equivalences in/of $(\infty,1)$-categories
A derivator is a refinement of a homotopy category which also includes information about homotopy limits and colimits. In some cases derivators can be used where one might otherwise need to use (∞,1)-categories, which can be helpful since their theory is much simpler and purely (2-)categorical.
Very similar structures were invented independently by Grothendieck (who introduced the name “derivator”), Alex Heller (who called his version “homotopy theories”), and Franke (who considered only the stable case and called his version a “system of triangulated diagram categories”). The definition given below combines elements from the work of all three.
The notion of derivator can be motivated in several ways.
Suppose we start from the perspective that what we really study in homotopy theory are (∞,1)-categories. The homotopy category of an (∞,1)-category (the 1-category obtained by setting all equivalent 1-morphisms equal) is a fairly coarse invariant, but for some purposes it is sufficient. On the other hand, sometimes we need more than merely the homotopy category, but doing everything with $(\infty,1)$-categories can be technically daunting. Frequently, all we need from an $(\infty,1)$-category that is not present in its homotopy category is to know that we have well-behaved constructions of homotopy limits and homotopy colimits.
A derivator is thus a compromise notion, containing more information than a homotopy category, but being easier to work with than an $(\infty,1)$-category. It consists of a homotopy category together with extra structure that enables one to compute with homotopy limits and colimits. Any $(\infty,1)$-category with sufficiently many limits and colimits has an underlying derivator, and working with these derivators suffices for a surprising number of things we may want to do with $(\infty,1)$-categories. However, derivators are an essentially 1-categorical notion, so we can study them using ordinary 2-category theory. Thus derivators provide a “truncated” version of higher category theory, which gives us the language to characterize higher category theory using only usual category theory, without any emphasis on any particular model (in fact, without assuming we even know any such model).
For instance, it turns out that we can also express many convenient universal properties in terms of derivators. A striking example is the theory of triangulated categories: if $A$ is an abelian category, its (bounded) derived category $D^b(A)$ is not defined by a universal property. A natural statement would be that, given a triangulated category, the category of additive functors $A\to T$ which send short exact sequences to distinguished triangles is equivalent to the category of triangulated functors $D^b(A)\to T$, but this statement is false, and in fact, does not even make sense (unless $A$ is semi-simple). But, in practice, everything behaves as if the above statement were meaningful and true. The ‘reason’ why this does not work is that the cone of a map in a triangulated category is not defined by a universal property. On the other hand, the cone of a morphism of complexes $X\to Y$ is canonically defined: this is the homotopy colimit of the diagram $0\leftarrow X \to Y$. In the world of derivators, we can remedy this situation and recover a suitable universal property.
On the other hand, we may ask: Why do we take for granted that the homotopy theory of spaces provides a good notion (among others) of ∞-groupoid? And why should we expect everything to be enriched in higher groupoids anyway? A priori this may seem arbitrary (although it certainly works very well). Instead we might ask: what is the mathematical structure over which everything is canonically enriched? How can we even correctly formulate such a question?
The notion of derivator provides a way to correctly formulate such a question, and the answer turns out to be mostly what we expect. Every type of “category theory” (at least, category theory without an a priori given enrichment) that we might want to do is automatically and uniquely enriched in homotopy types, i.e. in the homotopy category of CW-complexes or Kan complexes. (For ordinary 1-category theory, this enrichment is trivial, i.e. factors through sets considered as discrete homotopy types.) A derivator is simply a structure with the characteristics of ordinary category theory: categories, functors, natural transformations, Kan extensions, Grothendieck fibrations. We can then show that any derivator acquires such an enrichment, so that homotopy types are a canonical enriching place for “category theories.”
Let Cat denote the 2-category of large categories (not necessarily even locally small), and let $Dia$ be some 2-category of small categories, thought of as diagrams. One common choice is the 2-category of all small categories (which generates $Cat$), but we might also choose the 2-category of finite categories. A prederivator with domain $Dia$ is a strict 2-functor
As usual, $(-)^{op}$ denotes the 1-cell dual of a 2-category. Thus, a prederivator is a “$Cat$-valued presheaf” on $Dia$. Prederivators form a 2-category $PDer$ whose morphisms are pseudonatural transformations and whose 2-cells are modifications.
