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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Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by "simplicial decomposition"

I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow: The argument works by showing that ...
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Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?

In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about ...
L. Xie's user avatar
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Examples of non-proper model structure

I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...
Simon Henry's user avatar
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When is the localization of all hypercovers equivalent to that of Čech covers?

In Theorem A.6 of Dugger's paper, it is shown that a few localizations are equivalent to the localization of Čech covers. The Nisnevich localization at all hypercovers is equivalent to the ...
L. Xie's user avatar
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Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization

Let $L_{Nis}(sPre(Sm_S))$ be the Nisnevich localization of the category of simplicial presheaves, how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times_S X\to X$ are closed under ...
L. Xie's user avatar
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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 ...
Matt's user avatar
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1 answer
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Proving a Kan-like condition for functors to model categories?

I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been ...
Julian Chaidez's user avatar
9 votes
2 answers
713 views

Non-small objects in categories

An object $c$ in a category is called small, if there exists some regular cardinal $\kappa$ such that $Hom(c,-)$ preserves $\kappa$-filtered colimits. Is there an example of a (locally small) ...
Peter Bonart's user avatar
3 votes
1 answer
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Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
Nicky's user avatar
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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. ...
Valery Isaev's user avatar
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Strøm model structure on nonnegatively graded chain complexes

Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes. The ...
Najib Idrissi's user avatar
3 votes
2 answers
216 views

Bousfield localization of a left proper accessible model category

What is known about the Bousfield localization of a left proper accessible model category by a set of maps ? (I mean not combinatorial which is already known)
Philippe Gaucher's user avatar
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The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces

Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g. N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from https://neil-strickland.staff.shef.ac.uk/courses/...
Joao Faria Martins's user avatar
7 votes
1 answer
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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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Existence of tensor product of infinity operads

I am trying to show, or find a reference, for the following fact: "Given O,P two infinity operads [in the sense of Lurie, HA, Definition 2.1.1.10], there always exist a tensor product". In other ...
Andrea Marino's user avatar
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On cofibrations of simplicially enriched categories

Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells. We have a canonical inclusion functor , $$i: C \...
F.Abellan's user avatar
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1 answer
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Compact objects in the $\infty$-category presented by a simplicial model category

Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-...
Saal Hardali's user avatar
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2 votes
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159 views

Homotopy colimits of simplicial objects

Given a simplicial combinatorial model category $\mathcal{M}$ and a simplicial diagram $F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$, is there a nice (i.e. explicitely computable) way of ...
Edoardo Lanari's user avatar
3 votes
0 answers
102 views

Right adjoint preserving (trivial) cofibrations between cofibrant objects

Consider a right Quillen adjoint which is not a categorical left adjoint which takes (trivial resp.) cofibrations between cofibrant objects to (trivial resp.) cofibrations between cofibrant objects. ...
Philippe Gaucher's user avatar
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Which set of compact objects generates the subcategory of a compactly generated stable model category?

I couldn't find any info on what set of compact objects generates the following subcategory: Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...
Doelt_k's user avatar
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1 answer
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Is the Hurewicz model category left proper?

A model structure is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. In the Hurewicz (or Strom) model structure on the category of topological spaces, weak ...
Dmitry Vaintrob's user avatar
4 votes
1 answer
369 views

Homotopy limit over a diagram of nullhomotopic maps

Let $I$ be a $\mathrm{Top}_*$-enriched poset and $X: I \to \mathrm{Top}_*$, and consider the homotopy limit $$ \underset{i \in I}{\mathrm{holim}}X(i), $$ where the maps $X(i) \to X(j)$ are ...
Niall Taggart's user avatar
5 votes
2 answers
474 views

Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?

I saw this result in A Model Category Structure for Differential Graded Coalgebras by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg ...
Victor TC's user avatar
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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}{\...
Pulcinella's user avatar
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4 votes
1 answer
316 views

Thomason fibrant replacement and nerve of a localization

The Thomason model structure on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \...
Martin Frankland's user avatar
12 votes
3 answers
691 views

On model categories where every object is bifibrant

Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one ...
Simon Henry's user avatar
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7 votes
2 answers
414 views

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 ...
Tintin's user avatar
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14 votes
2 answers
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Example of non accessible model categories

By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's ...
Philippe Gaucher's user avatar
2 votes
0 answers
135 views

Why is a homotopy limit of a cosimplicial space not the ordinary limit?

