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

A THOMASON-LIKE QUILLEN EQUIV ALENCE BETWEEN QUASI-CA TEGORIES AND RELA TIVE CA TEGORIE S C. BAR WICK AND D.M. KAN Abstract. W e describe a Quillen equiv alence bet wee n quasi-c ate gories and r elative ca te gories whic h is surprisi ngly similar to Thomason’s Quillen equiv- alences b etw een simplicial sets and c ate gories . 1. Introduction In [JT] Joyal and Tierney c o nstructed a Quillen equiv alenc e S ← → s S betw een the Joyal structure on the ca tegory S of small simplicia l sets and the Rezk structure on the ca tegory s S o f s implicia l spac e s (i.e. bi-simplicial se ts ) and in [BK] we describ ed a Q uillen equiv alence s S ← → Rel Cat betw een the Rezk str ucture on s S and the induced Rezk structure on the categor y RelCat of rela tive categor ies. In this no te we o bserve that the r esulting comp osite Quillen equiv a lence S ← → RelCat admits a description which is almost identical to that of Thomaso n’s [T] Q uillen equiv alence S ← → Cat betw een the class ical structur e on S and the induced on o n the categor y Cat of small catego r ies, as reformulated in [BK, 6.7 ]. T o do this we recall from [B K, 4.2 and 4.5 ] the notion o f 2. The two-f ol d subdivision of a rela tive poset F or every n ≥ 0, let ˇ n (resp. ˆ n ) denote the rela tive p oset which has as under lying catego ry the categor y 0 − → · · · − → n and in which the weak eq uiv alences a re only the identity maps (resp. al l maps ). Given a re la tive p oset P , its terminal (resp. ini tial ) sub di vision then is the relative p oset ξ t P (resp. ξ i P ) which has (i) as obje ct s the monomorphisms ˇ n − → P ( n ≥ 0) Date : Octob er 29, 2018. 1 2 C. BAR WICK AND D.M. KAN (ii) as maps ( x 1 : ˇ n 1 → P ) − → ( x 2 : ˇ n 2 → P ) (resp. ( x 2 : ˇ n 2 → P ) − → ( x 1 : ˇ n 1 → P )) the commutativ e diagra ms o f the for m ˇ n 1 / / x 1 A A A A A A A A ˇ n 2 x 2 ~ ~ } } } } } } } } P and (iii) as we ak e quivalenc es those of the ab ov e diagr ams in which the induced map x 1 n 1 − → x 2 n 2 (resp. x 2 0 − → x 1 0) is a weak equiv a le nce in P . The tw o-fold sub divisi on of P then is the rela tive p oset ξ P = ξ t ξ i P . 3. Conclusion In view of [JT, 4.1] a nd [BK, 5 .2 ] we now c an state: (i) the left adjoint in the ab ove c omp osite Qu il len e quivalenc e S ← → RelCat is the c olimit pr eserving functor which for every inte ger n ≥ 0 sen ds ∆[ n ] ∈ S to ξ ˇ n ∈ Re lCat and t he right adjoint sends an obje ct X ∈ R elCat to t he simplicial set which in dimension n ( n ≥ 0 ) c onsists of the maps ξ ˇ n → X ∈ Rel Cat . while, in view of the fact that Cat is cano nic a lly isomor phic to the full subc ategory d Cat ∈ RelCat spanned by the re la tive categor ies in which every map is a we ak e quivalenc e and [BK, 6.7], (ii) the left adjoint in Thomason ’s Quil len e quivalenc e S ← → d Cat is the c olimit pr eserving functor which, for every inte ger n ≥ 0 , sends ∆[ n ] ∈ S to ξ ˆ n ∈ d Cat ( n ≥ 0) while the right adjoint sends an obje ct X ∈ d Cat to the simplicia l set which in dimension n ( n ≥ 0 ) c onsists of the maps ξ ˆ n − → X ∈ d Cat QUASI-CA TEGORIES AND RELA TIVE CA TEGORIES 3 References [BK] C. Barwi c k and D. M Kan, R elative c ate gories; another mo del for the homotopy the ory of homotopy the ories, Part I: The mo del structur e , T o appear. [JT] A. Joy al and M. Tierney, Quasi-c ate gories vs Se gal sp ac es , Categories in algebra, geometry and mathematical physics, Contemp. Math., vol. 431, Amer. Math. Soc., Providen ce, RI, 2007, pp. 277–326. [T] R. W. Thomason, Cat as a close d mo del c ate g ory , Cahiers T opologie G ´ eom. Diff´ eren tiell e 21 (1980), no. 3, 305–324. Dep ar tment of Ma thema tics, Massachusetts Institute of Technology, Cambridge, MA 02139 E-mail addr ess : clarkbar @math.mit.e du Dep ar tment of Ma thema tics, Massachusetts Institute of Technology, Cambridge, MA 02139