Another common convention is to use the double dual $Dia^{coop}$ which reverses both 1-cells and 2-cells, although confusingly sometimes in the literature this double dual is still denoted “$Dia^{op}$”. The motivation for the latter choice seems to be that then $D(A)$ is the category of “presheaves on $A$ with values in $D$” instead of the category of “diagrams of shape $A$ in $D$.” We have chosen the convention above since the main purpose of a derivator is a calculus of homotopy limits and colimits, and it is more usual to take limits and colimits of covariant functors. However, since $Dia^{coop}$ is isomorphic to $Dia^{op}$ via the 2-functor $(-)^{op}$ (as long as $Dia$ is closed under opposite categories in $Cat$), there is really very little difference.
There are two main motivating examples. Firstly, any category $C$ defines a “representable” prederivator by the assignation $A\mapsto Hom(A,C)$, sending $A\in Dia$ to the category of functors $A\to C$. This defines an embedding of $Cat$ into $PDer$.
Secondly, if $C$ is any category with weak equivalences, there is a prederivator $Ho(C)$ which sends $A\in Dia$ to the localization/homotopy category $Ho(C)(A)$ of $Hom(A,C)$ relative to the objectwise weak equivalences (as we allowed categories in $Cat$ to have large hom-sets, these localization exist). In general, this is a non-representable prederivator, although of course if the weak equivalences are just the isomorphisms, it reduces to the representable case. Note that to construct it, we don’t need anything besides ordinary (2-)category theory.
A derivator is a prederivator $D$ which satisfies a list of axioms. These axioms are of two sorts.
The first set of axioms says that there exist well-behaved homotopy limits and colimits, and more generally homotopy Kan extensions. Specifically, we require the following.
(Der3) For any functor $u : X\to Y$ in $Dia$, the inverse image functor $u^*:D(Y)\to D(X)$ admits a left adjoint $u_!$ and a right adjoint $u_*$. These can be understood as homotopy Kan extensions
(Der4) For any comma square
in $Dia$, the Beck-Chevalley transformations
are isomorphisms. Intuitively, this says that the Kan extensions in question are pointwise. In the presence of the second set of axioms, it suffices to require this when $C$ is the terminal category (for the first case) and when $B$ is so (for the second).
The second set of axioms are “sheaf” conditions. Of course, we cannot assert that $D$ is exactly a sheaf (in the appropriate 2-categorical sense), since the terminal category in $Dia$ is 2-categorically dense and so any sheaf on it is representable (and represented by $D(1)$), whereas we want to also allow non-representable derivators. But we do need some sheaf-like properties in order to do category theory. All of the following axioms can be understood as asserting that for some covering family $\{Y_i \to X\}$ in $Dia$, the canonical map $D(X) \to Desc(D,\{Y_i\})$ from $D(X)$ to the category of descent data for the covering, while not an equivalence, has some weaker good properties. They can also be understood as 2-categorical sketch conditions.
The standard axioms are:
(Der1) $D\colon Dia^{coop} \to Cat$ takes coproducts to products. Sometimes we require this only for finite coproducts. In particular, we have $D(\emptyset) = 1$.
(Der2) For any $X\in Dia$, consider the family of functors $x\colon 1\to X$ determined by the objects of $X$. Then the induced functor
is conservative (though not generally faithful). This in turn implies that the same is true for any jointly essentially surjective family of functors.
(Der5) For any $X\in Dia$, if $I$ denotes the interval category, then the induced functor
is essentially surjective and full (though again, it is not generally faithful). Sometimes it is convenient to assume this property when $I$ is any (perhaps finite) free category.
Mike Shulman: It’s clear to me that these are desirable requirements, which are moreover satisfied by all derivators of the form $Ho(C)$, but I would really like a conceptual explanation for why these axioms are sufficient.