I've been trying to compute a homotopy limit of a cosimplicial object $X: \Delta \to \mathscr{M}$, where $\mathscr{M}$ is some simplicial model category, we may take it to be spaces. I'm wondering ...
Maanroof's user avatar
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11 votes
2 answers
697 views

What are the advantages of simplicial model categories over non-simplicial ones?

Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a ...
Tim Campion's user avatar
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10 votes
0 answers
320 views

$\Gamma$-sets vs $\Gamma$-spaces

I know that every $\Gamma$-space is stably equivalent to a discrete $\Gamma$-space, i.e. a $\Gamma$-set. For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $\Gamma$-Groups, ...
Simon Henry's user avatar
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16 votes
2 answers
893 views

Counter-example to the existence of left Bousfield localization of combinatorial model category

Is there any known example of a combinatorial model category $C$ together with a set of map $S$ such that the left Bousefield localization of $C$ at $S$ does not exists ? It is well known to exists ...
Simon Henry's user avatar
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7 votes
1 answer
476 views

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 ...
Tim Campion's user avatar
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3 votes
1 answer
388 views

Definition A.3.1.5 of Higher Topos Theory

I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition of a $\mathbf{S}$-enriched model category, where $\mathbf{S}$ is a monoidal model category. But in the book model ...
Frank Kong's user avatar
8 votes
2 answers
406 views

For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?

How explicit are the model structures for various categories of spectra? Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit ...
Tim Campion's user avatar
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3 votes
1 answer
104 views

Monoidalness of a model category can be checked on generators

If $C$ is a cofibrantly generated model category which is also monoidal biclosed, then to check that $C$ is a monoidal model category, it suffices to check that the Leibniz product of generating ...
Tim Campion's user avatar
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8 votes
2 answers
355 views

Localization, model categories, right transfer

Suppose that we have a locally presentable category $M$ and $N$ is a locally presentable full subcategory of $M$. Both categories are complete and cocomplete. Lets suppose that we have an adjunction $...
num's user avatar
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2 votes
0 answers
54 views

Simplicial models for mapping spaces of filtered maps

Let $S$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. Hence $SIMP(S,K)$ is Kan. Suppose that ${...
Joao Faria Martins's user avatar
3 votes
1 answer
151 views

Simplicial models for fibrations between mapping spaces

Let $S,T$ and $K$ be simplicial sets with $K$ Kan. Given simplicial sets $S$ and $K$, we let $SIMP(S,K)$ be the internal hom in the category of simplicial sets. We let $|S|$ denote the geometric ...
Joao Faria Martins's user avatar
2 votes
1 answer
269 views

simplicial objects in a model category

Suppose that we have a (combinatorial if necessary) model category $M$, and let $F:\Delta^{op}\rightarrow M$ a simplicial object in $M$, such that for any natural number $n$, $F([n])$ is a fibrant ...
Paris's user avatar
  • 707
6 votes
1 answer
202 views

Simplicial localization of the cofibrant-fibrant objects

Let $M$ be a model category. I don't assume that $M$ has functorial factorizations or that $M$ is simplicial. Write $M^{c}$ (respectively, $M^{cf}$) for the full subcategory of $M$ on the cofibrant ...
Reid Barton's user avatar
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3 votes
1 answer
184 views

Cofibrations of functors

Let $\cal M$ and $\cal N$ be model categories, $S,T:\cal M\to N$ functors, and $\alpha:S\to T$ a natural transformation. Say that $\alpha$ is a <blank> cofibration if for any cofibration $i:A\to B$...
Mike Shulman's user avatar
5 votes
1 answer
606 views

Why is the straightening functor the analogue of the Grothendieck construction?

In classical category theory, there is the notion of functor (co)fibered in groupoids. Furthermore, via Grothendieck construction we have an equivalence between pseudo functors into the category of ...
Oscar P.'s user avatar
  • 559
6 votes
1 answer
481 views

Model structure on the category of topological groups

Consider the category $TopGr$ of topological groups. I want to know that this is a model category (can one understand its model structure by understanding a model structure on the category of enriched ...
Fat ninja's user avatar
  • 393
11 votes
1 answer
480 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 ...
Mike Shulman's user avatar
6 votes
1 answer
240 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})...
Liam Keenan's user avatar
0 votes
1 answer
181 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 ...
user09127's user avatar
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5 votes
2 answers
382 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}...
Lao-tzu's user avatar
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6 votes
1 answer
307 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, $1$-...
Zhaoting Wei's user avatar
  • 8,657
6 votes
0 answers
246 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,...
Pedro's user avatar
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