nLab
enriched derivator

Idea

If a derivator is regarded as a “shadow” of an (,1)(\infty,1)-category, then an enriched derivator is an analogous shadow of an enriched (,1)(\infty,1)-category.

Peter LeFanu Lumsdaine: What does “enriched (,1)(\infty,1)-category” mean here? — just one of the models for (,1)(\infty,1)-categories as (Top-, Kan-, SSet-)enriched categories, or actually some further idea of “(,1)(\infty,1)-category enriched in something”? (The latter sounds an interesting idea, but I’m not quite sure how to imagine it!)

Mike Shulman: I had the latter in mind. Just as a 1-category can be enriched over a monoidal 1-category, an (,1)(\infty,1)-category should be enrichable over a monoidal (,1)(\infty,1)-category. The “default” enrichment will be over \infty-groupoids (which one could express in any particular model), just as the “default” enrichment for 1-categories is over sets.

Definitions

There are several different things in which one could try to enrich a derivator. It would be nice to be able to remain completely in the world of derivators by enriching a derivator over a monoidal derivator. However, it seems unlikely that the naive notion of monoidal derivator (a pseudomonoid in the 2-category of derivators) contains enough information to make this feasible, so we either have to augment that notion somehow, or enrich over something else. At present, the latter is easier.

Enrichment over a monoidal (,1)(\infty,1)-category

The following is tentative.

Let VV' be a bicomplete closed symmetric monoidal (∞,1)-category, and let VVV\subset V' be a reflective sub-(∞,1)-category which is a 1-category and which is closed under the monoidal structure and the internal-hom. For example:

  • V=GpdV'=\infty Gpd with cartesian product, V=SetV=Set.
  • V=Gpd *V'= \infty Gpd_* (pointed \infty-groupoids) with smash product, V=Set *V=Set_*.

We can therefore talk about VV-enriched categories. Moreover, since VV is bicomplete and closed, the category VV-CatCat is again closed symmetric monoidal, and hence enriched over itself.

Definition

A (V,V)(V,V')-enriched prederivator is a VCATV CAT-enriched functor

D:VCat opVCAT. D\colon V Cat^{op} \to V CAT.

(or replacing VCatV Cat by some subcategory of it.)

For instance, when (V,V)=(Set,Gpd)(V,V') = (Set, \infty Gpd) then this is precisely the ordinary notion of prederivator. We define this by reference to VV, rather than VV', in order that the definition of prederivator can remain a purely 1-categorical notion, without referring explicitly to any (,1)(\infty,1)-category.

We now need to specify the axioms which an enriched prederivator should satisfy to be called an enriched derivator. Most of them are easy by analogy:

  • (Der1) DD takes coproducts to products.

  • (Der2) If 11 denotes the unit VV-category, then for any VV-category AA the family of functors D(A)D(1)D(A) \to D(1) is jointly conservative.

  • (Der3) For any VV-functor u:ABu\colon A\to B, the VV-functor u *:D(B)D(A)u^*\colon D(B) \to D(A) has a left and a right adjoint u !u_! and u *u_*.

  • (Der5) For any object xVx\in V, let I[x]I[x] denote the xx-fattened interval category, with two objects 00 and 11 and hom(0,1)=xhom(0,1)=x. Then for any VV-category AA, the induced functor D(A×I[x])D(A) I[x]D(A\times I[x]) \to D(A)^{I[x]} is essentially surjective and full.

The last, and most important, axiom, involves a characterization of the appropriate homotopy exact squares of VV-categories. Here is finally where the larger (,1)(\infty,1)-category VV' enters the picture. If AA is a VV-category and F:AVF\colon A\to V and G:A opVG\colon A^{op}\to V are VV-functors, then we can take their tensor product G AFVG\otimes_A F \in V, but we can also regard them as landing in VV' and take their homotopy tensor product G A hFVG \otimes^h_A F \in V', and these will generally give different results.

  • (Der4) Given any square
    A f B u v C g D \array{ A & \overset{f}{\to} & B \\ ^u\downarrow & & \downarrow ^v \\ C & \underset{g}{\to} & D}

    of VV-categories such that for any objects cCc\in C and bBb\in B, the induced map

    C(,c) A hB(b,)D(v(b),g(c)) C(-,c) \otimes^h_A B(b,-) \to D(v(b),g(c))

    is an equivalence in VV', then the induced transformation

    u !f *g *v ! u_! f^* \to g^* v_!

