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simplicial localization of a homotopical category

Context

Homotopy theory

(,1)(\infty,1)-Category theory

Contents

Idea

We give two constructions of the simplicial localization of a homotopical category.

Preparations

We first need the following preparations:

Let T\mathbf{T} be the zigzag category. We define the categorical realization of a zigzag t=(n,t +,t )t=(n,t_+,t_-) to be the category [t][t] generated freely by the following quiver data:

  • [t][t] has n+1n+1 vertices t 0,...,t nt_0,...,t_n
  • If it +i\in t_+, there is an edge g i:t i1t ig_i:t_{i-1}\to t_i
  • If it i\in t_-, there is an edge g i:t it i1g_i:t_i\to t_{i-1}

Let CC be a homotopical category with class of weak equivalences WW. Consider the functor category Hom Cat([t],C)Hom_{Cat}([t],C). We define the category of restricted tt-zigzags C tC^{t} to be the subcategory whose objects are zigzags FF such that all maps F(t i)F(t i1)WF(t_i)\to F(t_{i-1})\in W, and where morphisms are given by object-wise weak equivalences (natural weak equivalences) such that the maps on the initial and terminal vertices are identities. Finally, we define C t(X,Y)C^t(X,Y) to be the full subcategory of C tC^t spanned by those objects FF such that F(t 0)=XF(t_0)=X and F(t n=Y)F(t_n=Y).

Let f:ttf:t\to t' be a morphism of T\mathbf{T}. Then we define the functor f *:C t(X,Y)C t(X,Y)f_*:C^t(X,Y)\to C^{t'}(X,Y) to be the functor sending a zigzag F:[t]CF:[t]\to C with connecting maps g ig_i lying between t i1t_{i-1} and t it_i (not necessarily t i1t it_{i-1}\to t_i) to the zigzag f *F:[t]Cf_*F:[t']\to C with connecting maps h jh_j defined to be the ordered composite of the maps g ig_i such that if 1(j)i\in f^{-1}(j) (Note: There should be a better combinatorial way to precisely specify this, but I’m having trouble doing so). Notice that if f 1(j)f^{-1}(j) is empty, this procedure yields the identity map. On morphisms, everything remains well-defined by the naturality condition (implying that the diagrams do in fact commute). This proves that the assignment C (t)(X,Y):TCatC^{(t)}(X,Y):\mathbf{T} \to Cat is functorial in tt.

Notice that this is not actually functorial in XX and YY, but for a map ZXZ\to X, we can give a functor C t(X,Y)C 1t(Z,Y)C^t(X,Y)\to C^{1\vee t}(Z,Y). Similarly, for a map YZY\to Z, we can give a map C t(X,Y)C t1(X,Z)C^{t}(X,Y)\to C^{t \vee 1}(X,Z). This is a special case of concatenation, which we will describe as follows:

We may concatenate two combinatorial zigzags t=(n,t +,t )t=(n,t_+,t_-) and t=(m,t +,t )t'=(m,t'_+,t'_-) in the obvious way by taking ttt\vee t' to be the combinatorial zigzag (n+m,t +n(t +),t n(t ))(n+m, t_+ \cup n(t'_+), t_- \cup n(t'_-)) where n(S):={n+i:iS}n(S):=\{n+i: i\in S\} is the map adding nn to each index. This is clearly functorial in both coordinates and defines a monoidal product :T×TT\vee:\mathbf{T}\times \mathbf{T}\to \mathbf{T}.

We will lift this to a functor XYZ t,t:C t(X,Y)×C t(Y,Z)C tt(X,Z)\vee^{t,t'}_{XYZ}:C^t(X,Y)\times C^{t'}(Y,Z)\to C^{t\vee t'}(X,Z). We send a pair of zigzags to their concatenation in the obvious way. On morphisms, we concatenate natural transformation diagrams, the result of which remains natural by the hammock-shape of our natural transformation diagrams (see below), so XYZ t,t\vee^{t,t'}_{XYZ} is a functor in both coordinates.

A 1 A 2 A 3 A n3 A n2 A n1 X Y B 1 B 2 B 3 B n3 B n2 B n1 \begin{matrix} &&A_1&\to&A_2&\leftarrow&A_3&\to&\ldots&\to &A_{n-3}&\leftarrow &A_{n-2}&\to& A_{n-1}&&\\ &\nearrow&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&\searrow&\\ X&&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&&Y\\ &\searrow&\downarrow&&\downarrow&&\downarrow&&&&\downarrow&&\downarrow&&\downarrow&\nearrow&\\ &&B_1&\to&B_2&\leftarrow&B_3&\to&\ldots&\to &B_{n-3}&\leftarrow &B_{n-2}&\to& B_{n-1}&&\end{matrix}

(Note that this picture is technically hiding the identity morphisms XXX\to X and YYY\to Y. We omit them to show the hammock-shape, as noted below).

We note that we must define everything pointwise for the moment, but it’s reasonable expect all of this data to lift to an actual composition functor in the colimit over T\mathbf{T}, perhaps after some effort.

With this data in hand, we may proceed to actual constructions of the simplicial categories that we want.

Before that, a short note on terminology: Traditionally, only the second construction is called the hammock localization, but this name comes from the shape of a natural transformation between two tt-zigzags with the same initial and terminal vertices (notice how much the figure above looks like a hammock!), but as we will see, this features in both constructions. We could say that both constructions are “hammock localizations”, and indeed, they produce weakly equivalent results.

