The model-categories tag has no wiki summary.
6
votes
1answer
189 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 ...
3
votes
0answers
93 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 ...
8
votes
0answers
129 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
104 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
150 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
243 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
110 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
171 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
101 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
141 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
202 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
279 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
114 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 ...
1
vote
0answers
104 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
146 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, ...
8
votes
1answer
274 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
129 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
155 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 ...
7
votes
1answer
301 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
225 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
136 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
139 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 ...
4
votes
1answer
307 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
165 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
187 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
416 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
203 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
260 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
229 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 ...
2
votes
1answer
137 views
Does Wolbert's derived equivalence between $E_*^R$-local $R$-modules and $R_E$-modules come from a Quillen equivalence?
Let $R$ be a ring spectrum (in the world of EKMM $S$-modules) and let $E$ be a smashing $R$-module. Denote by $R_E$ the $E_*$-localization of $R$. By a theorem of Wolbert (Theorem 2 in Classifying ...
8
votes
1answer
266 views
Is the localisation of a product of categories the product of the localisation?
Let $\cal C, \cal D$ be model categories. Hovey says in his monograph "Model Categories" that the homotopy category $\operatorname{Ho}(\cal C \times D)$ is isomorphic to $\operatorname{Ho}(\cal C) ...
5
votes
0answers
123 views
Refined cofinality theorem for homotopy limits of spaces
If $C$ is a simplicial model category, $F\colon I\longrightarrow J$ a functor between small categories, and $X\colon J\longrightarrow C$ a diagram, the canonical map $holim_JX\longrightarrow ...
4
votes
0answers
196 views
Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?
Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...
7
votes
2answers
262 views
Where should I search for resolutions?
In my research, to test some conjectures or just to illustrate some facts, I often need to compute some explicit examples of derived functors (in the sence of Quillen's model categories). Mainly I ...
4
votes
1answer
224 views
Quillen equivalence of diagram categories
Let $C$ be a model category and $B$ a direct category. By Theorem 5.1.3 in Mark Hovey's book Model categories, there is a model category structure on the diagram category $C^B$ such that weak ...
3
votes
1answer
167 views
Bousfield localization before and after taking homotopy
The ncatlab says:
Under suitable conditions it should be true that for $C$ a model category whose homotopy category $\mathrm{Ho}(C)$ is a triangulated category the homotopy category of a left ...
8
votes
1answer
282 views
Unicity up to homotopy of simplicial enrichments
On the one hand, in their paper Simplicial structures on model categories and functors, Rezk, Schwede and Shipley proved that a simplicial model category structure on a given model category is unique ...
3
votes
0answers
160 views
Cisinski-Deglise model structure on chain complexes
Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and Déglise define the ...
5
votes
2answers
477 views
Homotopy limit-colimit diagrams in stable model categories
It is shown in Remark 7.1.12 of (a newer version of) Mark Hovey's book Model Categories that, in a stable model category, homotopy pullback squares coincide with homotopy pushout squares. The argument ...
4
votes
1answer
221 views
Is there a class of simplicial sets whose weak homotopy type is preserved by symmetrization?
We have the categories, $S$, of simplicial sets and $SS$, of symmetric simplicial sets (whose simplices are unordered). There are functors:
$H:S\to SS$ forgetting the ordering on simplices and
...
7
votes
0answers
124 views
Fibrations of orthogonal G-spectra and fixed points
There are at least two fixed point functors that characterize stable equivalences of orthogonal G-spectra: the geometric fixed points and the naive fixed points of a fibrant replacement.
Is this true ...
5
votes
0answers
140 views
Cellular model structures on continuous functors
The category of enriched functors from finite based CW complexes to based topological spaces
has a projective model structure. The fibrations
are the objectwise Serre fibrations and the weak ...
5
votes
2answers
373 views
A category with weak equivalences which is not a model category
I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ ...
6
votes
5answers
549 views
What is a good basic reference on model categories?
I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for ...
1
vote
0answers
140 views
Model structure on the category of chain complexes in an abelian category gives rise to the derived category
Let $\mathcal A$ be an abelian category with enough projectives. Consider a morphism $f \colon X^\bullet \longrightarrow Y^\bullet$ in the category $C(\cal A)$ of chain complexes in $\mathcal A$.
Is ...
4
votes
1answer
277 views
Homotopy left-exactness of a left derived functor
Let
$$
F: \mathcal{C} \leftrightarrows \mathcal{D} :G
$$
be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors
$$
\mathbb{L}F: ...
4
votes
1answer
118 views
Equivariant versus retractive spaces: a reference request
Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the ...
3
votes
1answer
151 views
Directed colimits of maps in a combinatorial model category
I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated ...
7
votes
1answer
437 views
Seeking errata for Berger-Moerdijk Axiomatic Homotopy Theory for Operads
The paper I'm referring to can be found here. It came out in 2003. I've been told by professors before that I should be careful relying on things from this paper, because it contained errors.
Is ...
6
votes
2answers
416 views
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 ...
