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3
votes
1answer
358 views

When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?

Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...
3
votes
1answer
298 views

How strong is the condition that an operad splits, i.e. O(n)=O(s)xO(n-s)?

I recently proved something for the operad $Comm$ valued in a model category $M$ and am trying to generalize it to other symmetric operads. I'm very new to operads, so please forgive me if there are ...
3
votes
0answers
177 views

Boardman Vogt W construction for modules over an operad

The W construction of Boardman and Vogt gives a cofibrant replacement for operads. In http://arxiv.org/abs/math/9907073, Salvatore describes a cofibrant replacement for algebras over an operad. Is ...
5
votes
1answer
571 views

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 ...
8
votes
1answer
486 views

How many model category structures are there on Top?

I recently started learning a little model category theory and in particular I found this nice exercise. I only know a little topology, but this prompted me to wonder how many model category ...
17
votes
2answers
533 views

Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...
6
votes
2answers
522 views

Connective spectra versus simplicial abelian groups - very basic question

Hello, I have very general , "introductory" questions (It is quite hard for me to seek for specific things in the algebraic topology literature). I guess that connective spectra have a model ...
3
votes
2answers
452 views

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 ...
1
vote
1answer
195 views

Technical question about cell complexes

Hello, I have a technical question. My terminology: I - set of standard inclusions $\partial I^n \to I^n$. I-cell (Relative Cell Complexes) - transfinite compositions of pushouts of maps in $I$. ...
1
vote
2answers
408 views

About a statement in Jardine and Goerss “Simplicial Homotopy Theory”

Hello, This probably just technical, but anyway: In "Simplicial Homotopy Theory" by Goerss and Jardine, chap. III, par. 2, after cor. 2.12, they describe a model structure on $Ch^{+}$, the category ...
4
votes
1answer
289 views

Transporting model structures via adjunctions

Hello, If $F$ is a left adjoint between $C$ and $D$, and $D$ has a model structure; We can define cofibrations and equivalences in $C$ to be those that are so after applying $F$. What are criterions ...
5
votes
1answer
376 views

Counterexample in cohomology for symmetric spectra?

Symmetric spectra are a particular model for spectra, introduced by Hovey, Shipley and Smith. They have the nice property that they have a well-behaved smash product. Our interest in spectra comes ...
1
vote
1answer
175 views

local model structure on simplicial presheaves

Hello, Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology. Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its ...
12
votes
1answer
529 views

When is the category of pro-objects a homotopy category?

For a category $C$, there is a category Pro-$C$ whose objects are cofiltered diagrams $I \to C$ and whose morphisms are given by $$ {\rm Hom}(\{x_s\},\{y_t\}) = \varprojlim_t\ \varinjlim_s\ {\rm ...
3
votes
2answers
198 views

Free Monoids in Closed Symmetric Monoidal Categories

There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated. Give some category $\mathcal{C}$ with the ...
3
votes
2answers
199 views

Fitting desired weak equivalences and cofibrations into a model category

Suppose I have a category $\mathbf{C}$ and classes of morphisms $\mathcal{W}$ and $\mathcal{C}$, and I would like to know that $\mathcal{W}$ and $\mathcal{C}$ are the weak equivalences and the ...
-1
votes
2answers
591 views

Alternative characterization of homotopy equivalence

Using the formalism of model categories its possible define the concept of homotopy as done here. If we take as model category $\mathbf{Top}$ having homotopy-equivalence as weak-equivalence and ...
8
votes
1answer
495 views

Hovey's unit axiom in monoidal model categories

Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following unit axiom not considered in other references (e.g. Schwede-Shipley): given a cofibrant ...
4
votes
2answers
288 views

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 ...
5
votes
2answers
495 views

Is the category of small categories locally presentable?

I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be ...
10
votes
5answers
1k views

Computations in $\infty$-categories

Direct to the point. Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that ...
4
votes
4answers
335 views

Find weak equivalences from fibrations and cofibrations

Let's suppose I have a category $\mathcal C$ with two weak factorisation systems $(C,F_W)$ and $(C_W,F)$, where $C_W\subset C$ and $F_W\subset F$. I would like to have a model structure on $\mathcal ...
6
votes
1answer
769 views

Does the right adjoint of a Quillen equivalence preserve homotopy colimits?

Call a diagram $E$ in a model category a homotopy colimit diagram if the morphism $$\mathrm{hocolim}~E\to \mathrm{colim}~ E$$ is a weak equivalence. A homotopy colimit is defined as the categorical ...
5
votes
1answer
358 views

Presheaves on a complete Segal space

Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which ...
5
votes
0answers
198 views

Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...
3
votes
3answers
408 views

Homotopic maps out of cofibration sequences

Consider a model category $\mathcal{C}$ and a sequence of cofibrations $0 \to X_0 \to X_1 \to X_2 \to \dots$ lying in $\mathcal{C}$. Let $X$ be the colimit of this sequence. Suppose furthermore that ...
7
votes
2answers
593 views

Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds. The real question here is, do simplicial object in a ...
1
vote
0answers
138 views

Can one recover an A-algebra from its cotangent complex?