It is easy to see that if $C$ has pointwise left and right Kan extensions along all functors in $Dia$, then its representable prederivator is a derivator. Somewhat more difficult to prove is that if $C$ is a model category (or more generally, has well-behaved homotopy Kan extensions), then the prederivator $Ho(C)$ is also a derivator. Thus the derivator encodes the notions of homotopy colimit and of homotopy limit. Note again that this way of seeing homotopy (co)limits does not use anything besides usual (2-)category theory.
It may sometimes be useful to consider prederivators which are like derivators, but which “do not have all limits and colimits.” Let us say that a semiderivator is a prederivator satisfying (Der1), (Der2), and (Der5).
If $D$ is a semiderivator, $x\in D(B)$ is a $B$-shaped diagram, and $v\colon B\to C$ is a functor in $Dia$, then a pointwise left extension of $x$ along $v$ is an object “$v_! x$” in $D(C)$, together with a morphism $x \to v^* v_! x$ which is initial in the comma category $(x / v^*)$ (this says that $v_! x$ is a “local” or “partial” value of the left adjoint $v_!$ at $x$, although the entire functor $v_!$ may not exist) which additionally satisfies the local Beck-Chevalley condition for any comma square as above. We have a dual notion of pointwise right extension.
We say a semiderivator is complete (resp. cocomplete) if it admits all pointwise right (resp. left) extension. Clearly a semiderivator is a derivator just when it is both complete and cocomplete.
Grothendieck’s definition of a derivator included only axioms (Der1), (Der2), (Der3), and (Der4).
Alex Heller’s definition of a homotopy theory included axioms (Der1), (Der2), (Der3), a weaker form of (Der4), and (Der5) for finite free categories. His pointed homotopy theories add the axiom of a pointed derivator, and his stable homotopy theories include a weaker version of the axiom now used for a stable derivator.
Franke’s definition of a system of triangulated diagram categories was irreducibly pointed, taking $Dia$ to consist of categories enriched over pointed sets. (See pointed derivator for the relationship of this approach to that of adding a “pointedness” axiom to an unpointed notion of derivator.) In this context, he assumed axioms analogous to (Der1), (Der2), (Der3), (Der4), and (Der5) for the interval category, plus the stability axiom. He also observed that if $Dia$ consists entirely of posets, then (Der4) follows from the other axioms.
Axioms (Der1)–(Der4) are clearly the easiest to motivate and the most obviously necessary. Axiom (Der5) is used in order to do things with morphisms in $D(*)$, by first lifting them to objects of $D(I)$. In particular, it is necessary to conclude that a stable derivator gives a triangulated category.
We now describe an “omnibus” theorem which is the main thing enabling us to compute with homotopy Kan extensions in a derivator.
For any functor $u\colon I\to J$ in $Dia$, we say it is a $D$-equivalence if the induced transformation $(\pi_I)_! (\pi_I)^* \to (\pi_J)_!(\pi_J)^*$ is an isomorphism, where $\pi_I\colon I\to *$ is the projection to the terminal category, and similarly for $\pi_J$. This means that homotopy colimits of constant diagrams of shapes $I$ and $J$ are equivalent. By the Yoneda lemma, this is equivalent to saying that
is an isomorphism for all $X,Y\in D(*)$, and by adjunction this is equivalent to saying that
is an isomorphism—i.e. that $u^*$ is fully faithful when restricted to the image of $\pi_J^*$. In particular, if $u^*$ is fully faithful, then $u$ is a $D$-equivalence.
In a representable derivator (i.e. in ordinary category theory), the colimit of a constant diagram of shape $I$ is a copower with the set of connected components of $I$. Thus, in a representable derivator, any functor that induces an isomorphism on sets of connected components will be a $D$-equivalence, and the converse is true as long as the category in question is not a preorder.
By contrast, in the derivator coming from a model category or $(\infty,1)$-category, the colimit of a constant diagram of shape $I$ is a copower with the nerve of $I$ regarded as an $\infty$-groupoid, so in this case any functor that induces a homotopy equivalence of nerves (a stronger condition) will be a $D$-equivalence. In fact, one can show:
(Alex Heller, Cisinski) A functor which induces a homotopy equivalence of nerves is a $D$-equivalence for any derivator $D$.
