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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 …

The claim would be true if it were the case that for any simplicial object $X_\bullet$ in some (cocomplete) category $C$, the maps $L_nX\to X_n$ from the latching objects are always monomorphisms. Th …

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 …

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 …

Just to make it clear: in this context "smash product" of $i$ and $j$ means the map $$ i\square j \colon (A\times B')\amalg_{A\times B} (A'\times B)\to A'\times B' $$ constructed from $i\colon A\to A …

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\ …

Just a remark about making the projective model structure on simplicial presheaves (on a category $C$) cartesian. In the projective model structure, representable objects are cofibrant, (and in some …

(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 …

Yes. Let $W$ be a complete Segal space, thought of as a simplicial "space" $(W_q)$. The fibrant objects of your model category will be the fibrations $f:X\to W$ such that for each simplicial operato …

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 $ …

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 …

Here's something easier. Let $C$ and $D$ be categories, and ask: is there an equivalence between $B\underline{\mathrm{Cat}}(C,D)$ (the classifying space of the functor category) and $\underline{\mathr …

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' …

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, $ …

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". …