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2
votes
1answer
57 views

A monoidal model structure on pointed spaces

Do the classes of pointed Hurewicz cofibrations, pointed Hurewicz fibrations and pointed homotopy equivalences give a model structure on pointed (compactly generated weak Hausdorff) topological spaces ...
6
votes
1answer
137 views

About a canonical model structure on topologically enriched categories

Consider the discrete combinatorial model structure on the category of $\Delta$-generated spaces: all maps are cofibrations and fibrations and the weak equivalences are the homeomorphisms. Applying ...
6
votes
0answers
144 views

About a zig-zag of Quillen adjunctions

I have the following situation: Three combinatorial model categories $\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$ such that all objects are fibrant (so no need to bother with the fibrant ...
7
votes
0answers
80 views

Explicit data for $E_n$-monoidal model and simplicial categories

The definition of a monoidal model category requires that the tensor product is biclosed (such is needed to ensure that the tensor product is derivable in both variables). Obviously, in the situation ...
4
votes
1answer
185 views

Simplicial model categories and simplicial equivalence

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow ...
4
votes
1answer
349 views

detecting weak equivalences in a simplicial model category II

The question is related to the question: detecting weak equivalences in a simplicial model category Suppose that we have a simplicial model category $M$ and denote by $M^{f}$ the full simplicial ...
0
votes
1answer
118 views

detecting weak equivalences in a simplicial model category

Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. ...
5
votes
1answer
162 views

Localization of a model category

Let $M$ be a very nice model category (cofibrantly generated, combinatorial or cellular and left proper simplicial model category). Let $f: X\rightarrow Y$ and $g: X\rightarrow Z $ be two morphisms ...
3
votes
2answers
107 views

Excellent monoidal model categories admit enriched fibrant replacement functors?

Let $\mathbf{S}$ be an excellent model category in which all objects are cofibrant, viewed as an $\mathbf{S}$-enriched category by its canonical self-enrichment. Then we know that there is an obvious ...
2
votes
0answers
106 views

Enriched homotopy colimit and space of paths

I work in the category of $\Delta$-generated spaces. Let $G$ be the group of strictly increasing continuous bijections from $[0,1]$ to itself (they necessarily take $0$ to $0$ and $1$ to $1$). I call ...
7
votes
1answer
202 views

Proposition in HTT on cofibrations of categories

Proposition A.3.3.9. in Higher Topos Theory is as follows: Let $S$ be an excellent model category and let $f:C\rightarrow C'$ be a cofibration of small $S$-enriched categories. Then (1) for every ...
2
votes
1answer
99 views

Two model structures of some category

Let $\mathscr{C}$ be a category and $(W,C,F)=$(weak equivalence,cofibration,fibration) is some model structure of $\mathscr{C}$. Another model structure of $\mathscr{C}$, $(W_{S},C,F_{S})$ are called ...
6
votes
0answers
120 views

Product-preserving fibrant replacement functor for the Joyal model structure

There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\...
5
votes
0answers
61 views

When is a bisimplicial set diagonal fibrant

Let $sSet^2$ be the category of bisimplicial sets. In the diagonal model structure on $sSet^2$ weak equivalences are diagonal weak equivalence (i.e.$ X \rightarrow Y$ is a weak equivalence if $dX \...
3
votes
2answers
262 views

Quillen equivalence, fibrant objects

Suppose that $G:M\leftrightarrow N: U$ is a Quillen equivalence between two model categories. Suppose that $a\in M$ is a fibrant object, is it true that there exists always a fibrant object $b\in N$ ...
4
votes
0answers
77 views

Classification of combinatorial model categories presentable by simplicial presheaves on a Reedy category

Dan Dugger proved that every combinatorial model category can be obtained up to Quillen equivalence as the localization of a model structure on simplicial presheaves on a small category $C$. Is there ...
5
votes
1answer
132 views

Model category of diagrams with the colimit detecting the weak equivalences

Let $I$ be a small category and $\mathcal{K}$ be a combinatorial model category. Is it known a model category structure on the functor category $\mathcal{K}^I$ such that a map of diagrams $D\to ...
3
votes
2answers
162 views

Best notation for fibrant/cofibrant replacement

In Quillen's original text on model categories (homotopical algebra) he uses $Q$ and $R$ to denote cofibrant and fibrant replacement respectively. This notation has been used by several other ...
15
votes
1answer
409 views