This was proven by Heller using the canonical enrichment of any derivator over $\infty$-groupoids. It also follows from Cisinski’s theorem that the nerve equivalences are the smallest basic localizer, once we verify that the $D$-equivalences are in fact a basic localizer—as we will now proceed to do.
Any functor in $Dia$ with a fully faithful left or right adjoint is a $D$-equivalence.
If $f\dashv g$ in $Dia$, then $g^* \dashv f^*$, and if $f$ is fully faithful, then the unit $1 \to g f$ is an isomorphism, and thus so is the unit $1\to f^* g^*$. Hence $g^*$ is also fully faithful and so $g$ is a $D$-equivalence. The other case is dual.
Let $W_D$ denote the class of $D$-equivalences in $Dia$. Then $W_D$ is saturated, in the sense that if $u\colon I\to J$ is a morphism in $Dia$ which becomes an isomorphism in $Dia[W_D^{-1}]$, then $u$ is a $D$-equivalence.
Fix some $X,Y\in D(*)$ and consider the functor $\Phi\colon Dia \to Set^{op}$ which sends $I$ to $D(I)(\pi_I^*X, \pi_I^*Y)$. Since $\Phi$ inverts $D$-equivalences, it factors through $Dia[W_D^{-1}]$. But if $u$ becomes an isomorphism in $Dia[W_D^{-1}]$, then it must be inverted by $\Phi$, but that is the definition of being a $D$-equivalence (as $X$ and $Y$ vary).
For any $D$, the class of $D$-equivalences is a basic localizer.
Saturation gives 2-out-of-3 property and closure under retracts. If $A$ has a terminal object, then $A\to 1$ has a fully faithful right adjoint and hence is a $D$-equivalence. Finally, given a triangle
to show that $u$ is a $D$-equivalence, it suffices to show that the transformation $v_! \pi_A^* \to w_! \pi_B^*$ is an isomorphism. By (Der2) it suffices to check this for any $c\in C$. We can then form the comma objects
and the transformation $c^* v_! \pi_A^* \to c^* w_! \pi_B^*$ factors as
using the analogous map for the functor $v/c \to w/c$. Therefore, if $v/c \to w/c$ is a $D$-equivalence, this composite is an isomorphism, and if this holds for all $c$, then by (Der2), $u$ is a $D$-equivalence.
Therefore, since the nerve equivalences are the smallest basic localizer, every nerve equivalence is a $D$-equivalence for any derivator $D$.
Now suppose given any square
in $Dia$ which commutes up to a specified 2-cell $\mu$. Given a derivator $D$, we say that this square is $D$-exact if the Beck-Chevalley transformations
are isomorphisms. (In fact, if one of these is an isomorphism, so is the other, since they are mates.) Thus, the derivator axioms say that all comma squares are exact.
Like the notion of exact square in ordinary category theory, this is a “functional” definition, but we can also give a more explicit characterization, using more or less the same argument. Given such a square as above and $i\in I$, $j\in J$, we write $(i/L/j)$ for the category whose objects are triples
The morphisms of $(i/L/j)$ are morphisms $\gamma\colon \ell \to \ell'$ in $L$ which make the evident triangles commute. Now there is a functor $r\colon (i/L/j) \to K(u(i),v(j))$ (the latter considered as a discrete category), which sends the above triple to the composite
in $K$.
A square as above is $D$-exact if and only if for all $i\in I$ and $j\in J$, the functor $(i/L/j) \to K(u(i),v(j))$ is a $D$-equivalence.
It is easy to verify that the horizontal or vertical pasting composite of exact squares is exact; hence if the given square is exact, then so is the composite
for any $j\in J$, where the left-hand square is a comma square. Conversely, if all of these squares are exact, then so is the given one, by (Der2). We play the same game by composing with comma squares on the top to conclude that the given square is exact if and only if all the induced squares
are exact. But the square
is exact, since it is a comma square, and by the universal property of a comma square, the square (2) factors uniquely through this one by a functor $r\colon (i/L/j) \to K(u(i),v(j))$, which is precisely the functor $r$ defined above. Specifically, we have $f = s r$ and $g = t r$. Therefore, the Beck-Chevalley transformation $g_! f^* \to (v j)^* (u i)^*$ is equal to the composite
where the second map is the Beck-Chevalley transformation for the comma square, which is an isomorphism. Thus, (2) is exact just when $t_! r_! r^* s^* \to t_! s^*$ is an isomorphism. But this says exactly that $r$ is a $D$-equivalence.
