on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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 $\mathbf{C}$.
For the purposes of the present entry, we understand under a category with weak equivalences the absolute minimum structure that may deserve to go by that name, namely a relative category:
Definition A relative category $(C,W)$ is a category $C$ equipped with a choice of wide subcategory $W$.
A morphism in $W$ are 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 model category structure on $RelCat$ is obtained from that on bisimplicial sets modelling complete Segal spaces in Theorem 6.1 of
It is shown in Meier that categories of fibrant objects are fibrant in this model structure.
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)
Lennart Meier, Fibration Categories are Fibrant Relative Categories, arxiv