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22
votes
2answers
940 views

Why is Voevodsky's motivic homotopy theory 'the right' approach?

Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...
3
votes
0answers
133 views

Can a functorial factorization be modified so that it fixes the initial object?

Consider a category $\mathcal C$ with a weak factorization system which is functorial. Let $*$ be an initial object. If $X\in \mathcal C$, denote by $\eta_X:*\to X$ the unique map. Using the given wfs,...
2
votes
1answer
151 views

Localization of a model category with respect to a class of maps

I am little bit lost with the following (standard?) problem in model categories. Suppose we have a Quillen adjunction between combinatorial model categories: $$L:M\leftrightarrow N: R $$ and let $(...
3
votes
1answer
113 views

Left lifting property and pushout

Let's say we are working in category $\mathcal{C}$, and that the three morphisms $ f: X \rightarrow X'$, $ g: Y \rightarrow Y'$ and $ h: Z \rightarrow Z'$ have the left lifting property with respect ...
5
votes
1answer
107 views

Inductive folk model structure on strict ω-categories

There is a paper of Lafont, Metayer, and Worytkiewicz [1] that constructs a model structure on the category of strict $\omega$-categories that they call the folk model structure. This model structure ...
7
votes
0answers
98 views

DG functors along which contractions can be lifted

For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...
1
vote
0answers
36 views

Maps having the right lifting property against cofibrations of compact spaces

I would like to know the properties of the maps that have the right lifting property against cofibrations of compact spaces. By definition, they are acyclic Serre fibrations, but I would hope to be ...
2
votes
0answers
39 views

sub relative cell complex

This is a question concerning the proposition 11.4.8 of P.Hirschhorn's book Model categories and their localizations. The proposition 11.4.8 is an analogous, in my opinion, to the well known ...
7
votes
0answers
141 views

Is $\text{DGA}^{-}$ a monoidal model category?

Let $\text{DGA}^{-}$ denote the category of non-positively graded differential graded algebras with differentials of degree $+1$. It is well-known that $\text{DGA}^{-}$ has a model structure with ...
2
votes
1answer
64 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
169 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
147 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
103 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
195 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
362 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
120 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
168 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
109 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
109 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
213 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
123 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
64 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
275 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
78 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
136 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
176 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
426 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
117 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
194 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 ...
7
votes
2answers
277 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\...
10
votes
0answers
170 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
203 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 ...
6
votes
1answer
197 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
186 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
160 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
399 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
213 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
329 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
222 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
227 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
506 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
282 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
190 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
433 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
70 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
154 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 ...