Given an A-algebra B, one can define the cotangent complex $L_{B/A}$ as $\Omega^1_{P/A}\otimes_PB$, where $P$ can be taken as the canonical resolution of $B$ associated to the pair of adjoint functor ...
33
votes
4answers
5k views

Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
4
votes
0answers
365 views

New model Structure on $E_{\infty}$-algebras?

Let $\mathbf{sSet}$ be the category of simplicial sets. Is it possible to put a new model structure on $\mathrm{E}_{\infty}$-algebra (of simplicial sets) such that the weak equivalences and fibrations ...
3
votes
2answers
189 views

A model structure on the category of “dualizing maps”

Let $C$ be the category with objects being maps $h:M\to A$ where $A$ is a commutative graded $\mathbb{Q}$-algebra (cdga), $M$ is a differential graded (dg) $A$-module and $h$ is an $A$-dg-module map, ...
12
votes
2answers
1k views

Acyclic models via model categories?

Recall the acyclic models theorem: given two functors $F, G$ from a "category $\mathcal{C}$ with models $M$" to the category of chain complexes of modules over a ring $R$, a natural transformation ...
10
votes
1answer
383 views

The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that: The cofibrations are the ...
4
votes
0answers
221 views

When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE ...
4
votes
3answers
773 views

univalent axiom as a property of a model category?

I am interested to understand the univalence axiom of Voevodsky; however, I know very little type theory. Thanks to response below, I now understand what is being univalent means for a morphism. A ...
4
votes
0answers
285 views

Equivariant sheaves and simplicial varieties

I would like to proof the following theorem: Let $\pi:X\rightarrow X/G$ be a principal $G$-bundle (say of varieties, Zariski locally trivial), then $\pi^*$ induces an equivalence between modules on ...
4
votes
0answers
362 views

Fiber sequences in proper model categories

I am confused about the notion of a fiber sequence (or dually a cofiber sequence) in a general pointed and proper model category $\mathcal{C}$. Following Hovey, we can define, like in topology, a map ...
4
votes
0answers
471 views

When are “diagrams of cofibrations” projectively cofibrant?

Let $P$ be a small category, and let $F:P\to M$ be a diagram in a left-proper combinatorial model category $M$. We say that $F$ is a diagram of cofibrations if for every object $p\in P$, $F(p)$ is ...
3
votes
0answers
169 views

Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor

Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
2
votes
1answer
325 views

Resolutions by Adapted Class of Objects and Model Categories

My question is about the construction of derived functor in the language of model categories. (As it is done for example the paper by Dwyer and Spalinski "Homotopy Theories and Model Categories".) ...
4
votes
0answers
166 views

Mapping into a geometric realization.

Suppose $S$ is a simplicial set, $X$ is a space, and we are given a map \[ f: \text{Sing}\,X\to S. \] When is is possible to produce a map $X\to |S|$? We can take ...
9
votes
1answer
792 views

Is the simplicial completion of a localizer always a bousfield localization of the injective model structure?

Background Recall (from Cisinski's Astérisque volume 308) that given a small category $A$, we define an $A$-localizer to be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the following ...
4
votes
1answer
288 views

Does adding degeneracies to a semi-simplicial diagram change the homotopy colimit?

Let $\Delta_{+}$ be the sub-category of the simplex category $\Delta$ containing only injective functions, and take $M$ to be a nice model category. I'll write $i \colon \Delta_{+} \hookrightarrow ...
2
votes
0answers
135 views

“non-Bousfield” localisations of model categories

When can I localise a model category by a set(or class) of morphisms, and how do I describe the localised model category ? By 'localise' I mean to find a localisation functor $q_S : M \rightarrow ...
1
vote
0answers
295 views

Is there a model category structure on non-negatively graded commutative chain algebras?

Let $\mathtt{DGA}$ be the category of non-negatively graded DG chain algebras, and $\mathtt{DGA}$* the category of non-negatively graded cochain algebras. Let $\mathtt{CommDGA}$ be the full ...
2
votes
0answers
292 views

simplicial deRham complex and model category structure

To every simplicial manifold is associated its simplicial deRham complex. Is there any literature that discusses explicitly to which extent this classical construction, regarded as a (contravariant) ...
4
votes
2answers
307 views

Simplicial presheaves that are colimits of themselves?

Suppose $C$ is a small category and $X_{\bullet}$ is a simplicial object in $C$. In particular, by composing with Yoneda $$y:C \to Set^{C^{op}}$$ $y(X)_{\bullet}$ is a simplicial presheaf. I believe ...
1
vote
1answer
142 views

Cofibrations and coequalisers in a proper model category

I have a proper model category and in it two coequalisers, $A_i \rightrightarrows B_i \to C_i$, $i=1,2$. I have a map of diagrams arising from maps $A_1 \to A_2$, $B_1 \to B_2$ where these two arrows ...
12
votes
2answers
646 views

Model category structure on categories enriched over quasi-coherent sheaves

Gonçalo Tabuada has shown that there is a Quillen model category structure on the category of small dg-categories, i.e. the category of small categories enriched over chain complexes (for a fixed ...
1
vote
0answers
131 views

end of a weak equivalence

I would like to get a concrete description of sufficient conditions for the end of a morphism in $\mathcal{C}^{J^{op}\times J}$ (which is a point-wise weak equivalence) to be a weak equivalence. In ...