# Tagged Questions

**2**

votes

**0**answers

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

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

**9**

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**0**answers

487 views

### Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...

**1**

vote

**0**answers

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 ...

**3**

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**0**answers

173 views

### What is a Homotopy between $L_\infty$-algebra morphisms II

I would like to continue on question I asking, what is a homotopy between
Lie infinity algebras, since I'm not satisfied in two directions:
1.) The naive approach to define a homotopy would be ...

**9**

votes

**1**answer

482 views

### What interesting homotopy invariants can I write down using the universal property of homotopy types?

I've recently been led to believe some version of the following statement:
Weak homotopy types, or equivalently $\infty$-groupoids (let me not commit myself to a particular model of these), are ...

**8**

votes

**1**answer

234 views

### Loops and suspensions of higher categories

Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ ...

**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 ...

**7**

votes

**1**answer

226 views

### Is there a Wall finiteness obstruction in other settings?

Let $\mathcal{S}$ be the $(\infty, 1)$-category of spaces. Then the compact objects of $\mathcal{S}$ are precisely the retracts of finite CW complexes. These are not the same as the finite CW ...

**5**

votes

**2**answers

264 views

### How does one Segal-subdivide a 2-category?

Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...

**4**

votes

**2**answers

735 views

### Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...

**4**

votes

**1**answer

427 views

### Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?

This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...

**20**

votes

**4**answers

1k views

### What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...

**4**

votes

**0**answers

248 views

### How to endow an n-fold Segal Space with a symmetric monoidal structure?

I would like to understand how I can endow an n-fold (complete) Segal Space with a symmetric monoidal structure. My question is basically the same as in this post: What is a symmetric monoidal ...

**6**

votes

**4**answers

682 views

### Is there a sense in which the homotopy theory of simplicial sets is the “paradigmatic” one?

I could not come up with a better title for my question.
What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):
the ...

**5**

votes

**0**answers

255 views

### Generating acyclic cofibrations for the various model structures in higher category theory

There are a number of model categories important in higher category theory, which provide a "presentation" of some $\infty$-category of $\infty$-categories. For example:
The Joyal model structure on ...

**2**

votes

**3**answers

900 views

### Exact sequences in homotopy categories

I am not really familiar with homotopical category theory, so please forgive me if I make rude mistakes. I know quite a bit of common category theory, as well as familiar with algebraic topology.
How ...

**4**

votes

**1**answer

274 views

### Writing an infinity groupoid as a colimit of sets

If we are given a simplicial set $X:\Delta^{op} \to Set$, we may regard it as a \emph{simplicial} simplicial set, i.e. a bisimplicial set by composing with the "constant" inclusion $Set \to ...

**8**

votes

**1**answer

391 views

### How do various notions of natural transformation relate to various notions of homotopy in $2Cat$?

In what follows, $2$-categories will be strict, and "$2$-functor" will mean "strict $2$-functor". (Please mention which terminological conventions you are using when answering.) I guess that the ...

**8**

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**0**answers

389 views

### Does the category of strict $2$-categories together with Dwyer-Kan equivalences provide a model for $(\infty,1)$-categories?

The question is the title.
In what follows, all $2$-categories and $2$-functors will be strict. Let $2-Cat$ denote the categories whose objects are $2$-categories and whose morphisms are ...

**2**

votes

**0**answers

142 views

### Universal polygraphic factorization of strict ω-categories relative to a cobase

Recall from 1 that a cofibration of strict ω-categories is a retract of relative $I$-cell complexes, where $I$ denotes the set of boundary inclusions $\partial D^n \hookrightarrow D^n$, where $D_n$ ...

**4**

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**0**answers

184 views

### Is the class of inner-anodyne morphisms right-cancellative with respect to the of the class of monomorphisms?

Recall:
Given a category $A$, and two classes of morphisms $S,S'$, we say that $S$ is right-cancellative with respect to $S'$ if for any pair of maps $f\in S, g\in S'$ such that $gf$ is defined, we ...

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**0**answers

221 views

### Completion of n-fold Segal spaces

During a reading course about Jacob Lurie's paper about Local TQFTs, I came across the notion of Segal spaces as models for $(\infty,1)$-categories. Looking at things like the "double nerve" for ...

**13**

votes

**3**answers

1k views

### Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...

**32**

votes

**4**answers

4k views

### Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...

**8**

votes

**1**answer

384 views

### Is there a “derived” Free $P$-algebra functor for an operad $P$?

Recall that an operad (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S_n$, with maps ...

**10**

votes

**1**answer

770 views

### The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...

**8**

votes

**1**answer

575 views

### Is the $\infty$-category of stable $\infty$-categories stable?

More generally, are there any remarkable properties enjoyed by the $\infty$-category of stable $\infty$-categories?

**7**

votes

**2**answers

531 views

### Historical Question about Simplicial Sets

I have a pretty easy historical question about simplicial sets. Unless I am mistaken, simplicial sets first came out of topology, explicitly from combinatorial topology and the study of simplicial ...

**7**

votes

**0**answers

292 views

### A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...

**6**

votes

**0**answers

581 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 ...

**5**

votes

**1**answer

744 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 ...

**4**

votes

**4**answers

729 views

### Homotopic quotients of simplicial sets as infinity-groupoids

Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of ...

**20**

votes

**2**answers

2k views

### What's the current state of the classification of not-fully-extended TQFTs?

Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some ...

**12**

votes

**1**answer

737 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 ...

**30**

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 ...

**17**

votes

**2**answers

1k views

### Why do my quantum group books avoid homotopical language?

I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.
Many notes closer to "Kontsevich ...

**9**

votes

**2**answers

1k views

### $(\infty,1)$-categories and model categories

I read several times that $(\infty,1)$-categories (weak Kan complexes, special simplicial sets) are a generalization of the concept of model categories. What does this mean? Can one associate an ...

**4**

votes

**1**answer

530 views

### references for models of stable infinity categories

There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...

**3**

votes

**2**answers

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 ...

**3**

votes

**1**answer

264 views

### omega-categories and n-fold complete segal spaces

Why are n-fold complete segal spaces or (\infty, n)-categories (which I'm unsure of how to distinguish from omega-categories) important for n >= 3? Why are they "badly behaved" for n >= 3? (Lurie ...