A Thomason-like Quillen equivalence between quasi-categories and relative categories

0.06 MB·English

by  C. Barwick

A THOMASON-LIKE QUILLEN EQUIVALENCE BETWEEN
QUASI-CATEGORIES AND RELATIVE CATEGORIES
1
C.BARWICKANDD.M.KAN
1
0
2 Abstract. We describe a Quillen equivalence between quasi-categories and
relative categories whichissurprisinglysimilartoThomason’sQuillenequiv-
n
alences between simplicial sets andcategories.
a
J

4

] 1. Introduction
T
A In [JT] Joyal and Tierney constructed a Quillen equivalence
. S ←→sS
h
t betweentheJoyalstructureonthecategoryS ofsmallsimplicialsetsandtheRezk
a
m structureonthecategorysS ofsimplicialspaces(i.e.bi-simplicialsets)andin[BK]
we described a Quillen equivalence
[
sS ←→RelCat
1
v between the Rezk structure on sS and the induced Rezk structure on the category
2 RelCat of relative categories.
7
In this note we observe that the resulting composite Quillen equivalence
7
0 S ←→RelCat
.
1 admits a description which is almost identical to that of Thomason’s [T] Quillen
0
equivalence
1 S ←→Cat
1
: between the classical structure on S and the induced on on the category Cat of
v
small categories,as reformulated in [BK, 6.7].
i
X
To do this we recall from [BK, 4.2 and 4.5] the notion of
r
a
2. The two-fold subdivision of a relative poset
For every n≥0, let
nˇ (resp. nˆ)
denote the relative poset which has as underlying category the category
0−→· · ·−→n
and in which the weak equivalences are only the identity maps (resp. all maps).
Given a relative poset P, its terminal (resp. initial) subdivision then is the
relative poset ξtP (resp. ξiP) which has
(i) as objects the monomorphisms
nˇ −→P (n≥0)
Date:January5,2011.
1
2 C. BARWICKANDD.M. KAN
(ii) as maps
(x : nˇ →P)−→(x : nˇ →P)
1 1 2 2
(resp. (x : nˇ →P)−→(x : nˇ →P))
2 2 1 1
the commutative diagrams of the form
nˇ // nˇ
1AxA1AAAAAA ~~}}}}}x}}2} 2
P
and
(iii) as weak equivalences those of the above diagrams in which the induced
map
x n −→x n (resp. x 0−→x 0)
1 1 2 2 2 1
is a weak equivalence in P.
The two-fold subdivision of P then is the relative poset
ξP =ξtξiP .
3. Conclusion
In view of [JT, 4.1] and [BK, 5.2] we now can state:
(i) the left adjoint in the above composite Quillen equivalence
S ←→RelCat
is the colimit preserving functor which for every integer n≥0 sends
∆[n]∈S to ξnˇ ∈RelCat
and the right adjoint sends an object X ∈ RelCat to the simplicial set
which in dimension n (n≥0) consists of the maps ξnˇ →X ∈RelCat.
while,inviewofthefactthatCatiscanonicallyisomorphictothefullsubcategory
d
Cat∈RelCat
spanned by the relative categories in which every map is a weak equivalence and
[BK, 6.7],
(ii) the left adjoint in Thomason’s Quillen equivalence
S ←→Cdat
is the colimit preserving functor which, for every integer n≥0, sends
∆[n]∈S to ξnˆ ∈Cdat (n≥0)
d
while the right adjoint sends an object X ∈Cat to the simplicial set which
in dimension n (n≥0) consists of the maps
ξnˆ −→X ∈Cdat
QUASI-CATEGORIES AND RELATIVE CATEGORIES 3
References
[BK] C. Barwick and D. M Kan, Relative categories; another model for the homotopy theory of
homotopy theories, Part I: The model structure,Toappear.
[JT] A.JoyalandM.Tierney,Quasi-categoriesvsSegalspaces,Categoriesinalgebra,geometry
and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providence, RI,
2007,pp.277–326.
[T] R. W. Thomason, Cat as a closed model category, Cahiers Topologie G´eom. Diff´erentielle
21(1980), no.3,305–324.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
MA02139
E-mail address: [email protected]
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
MA02139