A derivator is pointed, or has a zero object, if each category has a zero object.
This simple definition hides a lot of structure.
If is a pointed derivator and is a full subcategory, we write for the full subcategory of on those diagrams such that is null.
Suppose is a sieve. Since is in particular fully faithful, so is by general arguments (see homotopy exact square). Moreover, since is a sieve, the square
is homotopy exact (where is the inclusion). Therefore is terminal for any , so that lands inside . Conversely, if , then the map is an isomorphism on (since is fully faithful) and on (since both are terminal there), hence is itself an isomorphism; so is in the essential image of .
The following theorem is stated as Proposition 7.4 of Heller’s paper “Stable homotopy theories and stabilization”.
Let denote the “codirected mapping cylinder” of , by which we mean the full subcategory of on the objects for and for (here is the interval category ). This is also the same as the cocomma object? . Write for the evident inclusion and projection.
Consider the adjunction
We claim that this restricts to an adjunction
where denotes the subcategory of on the objects . Certainly takes into , since is the composite . On the other hand, the square
is homotopy exact, so takes into .
Now since is the inclusion of a sieve, by the previous lemma, identifies with . Therefore, the inclusion can be identified with the restricted right adjoint , which therefore has a left adjoint . Explicitly, the left adjoint is .
The proof of coreflectivity is of course dual, using the directed mapping cylinder instead of the codirected one.
Note that if is another full inclusion and
commutes, then we have a restriction , which has a left adjoint given by the composite
where denotes the reflection. It also has a right adjoint defined dually.
Define a category with zeros to be a category equipped with a full subcategory . By the preceeding remarks, any pointed derivator gives rise to a contravariant pseudofunctor defined on the 2-category of categories with zeros (and functors preserving the specified subcategories), all of whose transition functors have both adjoints. It is natural to look for exactness conditions, analogous to (Der4) and the characterization of homotopy exact squares, which apply to these adjoints.
Since the Beck-Chevalley transformation relating composites of these adjoints is the composite of the corresponding transformation for the unpointed diagram and a Beck-Chevalley transformation relating the restriction functors to the reflections , we clearly need to study when restriction commutes with these reflections. This is the purpose of the following definition and lemma.
A functor between categories with zeros is locally null-final if for every , every , and every morphism , the category of triples such that has a contractible nerve.
is an isomorphism.
Recall that is computed as the composite , where is the codirected mapping cylinder of . We can therefore factor the above square as
It suffices, therefore, to show that the squares
are homotopy exact.
For the first square, consider first a and and a in . The category of triples which compose to is contractible, since is an initial object. Second, we should consider a and an , and a in — but by definition of , no such morphism can exist. Thus, the first square is always homotopy exact.
In fact, however, the preceeding argument is unnecessary, because we have seen that with their targets restricted as above, the functors are equivalences, and the mate of an isomorphism with respect to any pair of adjoint equivalences is again an isomorphism. Note that the fact that is an equivalence was already used in making the assertion that is a left adjoint to , and therefore the inverse of the top square already factors into the definition of the transformation appearing in the statement of the lemma. (This may reassure a reader who was worried about the fact that the canonical transformation in the first square goes the “wrong direction”.)
For the second square, consider first an and a , and a . The category we must investigate containts two types of objects:
By definition of , the morphisms between these objects are the obvious ones, except that there are no morphisms from the second type to the first. Now the full subcategory on the first type of object is coreflective, since is coreflective in . And that full subcategory is contractible, since is a terminal object. Thus, since adjunctions induce homotopy equivalences of nerves, the category in question is also contractible.
Finally, consider the case of an and a , and a . The category in question consists of triples which compose to (since there are no morphisms in from to anything in the image of ). But this is precisely the category asserted to be contractible in the assumption that is locally null-final.
If a square
of categories with zeros has the properties that
it is homotopy exact as a square of categories, when the full subcategories are ignored, and
then for any pointed derivator, the induced transformation
is an isomorphism.
By definition, the displayed functors and are obtained by applying the and of a derivator followed by the reflection into the relative diagram categories. Thus the given square factors into
in which the first square is an isomorphism by the first assumption, and the second square by the second assumption and Lemma 2.
For example, we can conclude:
If is a fully faithful and locally null-final functor between categories with zeros, then for any pointed derivator , the functor is fully faithful.
satisfies the hypotheses of the previous theorem.
Theorem 2 is not best possible, however. Its two conditions characterize when the two squares into which the Beck-Chevalley transformation factors are separately isomorphisms. However, it might happen that the composite is an isomorphism even though one or the other of the transformations is not separately an isomorphism.
In particular, the condition of “local null-finality” on the bottom morphism uses no information about the categories and . If we know some things about them, then we can correspondingly weaken this condition.
Suppose given a square
of categories with zeros, where is equipped with a full subcategory of its category of zeros. Write , and suppose the following.
The square is homotopy exact as a square of categories, when the categories of zeros are ignored.
For any pointed derivator , the functor maps into . For instance, this is the case if for each , the category has a terminal object lying in .
For each and and morphism in , the following category has a contractible nerve: its objects are triples such that , and its morphisms are morphisms in commuting with the given morphisms and such that either or .
For every and and in , the category of triples such that has a contractible nerve (“relative local null-finality”).
Then for any pointed derivator, the induced transformation
is an isomorphism.
(Sketch) Because of the second condition in the theorem, instead of the reflection it will suffice to consider the reflection . We use the third condition to give an alternate way to compute this reflection, and the fourth condition to ensure that restriction along commutes with this reflection.
Let be the codirected mapping cylinder of . If are as before, then identifies with the subcategory of consisting of diagrams which are zero on the copy of that is the target end of the mapping cylinder and on the copy of inside the copy of that is the source end; call this subcategory .
