Questions tagged [model-categories]
A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
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Existence of homotopy limits and colimits in model categories
I am not an expert, thus I apologize if my question is very naive. Let $\mathsf{M}$ be a model category (I do not assume any functoriality on the factorization),
Q1. Is there a reference where it is ...
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Direct proof that the model category of cdgas is left proper
Let $k$ be a field of characteristic $0$. The projective model structure on the category $cdga$ of commutative differential graded $k$-algebras is proper. Since this model structure is transferred ...
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Maximal Cisinski model structure on simplicial sets
This is a very simple question coming from the observation that every (pre)sheaf category has the maximal Cisinski model structure on it. This is the Cisinski model structure with the smallest class ...
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Homotopy function complex for quasi-categories
The model structure on sSet for quasi-categories (the Joyal model structure) is not enriched over sSet with the Quillen model structure, so ordinary internal Hom of simplicial sets is not a correct ...
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On the difference between a projective chain complex and a level-wise projective chain complex
Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it ...
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An explicit description of Lawvere's segment in the category of simplicial sets
In any presheaf topos, there exists an object called Lawvere's segment, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h_a: x\in ...
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Trees in chain complexes
$\DeclareMathOperator{\Ch}{\mathit{Ch}}$Let $\Ch_\mathbb{Q}$ denote the model category of chain complexes over rational numbers. Let $T_\ast$ be a tree in $\Ch_{\mathbb{Q}}$ with $n$ vertices.
How to ...
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Is the Thomason model structure the optimal realization of Grothendieck's vision?
In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy ...
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Model categories as a tool to resolve size issues for localizing categories
I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
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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....
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When is the model structure on functors correct, i.e. when does localization commute with taking functor categories?
Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with ...
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Model category structure on spectra
I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.
Let $S$ be a finite dimensional Noetherian scheme ...
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Fiber vs homotopy fiber in model categories: simple question
I have a concrete problem with the homotopy fiber and I am getting lost with the
literature. I state my question and, to avoid confusions, I state downwards the
definitions I am using.
Let $C$ be a ...
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Localizations of model categories and $\infty$-categories
I am interested in the relation between Bousfield localizations of model categories and localizations of $(\infty,1)$-categories.
According to Hirschhorn's book we can form the left Bousfield ...
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Is it possible to compute André-Quillen cohomology by resolving the module variable?
Let $A\to B$ be a morphism of commutative rings. Let $\mathcal C$ be the category of commutative $A$-algebras augmented over $B$. Let $\mathcal M_B$ denote the category of $B$-modules. The cotangent ...
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Explicit generating acyclic cofibrations and right properness of a model category
Let $\mathcal{C}$ be a cofibrantly-generated model category. My impression is that the following two conditions are highly correlated:
$\mathcal{C}$ is right proper.
There is an explicitly-...
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Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes
It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the Dold-...
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The Quillen model structure on simplicial sets as a Bousfield localization
Starting with the trivial model structure on the category of simplicial sets (that is the weak equivalences are exactly the isomorphisms and the cofibrations and fibrations are arbitrary maps), is it ...
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A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
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Fibration of Batanin/Leinster $\omega$-groupoids
Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster?
I tried to search on Google and in Higher Operads, Higher Categories of Tom ...
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Finite homotopy limits commute with sequential homotopy colimits
I would like to know for what kind of model category finite homotopy limits commute with sequential homotopy colimits. Would cofibrantly generated and finitely locally presentable be enough? It seems ...
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Is there a cheap proof that (homotopy) endomorphisms are functorial?
This is, in some sense, the homotopy version of this question.)
If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of ...
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Is the canonical model structure on strict $\infty$-Cat left proper?
Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in A folk model structure on omega-cat left proper ?
All ...
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Inducing a model structure using a cosimplicial object
In a recent paper, Hiroshi Kihara induced a model structure on the category of diffeological spaces. He generates the classes of fibrations, cofibrations, and weak equivalences by constructing a ...
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Almost transferred model structures
Let $F : \mathcal{C} \leftrightarrows \mathcal{D} : U$ be a Quillen adjunction between cofibrantly generated model categories. The model structure on $\mathcal{D}$ is called transferred if $U$ ...
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Reference request - CDGA vs. sAlg in char. 0
Hello,
Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or ...
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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 ...
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Internal hom in $(\infty,2)$-categories
Let $X,Y$ be two $(\infty,2)$-categories, viewed as two fibrant objects in $\mathrm{Fun}(\Delta^{op},\mathrm{Set}_\Delta)$ with the complete Segal model structure (one uses the Joyal model structure ...
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Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?
Consider the category C of pointed simplicial sets.
The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C
models the suspension and loop functors on the underlying ∞-category of C.
There is another ...