Homotopy theories of operads

I know of three homotopy theories of colored operads. The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak ...
7
votes
1answer
111 views

Homotopy pushout independent of factorization and symmetric in cofibration category

$\require{AMScd}$In Algebraic homotopy, Baues defines the notion of homotopy pushout in a cofibration category in the following way: a commutative diagram \begin{CD} A @>k>> C \\ @AfAA @AAhA\...
5
votes
0answers
192 views

Flat Model Structure on $\mathbf{Ch}(\mathbf{Mod}(\mathcal{O}_X))$ computes pullback / pushforward

Given a ringed space $(X,\mathcal{O})$ of can construct the flat model structure on chain complexes of $\mathcal{O}$-modules: Weak equivalences are quasi-isomorphisms The fibrations are epimorphisms ...
6
votes
2answers
274 views

Are cofibrations accessible?

The category of fibrations in a combinatorial model category is accessible, accessibly embedded in the arrow category. How about the cofibrations? More generally, let $C$ be a locally presentable ...
3
votes
0answers
144 views

Comparing two simplicial resolution in a model category

Suppose that is a nice model category $\mathbf{M}$ (cofibrantly generated,...). Suppose we have a two diagrams $$F,G: \Delta^{op}\rightarrow \mathbf{M} $$ and a natural transformation $\nu: F\...
9
votes
0answers
166 views

Is there an “injective version” of the Bergner model structure?

The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions. ...
6
votes
1answer
192 views

Homotopy limit of model categories in the category of categories

Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or ...
5
votes
0answers
142 views

Extending a functor up to homotopy

Consider a functor between small categories $i:\mathcal{A}\to \mathcal{B}$ which is faithful, not full, and bijective on objects. Consider a functor $F:\mathcal{A}\to \mathcal{M}$ where $\mathcal{M}$ ...
3
votes
0answers
54 views

Calculating the intersection of the saturations of a decreasing sequence of morphisms

I have a sequence $W_0\supset W_1\supset \ldots$ of classes of morphisms in a presentable $(\infty,1)$-category $\mathcal{M}$. I can present this $(\infty,1)$-category as a model category, but I'm ...
5
votes
1answer
182 views

How are simplicial sets with Quillen model structure a simplicial model category?

I got very lost in checking that simplicial sets with Quillen model structure are indeed a simplicial model category. Recall that a model category $\mathcal{M}$ is simplicial if it is enriched in $\...
7
votes
1answer
176 views

Left Bousfield localization without properness, what is known?

I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those ...
5
votes
1answer
153 views

Reference request for statement on nlab: Reedy (co)fibrancy of (co)simplicial objects

On the nlab http://ncatlab.org/nlab/show/Reedy+model+structure#fibrant_and_cofibrant_objects it is claimed that a simplicial object in a model category, in which all monomorphisms are cofibrations, ...
5
votes
3answers
394 views

Transporting a model category structure along a left adjoint

There is a well-known theorem for transporting a model category structure along a left adjoint $F:\mathcal{M}\to \mathcal{N}$ which is explained here and which is due to Sjoerd Crans. The difficult ...
4
votes
1answer
206 views

Homotopy limits of Simplicial sets

Consider a small diagram $X_\bullet$ of simplicial sets. Under what sufficient conditions (apart from $X_\bullet$ being injective fibrant), is the map $\text{lim }X_\bullet \to \text{holim } X_\bullet$...
4
votes
3answers
325 views

Deriving the functor $ \int_{\Gamma} F(-,-)$

Suppose that $C$, $D$, and $E$ are combinatorial model categories, so that for any category $\Gamma$, the functor categories $C^{\Gamma}$, $D^{\Gamma}$, and $E^{\Gamma}$ have both the projective and ...
1
vote
1answer
199 views

Euclidean model structure on multipointed $d$-spaces

I use the notation of this question. A non-decreasing continuous bijection from $[0,a]$ to $[0,b]$ where $a,b\geq 0$ are two real numbers is denoted by $[0,a] \cong^+ [0,b]$. If $\phi:[0,a]\to U$ and $...
10
votes
1answer
224 views

Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent to a simplicial such?

Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent (by a zig-zag of symmetric monoidal Quillen equivalences) to a symmetric monoidal combinatorial ...
15
votes
3answers
489 views

Do combinatorial model categories and Quillen adjunctions model presentable $\infty$-categories?

Let $Q$ be the homotopical category of combinatorial model categories and left Quillen functors, with left Quillen equivalences for weak equivalences. Let $\mathbf Q$ be the corresponding $\infty$-...
13
votes
1answer
279 views

Weak complicial sets: Are the morphisms too strict?

In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified ...
5
votes
0answers
187 views

What good is a cofibration of categories?

I'm working on a problem where I have a cofibration of $V$-enriched categories $f: A\to B$, and would like to study the induced functor on presheaves $[B,V] \to [A,V]$. We can assume $V$ is a ...
13
votes
2answers
423 views

What is a model category from an $\infty$ point of view?

A number of different models for $\infty$ categories can seen to have analogs in $\infty$-category theory. For example: Quasicategories: $\Delta \subseteq \mathrm{Cat}_{(\infty, 1)}$ is (?) a dense ...
1
vote
0answers
69 views

A group object keeps so when passing to the homotopy category?

Let $\mathscr{M}$ be a model category and let $G$ be a group object in $\mathscr{M}$. Then is it also a group object in the homotopy category $\mathsf{Ho}\mathscr{M}$?
5
votes
0answers
150 views

Homotopy pullback of Quillen equivalence

Julie Bergner defined the homotopy pullback of a diagram of model categories, in Homotopy fiber products of homotopy theories, and Homotopy limits of model categories and more general homotopy ...
2
votes
1answer
105 views

A model category structure on chain complexes

The wikipedia claims that there is a model category structure of the category of arbitrary chain-complexes of R-modules which is defined by: weak equivalences are chain homotopy equivalences of chain-...
5
votes
0answers
139 views

Resolution of Simplicial Commutative Rings

I have just started learning some derived algebraic geometry. I was told that (if $ \mathrm{char}(\mathbb{K})=0 $) using commutative differential graded algebras in negative degree (for short $ \...
4
votes
0answers
89 views

Does a fibrant simplicial set give fibrant diagram

If $Y$ is a fibrant simplicial set and $\Delta^{\bullet}$ is the cosimplicial simplicial set, is $Y(\Delta^{\bullet})$ (i.e. $n^{th}$ simplicial set is $n \mapsto Y(\Delta^{n})=Hom(\Delta^{n},Y)$) a (...
10
votes
2answers
360 views

When does the forgetful functor from algebras over a monad commute with homotopy geometric realizations?

Let $\mathcal{C}$ be a combinatorial model category and $\mathrm{T}$ a monad on $\mathcal{C}.$ Assume that the model structure on $\mathcal{C}$ lifts to a model structure on the category of $\...
4
votes
0answers
89 views

Building conilpotent coalgebras from co-square-zero-extensions

Let $\mathrm{K}$ be a field of char. 0. Given a chain complex $\mathrm{X} $ over $\mathrm{K}$ denote $\mathrm{E}(\mathrm{X})$ the co-square-zero-extension on $\mathrm{X}, $ i.e. the cocommutative ...
5
votes
2answers
171 views

Combinatorial proof that some model categories are monoidal/enriched?

I'm looking for examples of proofs that some Quillen model categories are monoidal, or enriched over an other model category, which are based on explicit computation of the "pushout product" of the ...
8
votes
3answers
197 views

Looking for generalization of projective model structure

If $\mathcal{M}$ is a cofibrantly generated model category and $\mathcal{C}$ is a small category, then we can give $\mathcal{M}^\mathcal{C}$ the projective model structure, in which weak equivalences ...
4
votes
0answers
168 views

Long exact sequence of Quillen derived functors

Let $F:\mathcal A\to \mathcal B$ be an additive right exact functor between abelian categories $\mathcal A$ and $\mathcal B$. Then for a short exact sequence $$ 0\to A\to B\to C\to 0 $$ in $\mathcal A$...
5
votes
0answers
154 views

“Strict” homotopy theory of topological stacks/orbifolds

If we fix a finite group $G$, there are two different useful homotopy theories on the set of $G$-equivariant topological spaces (which are CW complexes, say). One, the "weak" homotopy theory, is given ...