    (and hence also its mate v *g *f *u *v^* g_* \to f_* u^*) is an isomorphism.

This should be compared with the characterizations of exact squares for ordinary (enriched) category theory and of homotopy exact squares.

A (V,V)(V,V')-enriched prederivator satisfying (Der1)–(Der5) is called a (V,V)(V,V')-enriched derivator.

Enrichment over a monoidal homotopical category

Note that all we really needed of VV' was a place for a well-behaved homotopy tensor product of VV-profunctors to live. Thus, we could replace it by anything else sufficiently good, such as a monoidal model category or a closed monoidal homotopical category. In this case, the homotopy tensor product can be computed as a “homotopy coend” or more explicitly as a two-sided bar construction.

Moreover, since the only dependence on VV' in the definition is via the homotopy tensor product of functors, the particular choice of model we make is essentially irrelevant for the resulting notion of derivator. In other words, the true input for a notion of enriched derivator is an ordinary monoidal category VV, together with “a homotopy theory in which VV is reflectively embedded.”

Examples

  • When (V,V)=(Set,Gpd)(V,V') = (Set,\infty Gpd), all the axioms of an enriched derivator are identical to those of an ordinary derivator, except for (Der4). The usual axiom (Der4) asserts the conclusion only for comma squares. However, Cisinski’s theorem characterizing homotopy exact squares shows that this implies the enriched version of (Der4) stated above (which is a priori stronger). To make this comparison with the way Cisinski’s theorem is usually stated, one has to observe that C(,c) A hB(b,)C(-,c) \otimes^h_A B(b,-) can be constructed as a two-sided bar construction, which in turn can (in this case) be identified with the nerve of a particular category.

  • Conjecturally, (Set *,Gpd *)(Set_*, \infty Gpd_*)-enriched derivators are the same as pointed derivators in the usual sense.

  • Also conjecturally, any VV'-enriched (,1)(\infty,1)-category should have an underlying (V,V)(V,V')-enriched derivator.

Properties

Just as an ordinary derivator encodes a well-behaved notion of [[homotopy limits], an enriched derivator encodes a notion of homotopy weighted limit. In order to recover this, we need to represent profunctors by their collages.

Specifically, let H:ABH\colon A⇸ B be a VV-profunctor, and AuH¯vBA \overset{u}{\to} \bar{H} \overset{v}{\leftarrow} B its collage. Then homotopy HH-weighted limits in an enriched derivator are computed by the composite v *u *v^* u_*, and similarly HH-weighted colimits are computed by u *v !u^* v_!.

Note that unlike in the ordinary unenriched case, it is not possible to compute all limits by Kan extending along functors to the terminal category. This is because of the presence and importance of weighted limits.

Homotopy Kan extensions are pointwise

Part of the intuition behind the usual axiom (Der4) is that it says that the homotopy Kan extensions which make up the structure of a derivator are all pointwise. For left Kan extensions along u:ABu\colon A\to B, pointwiseness means that each object (Lan uX)(b)(Lan_u X)(b) is calculated as a suitable colimit, which in the unenriched case can be expressed as an ordinary colimit over the comma category u/bu/b – hence why we assert that comma squares are exact.

In the enriched situation, the (co)limit involved in the notion of “pointwise Kan extension” is irreducibly a weighted limit, and not constructible as a comma object. However, the Kan extensions in an enriched derivator are still pointwise in a suitable sense; we can see this as follows. Let u:ABu\colon A\to B be a VV-functor, let bBb\in B, let B(u,b):1AB(u,b)\colon 1 ⇸ A denote the representable profunctor, and let HH be its collage. Then the square

A A u H B\array{ A & \to & A \\ \downarrow & & \downarrow^u\\ H & \to & B }

satisfies the hypothesis of the enriched (Der4). (This is just the homotopical version of Yoneda reduction: B(u,b) A hAB(u,b)B(u,b) \otimes_A^h A \simeq B(u,b).) Therefore, if we left extend along uu, then restrict to HH and further restrict to bb, we get the same thing as if we just extended to HH and restricted to bb — but the latter is exactly the process of taking the B(u,b)B(u,b)-weighted colimit. Thus, the evaluation of u !Xu_! X at bb is computed as a weighted colimit, and hence the Kan extension is “pointwise.”

Revised on May 26, 2011 06:08:49 by Mike Shulman (71.136.238.9)