A square is said to be homotopy exact if it is $D$-exact for all derivators $D$ (or, equivalently, for all $(\infty,1)$-categories, or for all model categories, or simplicially enriched categories).
A square is homotopy exact if and only if for all $i\in I$ and $j\in J$, the functor $(i/L/j) \to K(u(i),v(j))$ induces a weak homotopy equivalence of nerves.
“If” follows directly from Theorem 1 and the previous theorem. Conversely, we can take $D$ to be the derivator of spaces ($\infty$-groupoids), where the $D$-equivalences are precisely the nerve equivalences.
For each $i\in I$, $j\in J$, and $\varphi\colon u(i) \to v(j)$ in $K$, let $(i/L/j)_\varphi$ denote the subcategory of $(i/L/j)$ consisting of those triples for which the composite (1) is equal to $\varphi$. Then the square is homotopy exact if and only if each category $(i/L/j)_\varphi$ has a contractible nerve.
This gives a convenient way to compute many homotopy limits and colimits, which works in any derivator, and a fortiori in any $(\infty,1)$-category or model category. Example applications can be found at homotopy exact square.
We can consider now the 2-category of derivators. Actually, there are several different ways to define such a 2-category, depending on whether we require morphisms to preserve homotopy colimits, limits, both, or neither. Let us write $Der_!$ for the 2-category whose morphisms preserve colimits. Thus:
its objects are derivators,
its 1-morphisms are pseudonatural transformations $F:D\to D'$ which commute with the functors $u_!$ (i.e. the canonical comparison maps for these are isomorphisms),
its 2-cells are the modifications between these.
We write $Hom_!(D,D') = Der_!(D,D')$ for hom-categories in this 2-category.
Now, given a small category $X$, there is a $2$-functor, defined by evaluating at $X$
Note that, for any (pre)derivator $D$, the category $D(X)$ is canonically equivalent to the category $Hom(X,D)$ of morphisms of prederivators from $X$ to $D$ (considering $X$ as a representable prederivator). Of course, $X$ will usually not itself be a derivator, but nevertheless this $2$-functor is representable in the 2-category $Der_!$.
Thus, there exists a derivator $\widehat{X}$, endowed with a morphism of prederivators $h:X\to \widehat{X}$ (called the Yoneda embedding), such that, for any derivator $D$, composing with h defines an equivalence of categories
In other words, the map $h:X\to \widehat{X}$ is the “free completion of $X$ by homotopy colimits” in the sense of derivators.
Furthermore, $\widehat{X}$ can be described rather explicitly: it is the derivator associated to the model category of simplicial presheaves on $X$. In particular, the usual homotopy theory of simplicial sets gives rise to the derivator $\widehat{*}$, where $*$ stands for the terminal category. Note that in order to conclude this, we didn’t take for granted that homotopy types should be important: its universal property is formulated purely with ordinary category theory.
From there, you can see that any derivator is canonically enriched in the derivator $\widehat{*}\simeq Ho(SSet)$: as $*$ acts uniquely on any prederivator, $\widehat{*}\simeq Ho(SSet)$ acts uniquely on any derivator (as far as we ask compatibility with homotopy colimits). Thus the homotopy hypothesis might be reformulated vaguely as: is there an algebraic model of $\widehat{*}$? Then one may guess that some notion of higher groupoid might do the job.
Viewing a derivator as a partway point between an $(\infty,1)$-category and its homotopy category, and recalling that $(\infty,1)$-categories are often presented by categories with weak equivalences and in particular model categories, we can construct derivators in two general ways.
If $(C,W)$ is a category with weak equivalences, then the representable prederivator defined by $D_C(X) = Cat(X,C) = C^X$ comes equipped with weak equivalences $W^X$ on each category $C^X$. We define the homotopy prederivator $Ho(C)$ by inverting these weak equivalences in each diagram category:
If $C$ is a homotopical category with a functorial three-arrow calculus?, then $Ho(C)$ is a prederivator satisfying axioms (Der1), (Der2), and (Der5).
