# Tagged Questions

**3**

votes

**0**answers

94 views

### Recognition principle for 2-categories (2-groupoids)

Given a 2-category (i.e. bicategory) $C$ there is a unitary geometrical nerve whose 0-simplices are objects of $C$, 1-simplices are 1-arrows of $C$, 2-simplices are 2-commutative triangles (in certain ...

**1**

vote

**0**answers

94 views

### The bar construction or the quotient for monoidal category action on a category

Given a monoid $C$ acting (from the right) on a set $M$, there is a bar construction giving a simplicial set or equivalently a translation/action category,
$$
N_\bullet (M\rtimes C)= \cdots M\times ...

**3**

votes

**1**answer

157 views

### What is the suitable setting for supercoherence with value in a bicategory?

It was J.F. Jardine established the so called supercoherence theory in Journal of Pure and Applied Algebra Volume 75, Issue 2, 18 October 1991, Pages 103–194. The result can be roughly stated as ...

**9**

votes

**2**answers

351 views

### Do Homotopy Fully Faithful Functors Push-out?

A (homotopy) fully faithful functor is a map of $\infty$-categories which induces weak equivalences on mapping spaces.
Are homotopy fully faithful functors preserved under (homotopy) pushout?
More ...

**7**

votes

**1**answer

261 views

### What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...

**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 ...

**2**

votes

**2**answers

349 views

### how to make the category of chain complexes into an $(\infty,1)$-category

Related to this question, I would like to know, if there is an explicit presentation
of the $(\infty,1)$-category of a model category of abelian chain complexes, but this time in terms of simplicial ...

**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 ...

**10**

votes

**0**answers

259 views

### What is the history of the notion of subdivision of categories?

A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...

**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 ...

**2**

votes

**0**answers

186 views

### On small generators of an infinity category

Suppose that $\mathcal{C}$ is an $\infty$-category with pullbacks and small coproducts. Suppose that $\mathcal{D}$ is a small subcategory for every object $C,$ the canonical map ...

**1**

vote

**1**answer

171 views

### local model structure on simplicial presheaves

Hello,
Let $\mathcal{C}$ be a (small) category equipped with a Grothendieck pretopology.
Let $sPSh(\mathcal{C})$ be the category of simplicial presheaves on $\mathcal{C}$, together with its ...

**3**

votes

**2**answers

306 views

### Cartesian cubes and groupoids

Given a groupoid $G,$ one can consider the canonical epimorphism $$G_0 \to G.$$ Since it is an epimorphism in the $2$-topos of groupoids, $G$ is the weak colimit of the corresponding Cech diagram ...

**6**

votes

**1**answer

317 views

### Detecting equivalences of (infinity) categories by nerves

I have two questions:
Is there a way to tell if a functor $F:C \to D$ between two small categories is an equivalence in terms of the map $$N(F):N(C) \to N(D)$$ between simplicial sets? More ...

**4**

votes

**1**answer

275 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

**0**answers

390 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 ...

**4**

votes

**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 ...

**1**

vote

**0**answers

185 views

### Proof for a simplicial morphism

Sorry for the title but I can't come up with another one...
...
Suppose X and Y are simplicial sets
and $f : X \rightarrow Y$ is a map
such that:
1.) f maps dimensions the right way,
that is ...

**11**

votes

**2**answers

1k views

### What is a symmetric monoidal $(\infty,n)$-category?

This question arose from reading Jacob Lurie's "Classification of topological field theories" paper. In that paper, he uses complete $n$-fold Segal spaces as a model for $(\infty,n)$-categories, but ...

**9**

votes

**2**answers

1k views

### Semi-simplicial versus simplicial sets (and simplicial categories)

Hi,
Let's denote by "semi-simplicial set" a simplicial set without degeneracies, i.e. a contravariant functor from the category $\Delta_{inj}$ of finite linearly ordered sets and order preserving ...

**9**

votes

**1**answer

374 views

### The weak equivalences in the covariant model structure

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that:
The cofibrations are the ...

**6**

votes

**1**answer

478 views

### What does the “category” of $(\infty,1)$ category look like.

One knows that in higher category theory, the category of $(\infty,n-1)$ categories is naturally an $(\infty,n)$ category ,(I use the word category to mean category in the correct weakened sense). ...

**4**

votes

**1**answer

325 views

### A “join” of ω-categorical simplices

Recall that the category of level trees $\mathcal{T}$ is defined to be the category $[\mathbb{N}^{op},\Delta_a]$, where $\Delta_a$ is the skeleton of the category of finite possibly empty linearly ...

**10**

votes

**1**answer

771 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 ...

**4**

votes

**2**answers

306 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 ...

**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 ...

**8**

votes

**1**answer

438 views

### Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?

If you take the nerve of a groupoid, you get a Kan complex.
Question:
Take a bicategory that has adjoints for 1-morphisms, which is one notion of 'weak' groupoid (if all 2-morphisms are ...

**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 ...

**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 ...

**5**

votes

**1**answer

338 views

### nerves of crossed complexes, group T-complexes and classifying spaces

A (reduced) crossed complex is, intuitively, a non-abelian complex of groups $\ldots \to G_2 \to G_1 \to G_0$ with a $G_0$ action in such that $G_1 \to G_0$ is a crossed module.
There are a couple of ...

**5**

votes

**1**answer

215 views

### What is the right way to define the nerve of an unbiased monoidal category?

I've been toying around with unbiased composition in higher categorical structures on and off for a while now. In particular, I've been playing around with unbiased monoidal 2-categories. One ...

**4**

votes

**1**answer

687 views

### The Join of Simplicial Sets ~Finale~

Background
Let $X$ and $S$ be simplicial sets, i.e. presheaves on $\Delta$, the so-called topologist's simplex category, which is the category of finite nonempty ordinals with morphisms given by ...

**8**

votes

**3**answers

777 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

**2**answers

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

**1**answer

209 views

### When are two natural transformations of infinity-categories equivalent?

Suppose
C and D are two ∞-categories (quasi-categories),
$F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), which is just the ...

**6**

votes

**2**answers

295 views

### Is every left fibration of simplicial sets with nonempty fibers a trivial kan fibration?

In Lemma 2.1.3.4 of Higher Topos Theory, the statement of the lemma requires that the fibers are not only nonempty but contractible. However, in the proof, I don't see where contractibility is ...

**14**

votes

**6**answers

1k views

### Simplicial homotopy book suggestion for HTT computations

I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble ...

**4**

votes

**3**answers

508 views

### Computation of Joins of Simplicial Sets

It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a boundary?, etc? and ...

**6**

votes

**3**answers

560 views

### Joins of simplicial sets

Why doesn't the join operation on the category of simplicial sets commute up to unique isomorphism? I mean, aren't products and coproducts commutative up to isomorphism? That leads me to conclude at ...