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homotopy exact square

Homotopy exact squares

Idea

A homotopy exact square is the analogue of an exact square which applies to homotopy Kan extensions, or equivalently to (∞,1)-Kan extensions. It is especially important in the theory of derivators, which provide a calculus for computing with homotopy Kan extensions whose primary tool is the use of homotopy exact squares.

Definition

Let AA, BB, CC, and DD be small categories, and consider a square of functors

A f B u v C g D\array{A & \overset{f}{\to} & B\\ ^u\downarrow & \swArrow & \downarrow^v\\ C& \underset{g}{\to} & D}

which is inhabited by a natural transformation (which might be an identity?). Let MM be either

We write M AM^A, M BM^B, etc. for the model categories, simplicial categories, (,1)(\infty,1)-categories, or homotopy categories of diagrams in MM of whatever shape. We write f *:M BB Af^*\colon M^B\to B^A, g *:M DM Cg^*\colon M^D\to M^C, and so on for precomposition functors, which are always homotopically meaningful, and we write u !:M AM Cu_!\colon M^A\to M^C, v !:M BM Dv_!\colon M^B\to M^D and so on for the homotopically meaningful notions of pointwise left Kan extension. Specifically:

  • If MM is a model category, then u !u_! denotes the left derived functor of pointwise Kan extension along uu.
  • If MM is a simplicially enriched category, then u !u_! is the coherent pointwise homotopy left Kan extension along uu, which may be defined explicitly in various ways, such as using a bar construction.
  • If MM is an (,1)(\infty,1)-category, then u !u_! denotes the pointwise (∞,1)-Kan extensions along uu.
  • If MM is a derivator, then u !u_! simply denotes the left adjoint of u *u^* (which is assumed to exist and to “be pointwise” by the derivator axioms).

Assume that MM is such that the relevant extensions u !u_! and v !v_! exist. Then there is a canonical Beck-Chevalley transformation

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

defined as the composite

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

and we say that the given square is MM-exact if this transformation is an equivalence. If the square is MM-exact for all MM, we say it is homotopy exact. Note that by the general calculus of mates, this is equivalent to requiring that the dual transformation

v *g *f *u * v^* g_* \to f_* u^*

is an equivalence, where f *f_* and g *g_* denote the analogous sort of right Kan extension.

Of course when we say “for all MM” we need to specify what sorts of MM we consider. However, we actually get the same definition regardless of whether we mean “for all model categories MM” or “for all simplicially enriched categories MM” or “for all (,1)(\infty,1)-categories MM” or “for all derivators MM”. This is a nontrivial theorem, especially in the case of derivators.

Remarks

Since any 1-category is a degenerate sort of (,1)(\infty,1)-category, any homotopy exact square is exact in the usual 1-categorical sense, but the converse is not true. This also implies that a square can be MM-exact for some particular MM without being homotopy exact. However, there exists a “universal” MM such that MM-exactness is equivalent to homotopy exactness, namely M=GpdM=\infty Gpd.

Characterization

Of course, the above definition is “functional”, while in practice we want some more combinatorial characterization which is easier to check. This can be done completely analogously to the characterization of ordinary exact squares using comma objects, except that at the last step we need to consider a more restricted notion of “equivalence” (i.e. a more restricted basic localizer).

The characterization is the following. Given bBb\in B and cCc\in C and φ:v(b)g(c)\varphi\colon v(b) \to g(c), let (b/A/c) φ(b/A/c)_\varphi denote the category whose objects are triples (a,α,β)(a,\alpha,\beta) with α:u(a)c\alpha\colon u(a)\to c and β:bf(a)\beta\colon b\to f(a) such that g(α)v(β)=φg(\alpha) \circ v(\beta) = \varphi, and whose morphisms are morphisms aaa\to a' making two triangles commute.

Theorem

A square is homotopy exact if and only if each category (b/A/c) φ(b/A/c)_\varphi has a contractible nerve.

Examples

Comma squares

Any comma square is homotopy exact. In other words, if A=(v/g)A=(v/g) is the comma category with ff and uu the canonical projections, then the square is homotopy exact. If in addition C=*C=* is the terminal category, then gg just picks out an object dDd\in D and AA is the comma category (v/d)(v/d); thus this says that (pointwise) homotopy Kan extensions can be computed pointwise as homotopy limits over such comma categories.

Fully faithful functors

If u:ABu\colon A\to B is a fully faithful functor, then the square

A id A id u A u B\array{A & \overset{id}{\to} & A\\ ^{id}\downarrow && \downarrow^u\\ A & \underset{u}{\to} & B}

is homotopy exact. This just says that the unit Id Au *u !Id_A \to u^* u_! is an isomorphism, i.e. that left (and equivalently right) homotopy Kan extensions along uu are “honest” extensions.

Final functors

A functor u:ABu\colon A\to B is a homotopy final functor if and only if the square

A u B * *\array{A & \overset{u}{\to} & B\\ \downarrow && \downarrow\\ *& \underset{}{\to} & *}

is homotopy exact. Homotopy exactness of this square says that for F:BMF\colon B\to M, the canonical map hocolim Au *Fhocolim BFhocolim_A u^*F \to hocolim_B F is an isomorphism, which is one equivalent definition of when uu is homotopy final. In this case, the characterization theorem reduces to saying that uu is homotopy final if and only if each comma category b/ub/u has a contractible nerve, which is a known characterization of homotopy final functors.

Pullbacks of fibrations

This example is due to Moritz Groth. Let p:CDp\colon C\to D be a Grothendieck opfibration, and suppose that

A u C v p B q D\array{ A & \xrightarrow{u} & C \\ ^v\downarrow && \downarrow^p\\ B & \xrightarrow{q} & D }

is a pullback in Cat. Then we claim that this square is homotopy exact.

For a proof, suppose given bBb\in B and cCc\in C and a morphism ϕ:p(c)q(b)\phi\colon p(c) \to q(b), and let XX be the category whose contractibility we must check. By definition, since A=B× DCA = B\times_D C, an object of XX consists of objects cCc'\in C and bBb'\in B such that p(c)=q(b)p(c') = q(b'), and morphisms α:cc\alpha\colon c\to c' and β:bb\beta\colon b' \to b such that q(β).p(α)=ϕq(\beta) . p(\alpha) = \phi.

Let YY be the full subcategory of XX consisting of those objects with b=bb'=b and β\beta the identity, so that an object of YY is an object cCc'\in C such that p(c)=q(b)p(c') = q(b) together with a morphism α:cc\alpha\colon c\to c' such that p(α)=ϕp(\alpha) = \phi. Then the inclusion YXY\hookrightarrow X has a left adjoint, which sends (b,c,α,β)(b',c',\alpha,\beta) to (b,q(β) !(c),q(β)¯.α,1 b)(b, q(\beta)_!(c'), \overline{q(\beta)}.\alpha, 1_b), where q(β)¯:cq(β) !(c)\overline{q(\beta)}\colon c' \to q(\beta)_!(c') is an opcartesian arrow over q(β)q(\beta). Therefore, the nerves of YY and XX are homotopy equivalent.

But YY has an initial object, namely (ϕ !(c),ϕ¯)(\phi_!(c), \overline{\phi}), where ϕ¯:cϕ !(c)\overline{\phi}\colon c \to \phi_!(c) is an opcartesian arrow over ϕ\phi. Thus the nerve of YY is contractible, and thus so is the nerve of XX.

Groth has also proved that homotopy-exactness of such squares can be used to replace that of comma squares in the definition of a derivator.

References

Revised on June 8, 2011 04:34:34 by Mike Shulman (71.136.238.9)