Simplicial localization via the Grothendieck Construction

By the Grothendieck construction, we may present the functor C t(X,Y):TCatC^t(X,Y):\mathbf{T}\to Cat as a fibered category GrC T(X,Y)GrC^\mathbf{T}(X,Y) over T\mathbf{T}.

We describe the resulting category GrC T(X,Y)GrC^\mathbf{T}(X,Y) explicitly as follows:

  • The objects are pairs (t,F)(t,F) with tTt\in \mathbf{T} and FC t(X,Y)F\in C^t(X,Y).
  • The morphisms are pairs (f,g):(t,F)(t,G)(f,g):(t,F)\to (t',G) where f:ttf:t\to t' and g:f *FGg:f_*F\to G.
  • Composition is given by the rule (f,g)(f,g)=(ff,gf *g)(f',g')\circ (f,g)=(f'\circ f,g'\circ f'_* g).

In the sequel, we will construct the strict 2-category GrCGrC, called the Grothendieck enrichment with the same objects as CC and with morphism categories GrC T(X,Y)GrC^\mathbf{T}(X,Y).

The first order of business is to show that the XYZ t,t\vee^{t,t'}_{XYZ} assemble first to a functor XYZ:GrC T(X,Y)×GrC T(Y,Z)GrC T(X,Z)\vee_{XYZ}:GrC^\mathbf{T}(X,Y)\times GrC^\mathbf{T}(Y,Z)\to GrC^\mathbf{T}(X,Z). After that, we must show that all XYZ\vee_{XYZ} assemble to an actual law of composition.

We define the functor XYZ:GrC T(X,Y)×GrC T(Y,Z)GrC T(X,Z)\vee_{XYZ}:GrC^\mathbf{T}(X,Y)\times GrC^\mathbf{T}(Y,Z)\to GrC^\mathbf{T}(X,Z) as follows: On objects (suppressing subscripts), we have (t,F)(t,F):=(tt,F t,tF)(t,F)\vee (t',F'):= (t\vee t', F\vee^{t,t'} F'). This is clearly functorial by way of the results from our earlier preparations.

We now define the strict 22-category GrC by specifying the following data:

  • The objects are simply the objects of CC.
  • The hom-categories are given by GrC T(X,Y)GrC^\mathbf{T}(X,Y).
  • The identity morphism id XGrC T(X,X)id_X\in GrC^\mathbf{T}(X,X) is the unique empty zigzag.
  • We define the law of composition to be \vee.

It’s clear from the definitions that \vee is associative and that id XF=Fid_X\vee F=F whenever the composition makes sense. Then GrCGrC is a 22-category.

We define the categorical nerve: N:2-CatSet Δ-CatN:2\text{-}Cat\to Set_\Delta\text{-}Cat to be the canonical functor arising from the ordinary nerve. That is, NN is the identity on objects, and sends GrC T(X,Y)ν(GrC T(X,Y))GrC^\mathbf{T}(X,Y)\to \nu(GrC^\mathbf{T}(X,Y)) where ν\nu denotes the ordinary nerve. This is well defined since ν\nu is a right adjoint and therefore commutes with products.

We call the simplicial category N(GrC)N(GrC) the simplicial localization of CC.

Theorem: The categorical nerve N(GrC)N(GrC) sending objects to objects and hom-categories to simplicial sets ν(GrC T(X,Y))\nu(GrC^\mathbf{T}(X,Y)) equips N(GrC)N(GrC) with the structure of a simplicial category. Further, the morphisms of the homotopy category of this simplicial category coincide with those of the homotopy category arising from the ordinary construction using equivalence relations. That is, Ho C(X,Y)π 0(N(GrC(X,Y)))Ho_C(X,Y)\cong \pi_0 (N(GrC(X,Y))) naturally in XX and YY, where π 0():=Hom Ho(SSet)(Ho(Δ 0),Ho())\pi_0(-):=Hom_{Ho(SSet)}(Ho(\Delta^0), Ho(-)).

Hammock Localization

We define L HC T(X,Y)L_H C^\mathbf{T}(X,Y) to be varinjlim Tν(C T(X,Y))\varinjlim_\mathbf{T} \nu(C^\mathbf{T}(X,Y)). We define the Hammock Localization of CC to be the simplicial category L HCL_H C defined by the following data:

  • The objects of L HCL_H C are precisely the objects of CC.
  • The hom-spaces are precisely the $L_H C^\mathbf{T}(X,Y)
  • Composition is given by ν()\nu(\vee)
  • The identities Id XId_X are those maps Δ 0C T(X,X)\Delta^0\to C^\mathbf{T}(X,X) classifying the unique empty zigzag

It remains to show that the ν( XYZ t,t)\nu(\vee^{t,t'}_{XYZ}) assemble to a morphism of simplicial sets XYZ:L HC T(X,Y)×L HC T(Y,Z)L HC T(X,Z)\vee_{XYZ}:L_H C^\mathbf{T}(X,Y)\times L_H C^\mathbf{T}(Y,Z)\to L_H C^\mathbf{T}(X,Z) “in the limit”, as it were. By the universality of colimits in SSetSSet and the fact that ν\nu is a right adjoint, we have that L HC T(X,Y)×L HC T(Y,Z)L_H C^\mathbf{T}(X,Y)\times L_H C^\mathbf{T}(Y,Z)\cong \varinjlim_\mathbf{T} \nu(C^\mathbf{T}(X,Y)\times C^\mathbf{T}(Y,Z)$.

Then to prove that everything assembles correctly, we may need to use some sort of coend, but I’m not too sure.

(…)

Revised on February 13, 2011 23:22:14 by Anonymous Coward (85.244.145.154)