Actually, it is enough to assume that, for every small category $X$, $C^X$ admits a (not necessarily functorial) three-arrow calculus, but in practice this only happens when we have a functorial three-arrow calculus for $C$.
If $C$ is a model category (complete and cocomplete, but possibly without functorial factorisations), then $Ho(C)$ is a derivator (except possibly for axiom (Der5)).
Axioms (Der1) is easy to check, and when $C$ has functorial factorisations, we have a functorial three-arrow calculus. Axiom (Der3) is also easy, since homotopy limits and colimits in categories with weak equivalences are derived functors of the usual limits and colimits, and so they supply left and right adjoints to derived pullback functors. (This shows moreover that homotopy limits and Kan extensions in a model category coincide with the notions of homotopy limit and Kan extension in its homotopy derivator, so that by working in $Ho(C)$ we really are studying the things we want to study.) Axiom (Der4) requires a bit of work; there is a proof for combinatorial model categories using the injective model structure in (Groth).
Note that if $C$ is any complete and cocomplete 1-category, we can equip it with its trivial model structure in which the only weak equivalences are isomorphisms. Then the above derivator $Ho(C)$ is the same as the representable prederivator $D_C(X)$, which can easily be proven to be a derivator directly.
In this connection, it is interesting to point out that a given homotopy category can admit multiple “enrichments” to a derivator. For instance, the homotopy category of the model category of chain complexes over a field is equivalent to the category of graded modules over that field, which is itself complete and cocomplete. Thus we have two distinct derivators, which have equivalent underlying homotopy categories $D(*)$.
If $C$ is an (∞,1)-category, it has a homotopy category $Ho(C)$ obtained by identifying equivalent 1-morphisms. If our $(\infty,1)$-categories are modeled by quasicategories, then $Ho(-)$ is the left adjoint of the nerve, often denoted $\tau_1$.
Since categories are in particular $(\infty,1)$-categories, for any category $X$ we have a functor (∞,1)-category $C^X$, and thus a homotopy category $Ho(C^X)$. We define the homotopy prederivator of $C$ by
If $C$ has limits and colimits in the $(\infty,1)$-categorical sense, then $Ho(C)$ should be a derivator.
We discuss precise versions of the idea that derivators indeed constitute a model of (∞,1)-category theory.
The 2-category of “locally presentable” derivators (see above) is equivalent to the localization of the 2-category of combinatorial model categories at the Quillen equivalences.
This is shown in (Renaudin).
Notice that combinatorial model categories model precisely the locally presentable (∞,1)-categories, as discussed there.
Special kinds of derivators:
The calculus of homotopy Kan extensions used in derivators:
Special limits and structures in derivators:
The term derivator is originally due to Grothendieck, introduced in Pursuing Stacks . The first fifteen chapters of a 2000 page manuscript of Grothendieck (in French) about derivators can be found at:
Independently, there is a version due to Alex Heller (who called them “homotopy theories”):
Apparently also independent is the development by Franke, who takes an enriched approach to the pointed case and also assumes stability:
Georges Maltsiniotis has written an introduction to the topic (in French):
He also gave a course (in English) in Seville, Sep 2010, and part 3 is on derivators:
Part of the above material is adapted from
Cisinski has also written a number of papers on the subject (in French), which can be found at his homepage.
Derivators were also recently used by Gonçalo Tabuada in a universal characterization of higher algebraic K-theory:
An introduction to some of the theory of pointed and stable derivators, in English, can be found in the paper:
An brief informal discussion of derivators as a 2-categorical tool for studying $(\infty,1)$-categories is contained in
In the paper
it is proven that the 2-category of “locally presentable” derivators is equivalent to the localization of the 2-category of combinatorial model categories at the Quillen equivalences. Thus in some sense derivators capture “all the information” about a combinatorial model category, hence also about a locally presentable (∞,1)-category.
An introductory discussion aimed towards stable derivators is also in