Now if the adjunction restricts to an adjunction , we will be able to compute the reflection as , as before. This will follow if the following square is exact:
where denotes the same full subcategory of as above. Exactness of this square can be verified to be equivalent to the third condition in the theorem.
Finally, modifying the proof of Lemma 2, we see that the fourth condition in the theorem implies commutativity of with this reflection . Therefore, the theorem follows as before.
If , then the second and third conditions of Theorem 3 are vacuous and it reduces to Theorem 2. At the other extreme, if , then the second and fourth conditions of Theorem 3 are vacuous and the third becomes ” maps into .”
A derivator is pointed if and only if whenever is a sieve in , the functor has a right adjoint , and dually whenever is a cosieve, the functor has a left adjoint .
The “if” direction of this is easy, since is always a sieve, while assigns the terminal object of . Thus, if has a further right adjoint, it must preserve colimits, and in particular its value on the unique object of must be an initial object (as well as a terminal one).
The functors and are sometimes referred to as extraordinary and co-extraordinary inverse image functors. Our proof shows that , where and are respectively the inclusion of in the codirected mapping cylinder of , and its projection to .
In the literature, the existence of these functors is often taken as the definition of when a derivator is pointed. The equivalence of the two definitions is a “super-difficult” excercise in Maltsionitis’ notes.
In a pointed derivator, we have a suspension functor defined as the composite
where denotes the category
and its full subcategory on .
Equivalently, the suspension can be defined as the composite
This follows because we have , and thus is isomorphic to , which by definition is . But is fully faithful since is, and so it already takes into . Thus is doing nothing, so is essentially just which appeared in our previous definition of . (The other functor essentially implements the equivalence .)
Similarly, we have a loop space object functor defined as the composite
where is the full subcategory of on , and it can equivalently be given as the composite
The description in terms of relative diagram categories makes it clear that .
We can also describe and in terms of the extraordinary inverse image functors. Since the square
is homotopy exact, the composite in our first definition is the same as . The other functor can in turn be decomposed as a composite , where is the inclusion of into the interval category , and is the inclusion into .
Therefore we have . But can be identified with and with the inclusion , while factors as so that . But finally, the square
is homotopy exact, so and thus
Dually, we can identify , from which it again follows directly that .
from small -enriched categories to large ones, satisfying analogues of the derivator axioms. Specifically, if is a -category, define to be with a zero object adjoined. (Note that zero objects are absolute colimits in -categories.) Then set , the “relative diagram category” as defined above, i.e. the full subcategory of on those diagrams which send the zero object to zero.
Any -functor automatically preserves zero objects (since an object of a -category is zero iff its identity is the basepoint of the pointed homset ). In particular, any -functor induces a relative functor , and hence a pullback . According to the general theory of relative diagram categories, this functor has left and right adjoints and , obtained from left and right extension along followed by reflection or coreflection from into . In fact, however, in this case the reflection/coreflection is unnecessary: since the comma category has a terminal object , the functor already maps into .
Thus we have a -analogue of the derivator axiom (Der3), existence of adjoints. For the enriched axiom (Der2), conservativity, observe that if is the unit -category having one object with , then . So the family of functors for any -category gives rise a family picking out the nonzero objects, but since any two zero objects are isomorphic, (Der2) for implies that the resulting family of functors are jointly conservative.
We can also conclude a version of the axiom (Der4) about exact squares from the above theorems about exactness for categories with zeros. Namely, if
is a comma square of -categories, then
is homotopy exact when considered as a square of ordinary categories. Moreover, the functor is always locally null-final. Thus, by Theorem 2, our ”-enriched derivator” satisfies the Beck-Chevalley condition for any comma square of -categories.
It remains to consider the axiom (Der1) regarding coproducts. The coproduct ”” of two -categories and , as -categories, is not quite a “disjoint” union, but rather includes zero morphisms in both directions from each category to the other. The inclusions of and are fully faithful, however, and the induced functors and are locally null-final; hence the left extensions and are also fully faithful.
By exactness with zeros, we can show that and send everything to zero. Therefore, if we define a functor by , then we have . On the other hand, by fully-faithfulness of and , for any the map is an isomorphism on objects of , and similarly for and ; hence as well. In particular, is an equivalence of categories, providing the -enriched version of (Der1).
What about (Der5)?
Conversely, given a functor as above satisfying the axioms as given above, we can define , where denotes the free -category on an ordinary category (i.e. adjoin a new zero morphism between each pair of objects). This will automatically satisfy axioms (Der3) (homotopy Kan extensions) and (Der2) (conservativity on objects) since does, and it satisfies (Der1) on coproducts since does and since .
Does (Der4) on Beck-Chevalley conditions carry back over? Does this require a “pointed version of Cisinski’s theorem”?
Finally, of course each category has a zero object, so is a pointed derivator. Thus, it seems that -enrichment is an essentially equivalent way to express the notion of pointed derivator. (This approach was used by Franke (see below).)
However, what is not immediately clear is that passing from to and back to again gives the same derivator. At first sight this seems to require a stronger theorem about homotopy exactness with zeros than is stated above, one in which the square need not be homotopy exact without zeros.
Every derivator can be made pointed in a universal way; given we define to be the full subcategory of which is terminal when restricted along . It requires a little work to show that this is a derivator. The main observation being that the inclusion has a left adjoint (the “mapping cone”), which can be constructed just as in the proof of Theorem 1. See III.5 of Heller’s memoir.
See derivator for general references. The pointed reflection is discussed in III.5 of
The definition of relative diagram categories is taken from:
The -enriched approach is taken as basic in