model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
The model category structure on the category of categories with weak equivalences is a model for the (∞,1)-category of (∞,1)-categories.
Every category with weak equivalences $C$ presents under Dwyer-Kan simplicial localization a simplicially enriched category or alternatively under Charles Rezk‘s simplicial nerve a Segal space, both of which are incarnations of a corresponding (∞,1)-category $\mathbf{C}$ with the same objects of $C$, at least the 1-morphisms of $C$ and such that every weak equivalence in $C$ becomes a true equivalence (homotopy equivalence) in $\mathbf{C}$.
By the work of Barwick and Kan, there is a model structure on relative categories presenting the (∞,1)-category of simplicial spaces.
For the purposes of the present entry, a category with weak equivalences means the bare minimum of what may reasonably go by that name:
Definition A relative category $(C,W)$ is a category $C$ equipped with a choice of wide subcategory $W$.
A morphism in $W$ is called a weak equivalence in $C$. Notice that we do not require here that these weak equivalence satisfy 2-out-of-3, nor even that they contain all isomorphisms.
A morphism $(C_1,W_1) \to (C_2,W_2)$ of relative catgeories is a functor $C_1 \to C_2$ that preserves weak equivalences.
Write $RelCat$ for the category of relative categories and such morphisms between them.
The subdivided nerve of Barwick-Kan induces a right transferred model structure on $RelCat$ such that there is a Quillen equivalence
Both functors preserve and reflect weak equivalences, and the adjunction unit and counit are natural weak equivalences. The simplicial nerve $N$ is naturally weakly equivalent to $N_\xi$.
Furthermore, this all remains true if the Reedy model structure on $sSet^{\Delta^{\op}}$ is replaced with any of its Bousfield localizations.
The model structure on RelCat is left proper. It is also right proper when the model structure on $sSet^{\Delta^{\op}}$ is right proper.
The existence and properness of the model structure and the Quillen equivalence is Theorem 6.1 of Barwick-Kan, as is the fact $N_\xi$ preserves and reflects weak equivalences.
Proposition 10.3 shows the adjunction unit is a natural weak equivalence. That the counit is a natural weak equivalence follows by considering the triangle identity, 3-for-2, and the fact $N_\xi$ reflects weak equivalences
Finally, by considering the diagram
we see $f$ is a weak equivalence iff $N_\xi K_\xi f$ is, and iff $K_\xi f$ is.
In particular, Rezk’s complete Segal space structure on bisimplicial sets is a Bousfield localization of the Reedy model structure, so we can define:
Let $(RelCat, Rezk)$ denote the model structure corresponding to the complete Segal spaces.
The localization $L(RelCat, Rezk) \simeq (\infty,1)Cat$ is the ∞-localization functor $(C,W) \mapsto L(C, W)$ inverting the weak equivalences
We will see below that we can compute this through the map $(C,W) \to (NC, NW)$ to marked simplicial sets. Suppose we’re given a fibrant replcaement $(NC, NW) \to Y^\natural$. Since $NC$ is a quasi-category, composition induces an equivalence for every quasi-category $Z$
thus $Y$ satisfies the universal property of $L(NC, NW)$.
It is shown in Meier that categories of fibrant objects are fibrant in this model structure.
One often uses the hammock localization $L^H : RelCat \to sSetCat$, where $sSetCat$ is given the Bergner model structure whose weak equivalences are the Dwyer-Kan equivalences: i.e. the local weak homotopy equivalences.
A functor $f$ in $RelCat$ is a Rezk equivalence iff $L^H(f)$ is a Dwyer-Kan equivalence.
This is main theorem 1.4 of Barwick-Kan.
The idea underlying marked simplicial is directly analogous to the idea underlying relative categories. In fact, the functor $RelCat \to sSet^+ : (C,W) \to (NC, NW)$ preserves and reflects equivalences, since
Let $R$ be a fibrant replacement in $sSetCat$. Then the natural transformation $(NC, NW) \mapsto NRL^H(C,W)^\natural$ of marked simplicial sets is a natural weak equivalence.
This is theorem 1.2.1 of Hinich 13
The compatibility of the various nerve and simplicial localization functors is in section 1.11 of
Clark Barwick and Dan Kan,
Relative categories: another model for the homotopy theory of homotopy theories (arXiv:math/1011.1691)
A characterization of simplicial localization functors (arXiv:math/1012.1540)
In the category of relative categories the Rezk equivalences are exactly the DK-equivalences (arXiv:math/1012.1541)
A Thomason-like Quillen equivalence between quasi-categories and relative categories (arXiv:math/1101.0772)
Partial model categories and their simplicial nerves (arXiv:math/1102.2512)
Vladimir HinichDwyer-Kan Localization Revisited,
Homology, Homotopy and Applications Volume 18 (2016) Number 1 (arXiv:1311.4128, doi:10.4310/HHA.2016.v18.n1.a3)
Lennart Meier, Fibration Categories are Fibrant Relative Categories, arxiv
Last revised on June 10, 2021 at 11:35:53. See the history of this page for a list of all contributions to it.