the two categories match nicely. To show that ss is homotopy final, we must show that

(Δ/A) op s Tw(A) op * *\array{(\Delta / A)^{op} & \xrightarrow{s} & Tw(A)^{op} \\ \downarrow & & \downarrow \\ * & \to & * }

is a homotopy exact square. For this, it suffices to show that it becomes homotopy exact when pasted with any comma square

Q f * f (Δ/A) op Tw(A) op.\array{Q_{f} & \overset{}{\to} & *\\ \downarrow & \swArrow & \downarrow^{f}\\ (\Delta / A)^{op}& \underset{}{\to} & Tw(A)^{op}.}

The objects of the category Q fQ_{f} are strings of composable n2n\ge 2 arrows whose composite is ff. Its morphisms are like those of (Δ/A) op(\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,))B(A(-,b),A,A(a,-)) over ff.

However, the simplicial map B(A(,b),A,A(a,))A(a,b)B(A(-,b),A,A(a,-)) \to A(a,b), with A(a,b)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 fQ_f has contractible nerve. This implies that

Q f * * * \array{ Q_f & \to & * \\ \downarrow & & \downarrow \\ * & \to & * }

is homotopy exact, and thus ss is homotopy final.

As a bar construction

Another classical construction of a coend as a colimit is as the coequalizer of a parallel pair

f:xyH(y,x) xH(x,x). \coprod_{f\colon x\to y} H(y,x) \;\rightrightarrows\; \coprod_{x} H(x,x).

This can be obtained in a straightforward way from the previous construction. If PprPpr denotes the walking parallel pair (10)(1 \rightrightarrows 0), then there is a functor q:A twPprq\colon A^{tw}\to Ppr sending each object of AA to 0, each arrow of AA 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:A twCp^* H \colon A^{tw} \to C along qq. Because qq 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 *Hp^*H is equivalently its left Kan extension to the point, the functoriality of Kan extensions means that colim A twp *H\colim^{A^{tw}} p^* H is isomorphic to colim PprLan qp *H\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:(Δ/A) opA op×Ap\colon (\Delta / A)^{op} \to A^{op}\times A be as above, and let q:(Δ/A) opΔ opq\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, PprPpr is a final, but not homotopy-final, subcategory of Δ op\Delta^{op}. The functoriality of homotopy Kan extensions in a derivator means that the homotopy colimit of p *HD((Δ/A) op)p^* H\in D((\Delta / A)^{op}) can equivalently be calculated as the homotopy colimit of q !p *HD(Δ op)q_! p^* H \in D(\Delta^{op}).

Note that since an object D(Δ op)D(\Delta^{op}) is a simplicial object of DD, it makes sense to call its colimit geometric realization. Moreover, the homotopy version of qq is also a discrete opfibration, and since pullbacks of fibrations are homotopy exact, homotopy Kan extensions along qq are also computed as colimits over its fibers. These fibers are also discrete, so we obtain a simplicial diagram of the following sort:

x 0f 1x 1f 2x 2H(x 2,x 0) x 0f 1x 1H(x 1,x 0) xH(x,x) \cdots \coprod_{x_0\xrightarrow{f_1} x_1 \xrightarrow{f_2} x_2} H(x_2,x_0) \underoverset{\to}{\to}{\underoverset{\leftarrow}{\leftarrow}{\to}} \coprod_{x_0\xrightarrow{f_1} x_1} H(x_1,x_0) \underoverset{\to}{\to}{\leftarrow} \coprod_{x} H(x,x)

This is a derivator version of the bar construction of HH. (A bar construction is perhaps the most classical construction of homotopy coends.)

As a weighted colimit

A last classical construction of a coend is as a weighted colimit of H:A op×ACH\colon A^{op}\times A\to C, weighted by the hom-functor hom:A op×ASethom\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 BB denote the collage of hom:A op×ASethom\colon A^{op}\times A \to Set regarded as a profunctor 1A op×A1 ⇸ A^{op}\times A, with inclusions u:1Bu\colon 1\to B and v:A op×ABv\colon A^{op}\times A \to B. We can therefore define the homotopy coend of HD(A op×A)H\in D(A^{op}\times A) to be u *v !Hu^* v_! H.

To show that this is the same as the previous definitions, we simply observe that there is a comma square

Tw(A) op r A op×A ! α v 1 u B\array{Tw(A)^{op}& \overset{r}{\to} & A^{op}\times A\\ ^!\downarrow & \swArrow_\alpha & \downarrow^v\\ 1& \underset{u}{\to} & B}

Thus, by one of the axioms of a derivator, u *v !Hcolim(r *H)u^* v_! H \cong \colim (r^* H).

Revised on March 21, 2012 15:32:58 by Mike Shulman (