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When does a cosimplicial object compute homotopy colimits?
Suppose I want to compute the homotopy colimit of a diagram of spaces. There is a simple way of getting a simplicial space from this diagram, and a theorem tells me that taking the geometric ...
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Projective/injective object in functor category
Let $\mathcal{C}$ denote the functor category $Fun(\textbf{Man} , \textbf{Ab})$, where $\textbf{Man}$ and $\textbf{Ab}$ denote the category of smooth manifolds and abelian groups respectively. I want ...
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On the natural (bigraded) homotopy groups of a simplicial object in a model category
$\def\mc{\mathcal} \def\sm{\wedge}$
This question stems from the Goerss-Hopkins paper Moduli Problems for Structured Ring Spectra. Let me begin by attempting to summarize the relevant framework -- ...
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[Reference Request] The Definition of Adjoint Functors between dg-categories
Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair?
In my mind $F\dashv G$ ...
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Is hammock localization a localization in the sense of Lurie?
In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant ...
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Cofibrations in the model structures for non-negative graded (commutative) DG algebras
Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}^{+}$ denote the category of non-negative graded DG algebras and $\mathtt{CDGA}_{k}^{+}$ denote the category of non-negative graded ...
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Cofibrant replacements of a given object in a combinatorial model category
In a combinatorial model category, every $\lambda$-filtered colimit is a homotopy colimit for $\lambda$ regular big enough. So for $\lambda$ regular big enough, every $\lambda$-filtered colimit of a ...
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Reference request for equivalences between different models of lax limits
There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...
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Cofibrancy of a right module over an operad
If I have a right module $M$ over an operad $\mathscr{O}$ in spaces, are there general methods to determine if $M$ is cofibrant with respect to the Reedy model structure? What if I know that my module ...
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$k$-linear $\infty$ stable categories and dg categories
This question is related to this question, where I asked about the relation between the derived category of a fiber product $Y \times_Z W$ and the push out of the diagram of derived categories one ...
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The model category structure on $\mathbf{TMon}$
I ask this question here since I asked it here on Math.SE, and got no answers after a week of a bounty offer.
I am trying to understand the homotopy colimit of a diagram of topological monoids, and ...
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Homotopy pullbacks and pushouts in stable model categories
There are lots of similar questions that have been answered on this topic (particularly Homotopy limit-colimit diagrams in stable model categories), but I have a specific question that I do not ...
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When does the constant diagram functor preserve fibrant objects in the injective model structure on diagram categories?
When does the constant diagram functor preserve fibrant objects in the injective model structure on diagram categories?
For example, this is the case when the index category of the diagrams is a ...
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Does the Monoid Axiom hold for k-spaces?
In “Algebras and Modules in Monoidal Model Categories” Schwede and Shipley introduced the monoid axiom. If a cofibrantly generated monoidal model category $M$ satisfies this axiom and some smallness ...
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Different definitions of homotopy colimits
I was reading about the definition of homotopy limits and colimits, and I have seen two different approaches in "Homotopy theories and model categories" by Dwyer and Spalinski, and in "...
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Presenting geometric morphisms by geometric morphisms
It's known that any $\infty$-topos $\mathcal{E}$ can be presented by a Quillen model category $\mathbf{E}$ that is itself a 1-topos. For instance, if $\mathcal{E}$ is a left exact localization of a ...
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Does the functor of Chevalley–Eilenberg cochains $ CE^\bullet:L_\infty\mathbf{Alg}^{op}\to \mathbf{dgAlg} $ map homotopy limits to homotopy colimits?
I was wondering whether the functor of Chevalley–Eilenberg cochains
$$
\operatorname{CE}^\bullet:L_\infty\mathbf{Alg}^\text{op}\to \mathbf{dgAlg}
$$
maps homotopy limits to homotopy colimits. Is still ...
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Simple, explicit, functorial cylinder object in CDGA
In the model category of graded commutative dg-algebras CDGA over $\mathbb{Q}$ (with the projective model structure) there is a simple, functorial construction of a path object given by tensoring with ...
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Generalization of familiar theorem about singular homology to general model category
I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this?
The second question is maybe related, I don't know. But anyway, given $U:\...
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When do zigzags of weak equivalences detect isomorphisms in the localization?
The usual way to prove that two model categories are equivalent is to construct a zigzag of Quillen equivalences between them, but is it always possible? We can ask a more general question.
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Why does the cotangent complex really have a distinguished triangle?
Associated to any ring maps $A\to B\to C$ there is the distinguished triangle
$$\mathbf{L}_{B/A}\otimes^L_BC\ \longrightarrow \ \mathbf{L}_{C/A} \ \longrightarrow \ \mathbf{L}_{C/B} \ \stackrel{+1}{\...