We describe a Quillen equivalence between quasi-categories and relative categories which is surprisingly similar to Thomason's Quillen equivalences between simplicial sets and categories.
In [JT] Joyal and Tierney constructed a Quillen equivalence S ←→ sS between the Joyal structure on the category S of small simplicial sets and the Rezk structure on the category sS of simplicial spaces (i.e. bi-simplicial sets) and in [BK] we described a Quillen equivalence sS ←→ RelCat between the Rezk structure on sS and the induced Rezk structure on the category RelCat of relative categories.
In this note we observe that the resulting composite Quillen equivalence S ←→ RelCat admits a description which is almost identical to that of Thomason’s [T] Quillen equivalence S ←→ Cat between the classical structure on S and the induced on on the category Cat of small categories, as reformulated in [BK,6.7].
To do this we recall from [BK,4.2 and 4.5] the notion of
For every n ≥ 0, let ň (resp. n) denote the relative poset which has as underlying category the category
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 ξ t P (resp. ξ i P ) which has (i) as objects the monomorphisms ň -→ P (n ≥ 0)
Date: October 29, 2018.
(ii) as maps
the commutative diagrams of the form
(iii) as weak equivalences those of the above diagrams in which the induced map
is a weak equivalence in P .
The two-fold subdivision of P then is the relative poset ξP = ξ t ξ i P .
In view of [JT,4.1] and [BK,5.2] we now can state:
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