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8
votes
2answers
676 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
399 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
301 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
635 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 ...
8
votes
1answer
293 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
213 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
528 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
175 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 ...
9
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
84 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
779 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
528 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
462 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
496 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
395 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
739 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 ...
12
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
465 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
370 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 ...
3
votes
1answer
603 views

Is geometric realization of the total singular complex of a space homotopy equivalent to the space?

Let $X$ be a topological space and let $|Sing(X)|$ be the geometric realization of the total singular complex of $X$. Then $|Sing(X)|$ is a CW complex with one cell for each non-degenerate singular ...
2
votes
1answer
124 views

Is there a notion of “good” distributor/profunctor for model categories?

When considering functors between model categories one possibility is to restrict ones attention to quillen adjunctions. But what about distributors? What are the natural distributors to consider ...
2
votes
0answers
119 views

non-degenarete tools to calculate a derived functor on a model category which is a poset?

Are there theorems (esp. computational tools) on model categories which survive and do not trivialise when its underlying category is a (quasi-)poset ? Are there tools that may help to calculate ...
30
votes
6answers
3k views

Non-examples of model structures, that fail for subtle/surprising reasons?

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...
14
votes
1answer
626 views

Is there a combinatorial way to factor a map of simplicial sets as a weak equivalence followed by a fibration?

Background on why I want this: I'd like to check that suspension in a simplicial model category is the same thing as suspension in the quasicategory obtained by composing Rezk's assignment of a ...
9
votes
2answers
565 views

When do the Reedy and injective model category structures agree?

Let $R$ be a Reedy category and consider the category $\mathcal{P}(R) = \mathbf{sSet}^{R^{\mathrm{op}}}$ of simplicial presheaves on $R$. When are the Reedy and injective model structures on ...
14
votes
1answer
983 views

Homotopy colimits/limits using model categories

A homotopy (limits and) colimit of a diagram $D$ topological spaces can be explicitly described as a geometric realization of simplicial replacement for $D$. However, a homotopy colimit can also be ...
11
votes
1answer
352 views

How to localize a model category with respect to a class of maps created by a left Quillen functor

Let $M$ and $N$ be "nice" model categories. I'm happy to have "nice" mean combinatorial model category. Consider a Quillen pair $$ L: M\rightleftarrows N: R.$$ I want the following result: There ...
1
vote
1answer
192 views

Equivalences in Model Categories

If $\mathcal M$ is a model category and I know that $A$ and $B$ are isomorphic in $\mathrm{Ho}(\mathcal M)$, is it guaranteed that there is a zig-zag of weak-equivalences in $\mathcal M$ connecting ...
8
votes
3answers
783 views

Infinity groupoid objects

I was wondering if there is a model-theoretic way of defining the infinity category of infinity-groupoid objects in a category $C$ (more generally, if $C$ is an infinity category itself, but, right ...
4
votes
2answers
518 views

Is model structure on CatSet unique?

On the category CatSet of usual set based categories, there is a "folk" model structure, as described on the first page of Model structures for homotopy of internal categories by T. Everaert, R.W. ...
45
votes
6answers
2k views

What are surprising examples of Model Categories?

Background Model categories are an axiomization of the machinery underlying the study of topological spaces up to homotopy equivalence. They consist of a category $C$, together with three ...
0
votes
1answer
269 views

Analogs of left, right, inner, and Kan fibrations in CGWH

It is a theorem that the category of compactly generated weakly Hausdorff (CGWH) spaces is Quillen equivalent to the category of simplicial sets with the Kan model structure. However, I know next to ...
14
votes
2answers
726 views

Homotopy Limits over Fibered Categories

Suppose I have a small category $ \mathcal{C} $ which is fibered over some category $\mathcal{I}$ in the categorical sense. That is, there is a functor $\pi : \mathcal{C} \rightarrow \mathcal{I}$ ...
14
votes
3answers
1k views

Is there an additive model of the stable homotopy category?

Is there a model category $C$ on an additive category such that its homotopy category $Ho(C)$ is the stable homotopy category of spectra and the additive structure on $Ho(C)$ is induced from that on ...
4
votes
2answers
365 views

Motivation for the covariant model structure on SSet/S

I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure? I'm sorry for not providing ...
2
votes
1answer
425 views

characterization of cofibrations in CW-complexes with G-action

Is there a condition for a $G$-equivariant map $X \to Y$ to be a cofibration of $G$-spaces? Here $X$ and $Y$ are CW complexes, the group $G$ is finite, and acts by cellular maps. I am using the model ...
18
votes
4answers
1k views

A Peculiar Model Structure on Simplicial Sets?

I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do ...
3
votes
1answer
277 views

transfinite composition of weak equivalences in sSet

Weak equivalences in the standard model structure on simplicial sets are allegedly closed under transfinite composition. What's a reference for that?
4
votes
2answers
419 views

Five lemma in HoTop* and arbitrary pointed model categories

Let $\textbf{HoTop}^*$ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in $\textbf{HoTop}^*$, i.e. pointed homotopy ...
2
votes
1answer
378 views

Equivariant map preserves stabilizer

Let $G$ be a group and $X$ a set equipped with a transitive right $G$-action. Further, let $c: X\to X$ be a $G$-equivariant map. Is it true that $\text{Stab}(x) = \text{Stab}(c(x))$ for all $x\in X$? ...
3
votes
2answers
709 views

Homotopy Pushouts via Model Structure in Top

As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the ...
12
votes
3answers
579 views

Model Structure/Homotopy Pushouts in topological monoids?

Let C be the category of topological monoids, that is, the category of monoids in (Top, $\times$). Can the model category structure on Top (Serre fibrations, cofibrations, weak homotopy ...
9
votes
2answers
731 views

Are non-empty finite sets a Grothendieck test category?

A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, ...
15
votes
3answers
1k views

Model category structure on Set without axiom of choice

There is a model category structure on Set in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, and the weak equivalences are ...
9
votes
3answers
660 views

What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
14
votes
4answers
2k views

Homotopy pullbacks and homotopy pushouts

I have a good grasp of ordinary pullbacks and pushouts; in particular, there are many categorical constructions that can be seen as special cases: e.g., equalizers/coequalizers, kernerls/cokernels, ...
4
votes
3answers
250 views

Are injective Omega-spectra the S-local objects of symmetric spectra for some class S?

I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on nLab) of the projective level model ...
4
votes
1answer
410 views

Local Joyal-simplicial presheaves?

It is well known that left Bousfield localizations of the global functor model category $Func(C^{op}, SSet_{standard})$ of functors with values in simplicial sets equipped with the standard model ...
5
votes
3answers
367 views

Abstract Relation between Presehaves and Simplicial Sets

Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf ...