Search Results

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 …

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 …

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". …

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 $ …

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\ …

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 …

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 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 …

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 …

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 $I$. …

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 …

The Kan condition isn't exactly like the sheaf condition: the Kan condition allows you to "glue" (as you put it) in certain cases, but the result is not unique. A better analogy to the Kan condition …

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 …

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 …