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Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where …

I don't know if this does what you want, but it's one thing I know how to do. If $X\to Y$ is a map between Kan complexes, then you can build a factorization using the path space construction. Thus, $ …

In the context of Goodwillie's paper, he's got an explicit natural transformation $f:holim_I(X)\to holim_J(X|_J)$, where $X:I\to Top$ is a functor to spaces, and $J\subset I$ is a subcategory of $I$. …

I can't think of a reference for this. But here is what I would do: Given any functor $\pi\colon C\to I$ (not necessarily fibered), there's a "homotopy right Kan extension" functor $$\lim{}^\pi \co …

Quillen gives a couple of sets of sufficient conditions for $s\mathcal{C}$ to be a model category, in his Homotopical Algebra book. Notably, this includes the case when $\mathcal{C}$ is a complete an …

Starting with your setup of two related weak factorization systems, let $A$ be the class of morphisms $f$ with the following property: there exists a factorization $f=qsi$ such that: $i\in C$ and $q …

I've been thinking about this question too! It seems that showing that the Reedy and injective model structures agree for presheaves on $R$ boils down to having a well-behaved notion of "degeneracy". …

This paper proves some things about left properness for categories of simplicial algebras. The context of the paper is in terms of "algebras for a simplicial algebraic theory", which certainly includ …

(I'll assume that in a general model category $\mathcal{C}$, $\mathrm{Cyl}(X)$ really means: a factorization of $A\to X$ into a cofibration $A\to \mathrm{Cyl}(X)$ followed by a trivial fibration $\mat …

The analogs are: Serre fibrations (map with lifting property with respect to $I^n\times 0\to I^{n+1}$) trivial Serre fibrations (map with lifting property with respect to $S^{n-1}\to D^n$) retracts …

I'm not aware that the model category you want has been constructed. But it seems like an interesting question. You should ask Julie Bergner if she has thought of anything along these lines. I don' …

The unit map $T\to \mathrm{Sing}|T|$ is far from being a trivial fibration; it's actually injective. Did you mean to say it's a trivial cofibration? It is a weak equivalence for any simplicial set $ …

If you have a category $C$, then you can consider the category of functors $Func(C,Sets)$ from $C$ to sets. This comes with the Yoneda functor, $C^{op}\to Func(C,Sets)$, and is a generally useful thi …

Is this a counterexample? $R$ is the poset category $1\to 0\to 2$. Nonidentity maps in $R^+$: $0\to2$, $1\to 2$. Nonidentity maps in $R^-$: $1\to 0$. There are no other maps in $R$. The map $1\ …

I don't have a complete reference (and like Tyler, I don't know exactly what result you want). But here are some observations: there is a functor $\mathrm{Ex}^\infty$, which replaces a simplicial s …