The model-categories tag has no wiki summary.

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votes

**1**answer

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### 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

**4**answers

2k 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

**2**answers

694 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 ...

**4**

votes

**2**answers

376 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

**0**answers

595 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

**1**answer

232 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 ...

**7**

votes

**1**answer

812 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

**2**answers

534 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

**4**answers

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

**2**answers

706 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

**1**answer

462 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

**1**answer

343 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

**1**answer

728 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

**1**answer

344 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

**1**answer

219 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

**3**answers

548 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

**1**answer

182 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 ...

**11**

votes

**3**answers

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

**0**answers

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

**0**answers

88 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

**2**answers

935 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

**1**answer

578 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

**2**answers

518 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

**2**answers

545 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

**1**answer

429 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 ...

**13**

votes

**1**answer

838 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 ...

**14**

votes

**3**answers

2k 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

**4**answers

491 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

**1**answer

385 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

**1**answer

659 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

**1**answer

127 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

**0**answers

123 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 ...

**32**

votes

**6**answers

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

**1**answer

706 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

**2**answers

660 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 ...

**17**

votes

**1**answer

1k 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 ...

**12**

votes

**1**answer

372 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

**1**answer

205 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 ...

**9**

votes

**3**answers

904 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 ...

**7**

votes

**2**answers

589 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

**6**answers

3k 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

**1**answer

274 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 ...

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votes

**2**answers

782 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 $\pi : \mathcal{C} \rightarrow \mathcal{I}$ ...

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votes

**3**answers

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 ...

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votes

**2**answers

394 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

**1**answer

447 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 ...

**19**

votes

**4**answers

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

**1**answer

316 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?

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votes

**2**answers

439 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

**1**answer

388 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 $x\in X$?
...