# Tagged Questions

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103 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

**1**answer

149 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 $Fun( \Delta ,Cat )$?
...

**3**

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170 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 ...

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**0**answers

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 ...

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votes

**1**answer

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

**1**answer

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 ...

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113 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 ...

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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**

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**1**answer

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 ...

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**1**answer

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 ...

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**2**answers

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 ...

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**0**answers

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 ...

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votes

**1**answer

164 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}_*$ ...

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**0**answers

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 ...

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votes

**2**answers

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

**1**answer

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 ...

**8**

votes

**1**answer

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) ...

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**0**answers

195 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 ...

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votes

**1**answer

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 ...

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**5**answers

547 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 ...

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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 ...

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votes

**1**answer

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 ...

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**1**answer

232 views

### Coequalizer in category of dg-algebras

It is known that there is a model structure on category of dg algebras (non-commutative over arbitrary commutative ring). In particular it is complete and co-complete category. My question is how to ...

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318 views

### The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...

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**1**answer

335 views

### Model structure on the category of small $A_\infty$ categories, hocolims.

I strangely could not find a reference for this. What are some (if any) model structures on the category of small $A_{\infty}$ categories, with weak equivalences quasi-equivalences. Same question in ...

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**1**answer

294 views

### When do reflexive coequalizers preserve weak equivalences?

In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is levelwise a weak ...

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**1**answer

295 views

### Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...

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112 views

### An explicit description of injective fibrations

If $M$ is a combinatorial model category and $C$ a small category, then the category $M^C$ admits an injective model structure in which the cofibrations and weak equivalences are levelwise. I would ...

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292 views

### Homotopical Galois theory of coverings

In the hope this won't turn into a trivial problem (I couldn't find a similar discussion here), here's my question.
I'm studying a little homotopical algebra in this article by Brown. You can easily ...

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votes

**1**answer

376 views

### Is there a notion of a “model category which admits left Bousfield localization?”

At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...

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199 views

### Induced model structure?

Let $H$ be a set with a binary operation $\cdot _H$ on it. To show that it is a group, one has to show that $\cdot _H$ is associative, find an identity element in $H$, and so forth; it might take ...

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votes

**2**answers

229 views

### Small model categories?

All the standard examples for model categories are large categories. Is it possible to have a small model category? Are there any interesting examples?
EDIT:
Since a complete small category is a ...

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**0**answers

236 views

### [Reference Request] The Definition of Adjoint Functors between dg-categories

Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair?
In my mind $F\dashv G$ ...

**3**

votes

**1**answer

233 views

### Model structure on category of endofunctors

Let $\mathcal C$ be model category, perhaps even cofibrantly generated. I don't assume that $\mathcal C$ is small. Recall that $End(\mathcal C)$ is the category of endofunctors on $\mathcal C$, with ...

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**1**answer

105 views

### Bibliographical reference needed (characterizing the weak equivalences of a model category)

I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that ...

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243 views

### Is the geometric realisation of a nerve equivalent to the classifying space of its categorisation?

The nerve functor $N:Cat\to sSet$ has a left adjoint, namely the categorisation $C$. In fact there is a natural isomorphism $\epsilon: CN\to Id$ and $N$ is a full embedding.So if I start with a ...

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votes

**1**answer

556 views

### Is there a cheap proof that (homotopy) endomorphisms are functorial?

This is, in some sense, the homotopy version of this question.)
If $C$ is a category with $iC$ the subcategory of isomorphisms, there is a functor $X \mapsto End(X)$ from $iC$ to the category of ...

**17**

votes

**2**answers

527 views

### Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...

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votes

**1**answer

275 views

### Transporting model structures via adjunctions

Hello,
If $F$ is a left adjoint between $C$ and $D$, and $D$ has a model structure; We can define cofibrations and equivalences in $C$ to be those that are so after applying $F$. What are criterions ...

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votes

**1**answer

512 views

### When is the category of pro-objects a homotopy category?

For a category $C$, there is a category Pro-$C$ whose objects are cofiltered diagrams $I \to C$ and whose morphisms are given by
$$
{\rm Hom}(\{x_s\},\{y_t\}) = \varprojlim_t\ \varinjlim_s\ {\rm ...

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votes

**2**answers

192 views

### Free Monoids in Closed Symmetric Monoidal Categories

There appear to be questions perhaps tangentially related to this that have been asked already. If so a reference and a close would be heartily appreciated.
Give some category $\mathcal{C}$ with the ...

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votes

**2**answers

282 views

### Fibration of Batanin/Leinster $\omega$-groupoids

Is there (defined somewhere) a notion of fibration between two weak $\omega$-groupoids in the sense of Batanin/Leinster?
I tried to search on Google and in Higher Operads, Higher Categories of Tom ...

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votes

**2**answers

472 views

### Is the category of small categories locally presentable?

I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be ...

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votes

**5**answers

1k views

### Computations in $\infty$-categories

Direct to the point.
Since now I've looked a lot of presentations of $\infty$-categories, but it seems that the only way to do explicit computations on these objects is via model categories. Is that ...

**4**

votes

**4**answers

326 views

### Find weak equivalences from fibrations and cofibrations

Let's suppose I have a category $\mathcal C$ with two weak factorisation systems $(C,F_W)$ and $(C_W,F)$, where $C_W\subset C$ and $F_W\subset F$.
I would like to have a model structure on $\mathcal ...

**5**

votes

**1**answer

339 views

### Presheaves on a complete Segal space

Let C be an $(\infty,1)$-category, incarnated as a complete Segal space, hence in particular a bisimplicial set. Is there a model structure on the slice category of bisimplicial sets over C which ...

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**0**answers

194 views

### Left adjoints to several inclusions of homotopy simplicial model categories

The left adjoint to the inclusion $sGrp\hookrightarrow sPtSet$ of the category of simplicial groups into the category of pointed simplicial sets is homotopy equivalent to the loop suspension functor ...

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votes

**3**answers

389 views

### Homotopic maps out of cofibration sequences

Consider a model category $\mathcal{C}$ and a sequence of cofibrations $0 \to X_0 \to X_1 \to X_2 \to \dots$ lying in $\mathcal{C}$. Let $X$ be the colimit of this sequence. Suppose furthermore that ...

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**2**answers

535 views

### Do simplicial objects in a Topos form a model category?

Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds.
The real question here is, do simplicial object in a ...

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**0**answers

215 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 ...