nLab
pullback in a derivator

Pullbacks in derivators

Idea

A pullback in a derivator is the generalization to the context of a derivator of the notion of pullback in ordinary category theory. Viewing a derivator as the “shadow” of an (∞,1)-category, the notion of pullback therein coincides with the notion of homotopy pullback in an (,1)(\infty,1)-category.

Definition

Let \square denote the category

a b c d \array{a & \to & b \\ \downarrow & & \downarrow \\ c & \to & d}

that is the “free-living commutative square”, and let LL be the full subcategory of \square on b,c,db,c,d, with inclusion u:Lu\colon L\to \square.

Let DD be a derivator and let XD()X\in D(\square) be a commutative square in DD. We say that XX is a pullback square, or is cartesian, if the unit Xu *u *XX\to u_* u^* X of the adjunction u *u *u^*\dashv u_* is an isomorphism. Since uu is fully faithful, so is u *u_*, so this is equivalent to saying that there exists some YD(L)Y\in D(L) such that Xu *YX\cong u_* Y.

If DD is merely a prederivator, then we can phrase the same definition by saying that XX has the universal property that u *u *Xu_* u^* X would, if the whole functor u *u_* existed.

The dual notion, of course, is a pushout or cocartesian square.

Properties

Pasting law

Using properties of homotopy exact squares, we can prove the “pasting law” for pullback squares in a derivator:

Lemma

Given a diagram

a b c d e f \array{a & \to & b & \to & c \\ \downarrow & & \downarrow & & \downarrow \\ d & \to & e & \to & f }

in which the right-hand square bcefbcef is a pullback, then the left-hand square abdeabde is a pullback if and only if the outer rectangle acdfacdf is a pullback.

The following proof should be compared and contrasted with the standard proof for pullbacks in 1-categories, and the quasi-categorical proof for pullbacks in (,1)(\infty,1)-categories. In particular, note that the statement for derivators is a generalization of both, since both 1-categories and (,1)(\infty,1)-categories give rise to derivators.

Proof

First of all, by “a diagram” in a derivator, we mean an object of D(X)D(X) for some suitable category XX. In the above case, XX is the category consisting of two commutative squares, as pictured above. We’ll write abcdefabcdef for this XX, and similarly cdefcdef for its lower-right L-shaped subcategory, and so on. We leave the verification of homotopy exactness of all squares to the reader.

Firstly, since the squares

cef bcef cdef abcdefandcdf acdf cdef abcdef\array{cef & \overset{}{\to} & bcef\\ \downarrow && \downarrow\\ cdef & \underset{}{\to} & abcdef} \qquad and \qquad \array{cdf & \overset{}{\to} & acdf\\ \downarrow && \downarrow\\ cdef& \underset{}{\to} & abcdef}

are exact, if we start from a cdefcdef-diagram and right Kan extend it to a full abcdefabcdef-diagram, then the right-hand square and outer rectangle must be pullback squares. Moreover, by composition of adjoints, right Kan extension from cdefcdef to abcdefabcdef is equivalent to first extending to bcdefbcdef and then to abcdefabcdef, and since the square

bde abde bcdef abcdef\array{bde & \overset{}{\to} & abde\\ \downarrow && \downarrow\\ bcdef & \underset{}{\to} & abcdef}

is exact, the left-hand square in such an extension must also be a pullback.

Now if we start with an abcdefabcdef-diagram, say FF, we can restrict it to a cdefcdef-diagram and then right Kan-extend it to a new abcdefabcdef-diagram. If u:cdefabcdefu\colon cdef \to abcdef is the inclusion, then this results in u *u *Fu_\ast u^\ast F, and we have a canonical natural transformation η:Fu *u *F\eta \colon F \to u_\ast u^\ast F (the unit of the adjunction u *u *u^\ast \dashv u_\ast). Since the square

cdef cdef cdef abcdef\array{cdef & \overset{}{\to} & cdef \\ \downarrow && \downarrow\\ cdef & \underset{}{\to} & abcdef}

is exact, the counit u *u *GGu^\ast u_\ast G \to G is an isomorphism for any GG, and in particular for G=u *FG=u^\ast F, from which it follows by the triangle identities that u *Fu *u *u *Fu^\ast F \to u^\ast u_\ast u^\ast F is also an isomorphism — i.e. the components of η:Fu *u *F\eta\colon F \to u_\ast u^\ast F at cc, dd, ee, and ff are isomorphisms. Now if the right-hand square of FF is a pullback, then the restrictions of FF and u *u *Fu_\ast u^\ast F to bcefbcef are both pullback squares; hence since the cefcef-components of η\eta are isomorphisms, so is the bb-component. And if the left-hand square of FF is a pullback, then we can play the same game with abdeabde to show that the aa-component of η\eta is an isomorphism, while if the outer rectangle is a pullback, we can play it with acdfacdf. Hence in both of these cases, η\eta itself is an isomorphism, since all of its components are — and thus the remaining square in FF is also a pullback, since we have shown that it is so in u *u *Fu_\ast u^\ast F.

Detection Lemma

The following lemma, which detects when squares occurring in a Kan extension are pullbacks or pushouts, is due to Jens Franke; see also Groth. We state it in terms of pushouts.

Lemma

Let f:KJf\colon K\to J be any functor and let i:Ji\colon \square \to J be injective on objects, with lower vertex i(1,1)=zi(1,1) = z. Suppose that zz is not in the image of ff, and that the induced functor Γ(Jz)/z\Gamma \to (J \setminus z)/z is a nerve equivalence (such as if it has an adjoint). Then for any derivator DD and any YD(K)Y\in D(K), the square i *f !Yi^* f_! Y is cocartesian.

Proof

Since f !=j !f¯ !f_! = j_! \bar{f}_! where f¯:K(Jz)\bar{f}\colon K\to (J\setminus z) is induced by ff and j:(Jz)Jj\colon (J\setminus z) \to J is the inclusion, it suffices to suppose that K=(Jz)K = (J\setminus z). Now what we want is to prove that the following square is homotopy exact:

Γ (Jz) J \array{ \Gamma &\to & (J\setminus z) \\ \downarrow && \downarrow\\ \square &\to & J}

Exactness is trivial at all objects of \square except (1,1)(1,1). In that case, we paste with another square:

Γ Γ (Jz) * (1,1) J \array{ \Gamma &\to& \Gamma &\to & (J\setminus z) \\ \downarrow & \swArrow & \downarrow && \downarrow\\ \ast &\underset{(1,1)}{\to} &\square &\to & J}

The left-hand square is a comma square, hence homotopy exact, so it suffices to show that the composite square is homotopy exact. But the comma object associated to the cospan *J(Jz)\ast \to J \leftarrow (J\setminus z) is (Jz)/z(J \setminus z)/z, and of course this comma square is also exact. And the composite square factors through this comma square by the functor Γ(Jz)/z\Gamma \to (J \setminus z)/z which is assumed a nerve equivalence; hence it is also homotopy exact.

References

See all references at derivator. Referred to particularly above are:

  • Jens Franke, Uniqueness theorems for certain triangulated categories with an Adams spectral sequence, K-theory archive
  • Moritz Groth, Derivators, pointed derivators, and stable derivators pdf

Revised on June 8, 2011 04:33:22 by Mike Shulman (71.136.238.9)