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6
votes
0answers
85 views

Fibrations of the injective model structure on G-simplicial sets

Let $G$ be a discrete group. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which ...
0
votes
0answers
59 views

Homotopy uniqueness of some coends

I am not very convinced about Hirschhorn's proof of Corollary 18.4.15 Here, $\mathscr{C}$ is a Reedy category and $\mathcal{M}$ is a simplicial model category. Furthermore, $X\otimes_{\mathscr{C}} K$ ...
2
votes
0answers
89 views

When is the product of an infinite family of simplicial sets also a homotopy product?

The homotopy product of an infinite family of simplicial sets can be computed by deriving the product functor sSetW→sSet, for example, by performing the componentwise fibrant replacement using Kan's ...
8
votes
0answers
127 views

Reference for maps whose pushouts are also homotopy pushouts

Consider a category C with weak equivalences, e.g., a model category. For the purposes of this question, let's say that a morphism f in C is an i-cofibration if any pushout (alias cobase change) ...
1
vote
0answers
57 views

Latching space functor in Reedy model strucutre

I am new to MO and I hope that this question is suitable for it. Following Hovey's "Model Categories" ( http://ericmalm.net/ac/projects/symmetric-spectra/hovey--model-cats.pdf ) to study model ...
5
votes
0answers
153 views

Can we “complete” model categories to compute derived functors in the usual way?

Suppose we have a functor between two model categories $F\colon \mathcal{C}\to \mathcal{D}$. Suppose we don't know that it is a part of a Quillen pair. Nevertheless, it can still happen that the ...
5
votes
1answer
234 views

A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea. Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
5
votes
2answers
254 views

Is the category of $G$-spaces a model category?

Let $G$ be a compact Lie group and $\mathcal{C}_G$ the category of $G$-spaces (ie. topological spaces endowed with continuous left $G$-actions). Is there a model category structure on $\mathcal{C}_G$ ...
13
votes
4answers
422 views

Fibrations and Cofibrations of spectra are “the same”

My question refers to a folklore statement that I have now seen a couple of times, but never really precise. One avatar is: "For spectra every cofibration is equivalent to a fibration" (e.g. in the ...
11
votes
2answers
311 views

The definition of Reedy category

The common definition of Reedy category seems to be this one that a Reedy category is a small category $R$ with two wide subcategories $R_+$ and $R_-$ and an ordinal-valued degree function on its ...
5
votes
1answer
209 views

Constructing a “geometric” model structure on Cat by localizing the “categorical” model structure

Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the ...
6
votes
1answer
298 views

Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case?

Suppose that $\mathcal{M}$ is a model category which is not combinatorial, does a homotopy limit in $\mathcal{M}$ correspond to a limit in the associated $\left(\infty,1\right)$-category? How about ...
5
votes
2answers
218 views

Morphisms every pushout of which is a weak equivalence

Let $M$ be a category equipped with a class of weak equivalences $W$. Is there a name for a morphism $f$ such that every pushout of $f$ (including, of course, $f$ itself) is a weak equivalence? For ...
1
vote
0answers
67 views

When can I compute the simplicial mapping space from a presheaf to a simplicial presheaf naively?

Suppose that $\mathscr{C}$ is is a small category, $Y$ is a presheaf (of sets) on $\mathscr{C},$ and $X_\bullet$ is a simplicial presheaf. There is a spectrum of simplicial model structures on ...
3
votes
0answers
79 views

How do you compute a homotopy colimit in a category of fibrant objects?

This question may be a bit vague, (so if suggested, can I make it community wiki), but I was wondering what techniques there exists for computing homotopy colimits in a category of fibrant objects. A ...
6
votes
0answers
116 views

When does a sheaf of categories represent a homotopy sheaf?

Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...
3
votes
0answers
99 views

On the preservations of certain colimits (by covers) under simplicial localization (of a category of fibrant objects)

Suppose I have a category of fibrant objects $\mathcal{C}$ (with weak equivalences $W$ and fibrations $F$) together with a subcanonical Grothendieck pre-topology $J$ whose covering families consist of ...
2
votes
0answers
107 views

Which reflexive coequalizer diagrams are projectively cofibrant?

Consider the walking reflexive pair category W, which consists of two objects 0 and 1 and three generating morphisms f: 0→1, g: 0→1, and h: 1→0 satisfying the relation fh=gh=id₁. Consider the ...
0
votes
0answers
86 views

Path objects in projective model structure

I want to know how path objects look like in the presheaf category $[\mathcal{B}(\mathbb{Z}/2\mathbb{Z})^{\text{op}}, \text{Gpd}]$. Note that this category is just groupoids equipped with involutions. ...
0
votes
0answers
74 views

Comparison of model structures

I am just learning some basics of model categories, so pardon me if my question is trivial. Suppose that $\mathfrak{C}$ is a category and there exists two model structures on $\mathfrak{C}$ (For ...
2
votes
1answer
58 views

dg-flat complexes and their characters

Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X_n$ is flat for each n and ...
6
votes
1answer
224 views

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 ...
4
votes
0answers
115 views

Formal DG-algebras

Sorry for this question but I really have difficulties with model categories. Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
9
votes
0answers
139 views

Homotopy theory of suplattices

In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ...
2
votes
0answers
119 views

