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0
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0answers
64 views

Reference for generalized model categories [closed]

I would like to know some references for "Stanculescu's notion" of Generalized model category. Motivation The idea is find some works that use the notion and increase my knowledge about it. ...
3
votes
2answers
178 views

How to simplify the proof of right-properness?

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback ...
5
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0answers
88 views

Continuous cohomology via model category

Is it possible to formulate notion of continuous cohomology in terms of model categories? If yes, then is there a reference for this?
8
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1answer
267 views

Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...
4
votes
0answers
102 views

What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, ...
27
votes
5answers
2k views

What is modern algebraic topology(homotopy theory) about?

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
2
votes
1answer
69 views

Homotopy limits of homotopically constant diagrams over contractible categories

I suspect that the following result should be true and more or less well known: Let $\mathcal{M}$ be a model category and $I$ a small category with contractible nerve. For every diagram $X: I \to ...
3
votes
1answer
221 views

Reference for t-structures on stable model categories

What kind of definitions of t-structures on stable model categories have been investigated in the literature? Of course, one can always define a t-structure on a stable model category as a ...
3
votes
0answers
117 views

Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...
14
votes
1answer
471 views

Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
9
votes
2answers
149 views

Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let ...
10
votes
1answer
487 views

Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering... ...
1
vote
1answer
89 views

dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...
2
votes
0answers
67 views

How to show these class of morphisms are perfect

In Lurie's book "Higher Topos Theory", a class $\mathsf{W}$ of morphisms in a category $\mathcal{A}$ is called perfect if Every isomorphism belongs to $\mathsf{W}$. it satisfies "2 out of 3 ...
1
vote
0answers
63 views

Cofibrancy of simplicial objects [duplicate]

Let $\mathcal{C}$ be a site. Consider $sPsh(\mathcal{C})$ be the equipped with the local projective model structure. Let $C_{\bullet}$ be a cofibrant object in $\mathcal{C}$ and let $y(C_\bullet)$ be ...
11
votes
1answer
348 views

Examples of differential cohomology in cohesive $\infty$ topos

I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public. The formulation of differential cohomology in ...
7
votes
1answer
169 views

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 ...
7
votes
1answer
188 views

Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too. Let $k$ be a field and $R$ a $k$-algebra. The stable ...
6
votes
2answers
317 views

Is dgCat a category or a 2-category?

Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has ...
3
votes
0answers
100 views

Reedy fibrant and cofibrant objects

Let $\mathcal{C}$ be a Reedy category, $\mathcal{M}$ be a model category. Then consider $Fun(\mathcal{C}, \mathcal{M})$, the category of diagrams from $\mathcal{C}$ to $\mathcal{M}$ equipped with the ...
13
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1answer
350 views

Fibrant-cofibrant models of Eilenberg-MacLane spectra

There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct ...
5
votes
1answer
59 views

Characterization of right properness using slice categories

I would like to know how to cite this theorem (which has a quite surprising consequence): A model category $\mathcal{M}$ is right proper if and only if for any weak equivalence $f:A\to B$, the ...
5
votes
1answer
80 views

Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer. Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...
2
votes
1answer
89 views

Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors $F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the ...
3
votes
1answer
156 views

Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension

I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...
11
votes
2answers
257 views

Quasicategories for non-simplicial model categories

If I have a simplicially enriched model category, then I can take the coherent nerve of the full subcategory of bifibrant opjects to obtain a quasicategory. If I have a model category that is not ...
8
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0answers
336 views

Standard model structures on $Top$

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ...
9
votes
1answer
289 views

Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...
4
votes
1answer
193 views

Stabilization of a generic pointed model category

Let $\mathcal C$ be a pointed model category. It is well-known that its homotopy category $\mathrm{Ho}(\mathcal C)$ is naturally a $\mathrm{Ho}(\underline{\mathrm{sSet}}_*)$-category, where ...
1
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0answers
65 views

Does it require Reedy fibrancy when we want the totalization to be weakly equivalent to the homotopy limit?

