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4
votes
0answers
226 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
786 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
295 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
372 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
496 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
172 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
338 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
167 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 ...
10
votes
1answer
812 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
299 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
141 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
310 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
297 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
308 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
665 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
132 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 ...
3
votes
1answer
262 views

geometric realization on $\mathbf{sTop}$

Is geometric realization $|\cdot|:\mathbf{Top}^{\mathbf{\Delta}^{\textrm{op}}}\rightarrow \mathbf{Top}$ a left Quillen functor? If so, under what model structure on ...
2
votes
1answer
285 views

Model categories and cellular maps

A question came up on MSE and it generated, for me, the following question: When looking at the maps of CW/cell/simplicial complexes do the cellular/simplicial maps have a model theoretic ...
12
votes
1answer
1k views

Do homotopy groups “always” commute with filtered colimits?

It is well-known that homotopy groups, of, say, simplicial sets, commute with filtered colimits. However, I could not find a reference for an analogous result for homotopy groups of spectra, or, ...
3
votes
2answers
499 views

Inner hom and geometric realization.

I would like to prove the following fact, which I learned from a previous MO question. Let $S_\cdot,T_\cdot\in\mathbf{sSET}$ be simplicial sets, and assume that $T_\cdot$ is Kan. Then there is a ...
6
votes
1answer
493 views

When does a cosimplicial object compute homotopy colimits?

Suppose I want to compute the homotopy colimit of a diagram of spaces. There is a simple way of getting a simplicial space from this diagram, and a theorem tells me that taking the geometric ...
16
votes
4answers
1k views

How canonical is cofibrant replacement?

Quillen's original definition of a model category included noncanonical factorization axioms, one being that any map can be factored into a cofibration followed by an acyclic fibration. More recent ...
8
votes
2answers
680 views

“Strøm-type” model structure on chain complexes?

Background The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by ...
3
votes
2answers
354 views

Model categories of simplicial objects

If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for ...
7
votes
0answers
590 views

An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point: Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with the contravariant model structure (cofibrations are monomorphisms ...
4
votes
1answer
230 views

Compatibility of classifying space with inner-hom?

Let $\mathbf{sTop}$ be the functor category $\mathbf{Top}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let $\mathbf{sCat}$ be the functor category $\mathbf{Cat}^{{\mathbf{\Delta}}^{\textit{op}}}$, and let ...
6
votes
1answer
783 views

Fibered/cofibered higher categories, relative model structures, slicing, and (∞,2)-category theory

Jacob Lurie defined a model structure on the category of marked simplicial sets sliced over a fixed simplicial set $S$ called the cartesian model structure. (For a definition, see here or HTT ...
12
votes
2answers
506 views

Pointed Hurewicz model structure

In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and ...
19
votes
4answers
1k views

Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
8
votes
2answers
697 views

Does the category of topological symmetric spectra satisfy the monoid axiom ?

In the paper "Symmetric spectra"by Hovey, Smith and Shipley, they say that they don't know if the monoid axiom holds for topological symmetric spectra. This paper was written in 1998 so I am wondering ...
6
votes
1answer
435 views

How does Berger-Moerdijk's relative Boardman-Vogt work?

In "The Boardman-Vogt resolution of operads in monoidal model categories," the authors construct factorizations of sufficiently nice operad maps $P\to Q$ into a cofibration followed by a weak ...
8
votes
1answer
320 views

Is the injective model structure on symmetric spectra Bousfield localizable?

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on ...
7
votes
1answer
685 views

Non standard (?) model category structure on (co)chain complexes.

Let $\cal{A}$ be an abelian category with enough projectives and $\mathbf{C}_+ (\cal{A})$ the category of bounded below chain complexes. Since Quillen (Homotopical algebra, 1.2, examples B), there is ...
10
votes
1answer
326 views

Is the model category of Complete Segal Spaces right proper?

Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories". Has ...
4
votes
1answer
216 views

Model category with formally smooth morphisms as fibrations?

Let's view the category of algebraic spaces as a full subcategory of the category of "spaces" over the opposite category of commutative rings, that is, the category of sheaves on $CRing^{op}$ in the ...
5
votes
3answers
544 views

A reference for Calculus of Functors for Model Categories

I am wondering where I might look to see what has been done in terms of Calculus of Functors for more general weak equivalences and Model Categories. I am at least aware of some of the extended ...
2
votes
1answer
176 views

In what generality does the following statement hold: A fibration is acyclic if and only if all fibres are contractible fibrant objects.

This may not be precise enough for MO, but I'll give it a go. Let $M$ be a symmetric closed monoidal model category with unit $u\in Ob(M)$. We define the vertices of an object $A$ to be points $x\in ...
10
votes
3answers
1k views

What is the “universal problem” that motivates the definition of homotopy limits/colimits (and more generally “derived” functors)?

The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan ...
2
votes
0answers
72 views

Reference for model structure on CosimplicialAbelianGroups

There is a standard (simplicial) model category structure on the category $Ab^\Delta \simeq Ch^\bullet_+(Ab)$ of co-simplicial abelian groups, whose fibrations are the degreewise surjections (and weak ...
2
votes
0answers
87 views

Is CosimplicialAlgebras left proper?

The model category structure on co-simplicial commutative $k$-algebras, $CAlg_k^\Delta$, with fibrations degreewise surjections: is it left proper?
8
votes
2answers
859 views

Example of a CW complex not homeomorphic to the realization of a simplicial set?

I've often heard that we can give examples of CW complexes that aren't homeomorphic to the realization of any simplicial set (although I haven't heard that there exist Kan complexes that aren't ...
3
votes
1answer
560 views

Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories

Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed. Here is the correct statement of (*): For any cofibration $f:A\to B$ and ...
5
votes
2answers
502 views

Characterizing the rationalization of spaces.

In the category of rational spaces, loop spaces split as products of Eilenberg-Mac Lane spaces and SUSPENSIONS split as wedges of (rational) spheres. I wonder if anything of the following form is ...
14
votes
2answers
530 views

How many model categories have the same weak equivalences?

There are many situations which arise where one might consider different Model categories with the same underlying category. For example in (left) Bousfield localization you start with a model ...
9
votes
1answer
422 views

Schemes as a model category

I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense. A natural temptation is to relate this machinery to ...
12
votes
1answer
798 views

A Model Category of Segal Spaces?

So in Julie Bergner's work on (infty, 1)-categories arXiv:0610239, she considers several model categories which model (infty, 1)-categories, which are known to be equivalent. I'm guessing that there ...
13
votes
3answers
1k views

What determines a model structure?

It is easy to prove that a model structure is determined by the following classes of maps (determined = two model structures with the mentioned classes in common are equal). cofibrations and weak ...
7
votes
4answers
484 views

An explicit description of Lawvere's segment in the category of simplicial sets

In any presheaf topos, there exists an object called Lawvere's segment, which can be described as the presheaf $L:A^{op}\to Set$ such that for each object $a\in A$, $L(a)=\{x\hookrightarrow\ h_a: x\in ...
3
votes
1answer
378 views

Slick verification of the model category axioms for Spaces and SSets with the q-model structure?

We choose our category of spaces to be compactly generated weak Hausdorff spaces for convenience, denoted $CGWH$. Questions: 1.) Is there any sort of slick argument to verify that CGWH with the ...