Questions tagged [model-categories]
The model-categories tag has no usage guidance.
11
votes
1answer
187 views
Which maps of simplicial sets geometrically realize to fibrations?
If $f:X\to Y$ is a Kan fibration of simplicial sets, then its geometric realization $|f| : |X|\to |Y|$ is (in some suitable convenient category of topological spaces, like compactly generated ones) a ...
5
votes
1answer
127 views
Quillen equivalent module categories
Let $f:A \rightarrow B$ be a weak equivalence of simplicial commutative rings. There is a Quillen pair $(-\otimes_{A}B, f_{\ast})$ which is an equivalence. In this situation, $(-\otimes_{A}B, f_{\ast})...
0
votes
1answer
142 views
A question about combinatorial model categories
I am currently reading the appendices of Higher Topos Theory, and I was puzzled by Lurie's proof of lemma A.2.6.7 (I can not make sense of the end of the proof.)
He uses this result to prove Jeff ...
5
votes
2answers
220 views
Limit of weak equivalences in a Bousfield localization
Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}\to X_n), (q_{n+1}: Y_{n+1}...
6
votes
1answer
207 views
Does the Dwyer-Kan model structure make dgCat a model $2$-category?
Let dgCat be the category of small dg-categories. The well-known Dwyer-Kan model structure makes dgCat a model category.
Now we consider dgCat as a 2-category, which objects small dg-categories, $1$-...
4
votes
0answers
169 views
$\mathrm{HH}$ and $\mathrm{HC}$ as two different Taylor expansions at the same point
Let us consider the functors from dg algebras to graded spaces given by Hochshcild homology and cyclic homology. Then it is well known that in degree $0$, both functors are given by $A\longmapsto A/[A,...
3
votes
0answers
133 views
filetered colimit of fibrant-cofibrant objects
Suppose that we have a $\lambda$-combinatorial model category $M$ (for some cardinal $\lambda$) such that any $\lambda$-filtered colimit of fibrant-cofibrant objects is always fibrant. My question is ...
6
votes
2answers
608 views
Theorem 2.1.2.2 Higher Topos Theory
At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
4
votes
0answers
267 views
Quillen pair, fibrant-cofibrant objects
This question is a follow-up of the question I have asked today : left quillen functor and fibrant objects
Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that ...
1
vote
1answer
124 views
left quillen functor and fibrant objects
Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that both model categories are combinatorial model categories.
Suppose that the functor $L$ (left adjoint) takes ...
9
votes
0answers
363 views
Is the Thomason model structure on Cat simplicial? Is it a monoidal model category?
The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e....
4
votes
1answer
107 views
Ore localization and model structures
The question is this:
Suppose C is a category, with a given multiplicatively closed set of morphisms S ⊆ C. The role of the denominator conditions on S is rather similar to the role of a Quillen ...
3
votes
1answer
173 views
derived functor that preserves weak equivalences
Suppose we have a functor $F:A\rightarrow B$ between model categories.
1- Assume that F takes weak equivalences to weak equivalences and cofibrations to cofibrations, can we define the derived ...
32
votes
2answers
3k 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
181 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
160 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
131 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
115 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
108 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
40 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
41 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
144 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
66 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
172 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
154 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
104 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
197 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
366 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
121 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
169 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
114 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
110 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
220 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
101 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
124 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
67 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
291 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
82 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
183 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
440 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
121 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\...
6
votes
0answers
197 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
279 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
145 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
176 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
208 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
208 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 $\...