This question arises when I am reading the last two Chapter of Hirschhorn's "Model categories and their localizations" In Part (2) of Theorem 19.8.4 of that book it says If ...
1
vote
0answers
81 views

Internal Hom on simplicial presheaves and the preservation of cofibrant objects

1)Let $\mathcal{C}$ be a cartesian closed small category. Let $\operatorname{Map}\: : \: sPsh(\mathcal{C})\times sPsh(\mathcal{C})\to sPsh(\mathcal{C})$ be the internal Hom of simplicial presheaves, ...
6
votes
2answers
491 views

On combinatorial and cellular model categories and infinity categories

I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...
6
votes
3answers
510 views

Higher refinement of Seifert-van Kampen theorem on the language of hocolim

I like the following version of SvKT. If $\Pi_1$ is the functor of fundamental groupoid and $(X_i)_{i\in I}$ is a diagram of spaces then $$\Pi_1({\sf hocolim}\: X_i)\simeq {\sf hocolim}\: ...
3
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0answers
88 views

About Quillen equivalences between Bousfield localizations

Let $\mathcal{M}$ be a locally presentable category equipped with two left proper and left determined combinatorial model structures $\mathcal{M}_1$ and $\mathcal{M}_2$. There exist two sets $S_1$ and ...
9
votes
1answer
252 views

Tensor products over operads and bar constructions

Let $O$ be an operad in spaces, $A$ an $O$-algebra and $R$ an right $O$-module. One can define $R \otimes_O A$ as the coequalizer of the two maps $ROA$ to $RA$. One can also define $B(R,O,A)$ (as in ...
0
votes
3answers
123 views

smash product of pointed spaces preserve weak equivalences

Consider the category of pointed simplicial sets with usual notion of weak equivalence. The question is does the functor $$Y \mapsto X\wedge Y$$ preserve weak equivalences? or at the very least does ...
3
votes
0answers
114 views

How can one “extend scalars” for (motivic) ring spectra and for modules over it?

Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a ring object in a symmetric stable model category); let $R$ be a flat associative commutative unital algebra ...
5
votes
0answers
124 views

Homotopy pullback preserving functor

In the paper http://arxiv.org/pdf/math/0101162.pdf, the authors claim during the proof of Prop. 4.2 that a functor $F:A \to B$ which preserves fibrations and weak equivalences preserves homotopy ...
3
votes
1answer
224 views

Model structure on non-negative differential graded algebras with homological grading

I was wondering if there exists a model structure on the category of non-negative differential graded algebras with homological grading. To be more precise: Let $Ch_{k}$ the model category of ...
13
votes
1answer
387 views

Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...
3
votes
2answers
282 views

How to show the following two definitions of homotopy monomorphism are equivalent?

Let $M$ be a model category. In Toen's The homotopy theory of dg-categories and derived Morita theory Page 11 it is written: a morphism $x \to y$ in a model category $M$ is called a homotopy ...
3
votes
1answer
119 views

A question about the morphisms in the homotopy category of dg-Cat

Let $dg-Cat$ denote the category of (small) dg-categories and $Ho(dg-Cat)$ denote the localization of $dg-Cat$ at quasi-equivalence. Using the model structure on $dg-Cat$ we can describe the morphisms ...
13
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2answers
422 views

A model category of abelian categories?

Let $\mathcal{M}$ be the following category: The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels. The morphisms are functors that preserve the ...
1
vote
0answers
204 views

Does the Boardman-Vogt tensor product of operads commute with their W-construction

I have absolutely no idea whether this is true or not but it could well be useful for me in the future if it is. If we have topological operads $\mathcal{P}$ and $\mathcal{Q}$ and we let $W$ denote ...
3
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0answers
110 views

Homotopy (co)limit (co)cones

Let $\mathscr{M}$ be a model category and let $\mathscr{I}$ be a small category. Consider any homotopy colimit functor ...
6
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2answers
196 views

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 ...
0
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0answers
48 views

Contraction of simplicial presheaves

Let $X,Y$ be two simplicial presheaves on a small category $\mathcal{C}$, let $*$ be the final simplicial presheaf. Consider the category of simplicial presheaves equipped with its projective model ...
2
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0answers
39 views

Comparing right and left quasi-representable bimodules

Let $\mathcal V$ be your favourite (closed, symmetric) monoidal model category. To fix ideas, set $\mathcal V = \mathrm{Ch}(k)$, the category of chain complexes over a fixed commutative ring. Given a ...
8
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1answer
293 views

When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian ...
6
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1answer
185 views

Technical lemma about frame and cosimplicial resolution

I'm reading the Hirschhorn's book model categories and their localization and I have a question about frames and resolutions. Following the book (definition 16.6.1) a cosimplicial frame on an object ...