Homotopy limits of weak diagrams

This question is somehow related to my question at http://math.stackexchange.com/questions/683915/derived-pseudo-functor and to the question here: A homotopy commutative diagram that cannot be ...
0
votes
1answer
164 views

Homotopy limit of a cosimplicial category

Consider the usual model structure on Cat (category of small categories). Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$? ...
1
vote
2answers
281 views

Equivalence of homotopy categories and model structure theory

I apologize if this is to basic for MO, but it just seems a bit to advanced for SE. Let $\text{Top}$ be the category of topological spaces and $\text{sSet}$ the category of simplicial sets. There is ...
2
votes
0answers
129 views

Cloven Kan fibrations

For each groupoid $G$ there is a monad such that the cloven fibred groupoids over $G$ are algebras for this monad. I am looking for an infinite dimensional counterpart: a monad on simplicial sets ...
3
votes
0answers
176 views

Invariants of groups that are invariant under passage to finite index subgroups

This question is mostly idle curiosity. Recall the following standard terminology: if $P$ is a property of groups, a group $G$ is said to be virtually $P$ if it has a subgroup of finite index which ...
5
votes
0answers
107 views

Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
1
vote
0answers
143 views

Terminology question in model category theory

This is a terminology question. That will help me to write down my papers. In Marc Olschok's PhD available here (I cannot find it anymore on the Internet so it is in my webpage, the original URL is ...
0
votes
1answer
222 views

right adjoint for pullback along fibration

Let $Grpd$ be the category of groupoids and $p:E\rightarrow B$ a fibration in the standard model structure on $Grpd$ (ie an isofibration). How do you prove that the pullback functor $p^{\star}:Grpd/B ...
2
votes
1answer
326 views

Homotopy groups of filtered homotopy limits

Let $X$ be the homotopy limit of a filtered system of simplicial sets $X_i$. When are the morphisms $\pi_j(X)\to \varprojlim \pi_j(X_i)$ surjective for all $j\ge 0$? This seems to be no problem when ...
3
votes
0answers
137 views

What are the Čech-local equivalences of (simplicial pre)sheaves?

Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield ...
2
votes
0answers
123 views

Model structure on stacks

does anyone know if there is a model structure on stacks where the cofibrations are the monomorphisms ? As far as I know usually to get a model structure on stacks one localizes a model structure on ...
5
votes
0answers
157 views

Weak equivalences of left Bousfield localizations

Suppose C is a complete and cocomplete category with two model structures (C0,F0,W0) and (C1,F1,W1) such that C0⊃C1, F0⊂F1, W0⊂W1. If necessary, the model structures can be assumed to be simplicial, ...
9
votes
1answer
285 views

Which statements and arguments of Hovey's “Model categories” fail without functorial factorizations of morphisms?

I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations ...
4
votes
1answer
163 views

Is the injective structure on unbounded chain complexes simplicial?

In Mark Hovey's article Model category structures on chain complexes of sheaves (arXiv:math/9909024) a model structure on the category $Ch(A)$ of unbounded chain complexes for a Grothendieck abelian ...
3
votes
1answer
170 views

Factorization of morphisms in a diagram category

Let us suppose that $I$ is a small category and $\mathcal{E}$ a combinatorial model category. Then there exists two Quillen equivalent combinatorial model category structures on the diagram category ...
8
votes
1answer
345 views

Model structure for cooperads and for coalgebras

I am studying the homotopy theory of (algebraic) operads and I came up with several questions I am unable to answer to. I would like to stress that I don't have applications in mind, I just would like ...
3
votes
2answers
237 views

Analogues of 'cone' distinguished triangles for pointed model categories?

For an additive $A$ and any morphism $f:X\to Y$ in $C(A)$ one has the following distinguished triangle in the homotopy category $K(A)$: $X\to Y\to Cone(f)\to X[1]$. What is the closest analogue of ...
3
votes
0answers
152 views

Jardine model structure as left Bousfield localization

This should be a really basic question, but I'm stuck on it. The question. I see written everywhere (for example here, or in the article [DHI] Hypercovers and simplicial presheaves of Dugger, ...
3
votes
0answers
160 views

Homotopy theory of acyclic categories

Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the ...
5
votes
1answer
395 views

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 ...
2
votes
1answer
190 views

Left homotopy in the quillen model structure

Let $ \mathscr{T} $ be the category of [insert technical conditions here] topological spaces. Equip $ \mathscr{T}$ with the quillen model structure. The category of based spaces $ \mathscr{T}_*$ ...
0
votes
0answers
192 views

is this a simplicial model category?

A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One ...
4
votes
2answers
432 views

On triangulated categories of pro-objects

Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories? I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...
5
votes
2answers
209 views

A model category which is an additive category

Let $\cal C$ be a model category which is also additive. Suppose that the homotopy category $\operatorname{Ho}\mathcal C$ is additive, for example this is true when the weak equivalences in $\cal C$ ...
8
votes
1answer
289 views

The state of the art in the rectification of homotopy-coherent structures

My question concerns rectification theorems for homotopy-coherent structures. As the meaning of this may be unclear, let me list a few examples of what I am thinking of: Cordier and Porter proved a ...
3
votes
1answer
240 views

Alternative model structure on retractive spaces

Background For a topological space $X$, let $R(X)$ be the category of retractive spaces over $X$. An object of this category is a space $Y$ equipped with maps $s: X \to Y$ and $r: Y \to X$ such that ...