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A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. It is not hard to show that every finite simplicial set has only a finite number of simplicies in …

Here is an extended comment regarding Charles answer, including some more references. In this paper, Theorem 2.5.9, it is shown that every model category (not necessarily a combinatorial one) has all …

It is known (follows for example from Proposition 4.2 of Simplicial Homotopy Theory by Goerss and Jardine) that a Kan-fibration can be defined as a map having the right lifting property with respect t …

The functor $F$ you are looking for can be described as follows. Given a model category $M$, with class of weak equivalences $W$, one may associate to it an $\infty$-category $M_\infty$, equipped with …

The following is taken from Section 2.3.2 of http://arxiv.org/abs/math/0207028 Let $C$ be a small simplicial category, $S\subseteq C$ a simplicial subcategory and $M$ a simplicial combinatorial mode …

Here is another family of examples of non-cofibrantly generated model categories which admit left Bousfield localization with respect to certain classes of maps. Let $C$ be a proper simplicial model c …

A finite simplicial set is a simplicial set having only a finite number of non degenerate simplicies. My question is: if $A$ and $B$ are finite simplicial sets, does this imply that the simplicial set …

A simplicial set is called finite if it has only a finite number of non degenerate simplicies (or equivalently, if its number of $n$-simplicies grows polynomially with $n$). In an answer to a previou …

If you believe Vopěnka's principle then any cofibrantly generated model category is Quillen equivalent to a combinatorial one and thus its underlying $\infty$-category is presentable. It follows that …

The following paper by Tom Leinster http://arxiv.org/abs/math/0002180 defines for any symmetric monoidal model category $M$ and any (symmetric) operad $P$ (in $Set$) the notion of an $\infty$-algebr …

There was a mistake in an earlier version of the paper that you mention. If you define $\pi_0$-fibrations and $\pi_0$-equivalences the way that you did, they do not give the structure of a category o …

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ar …

The Gelfand duality says that $$X\to C(X)$$ is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras …