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-1
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1answer
74 views

Homotopy Groups for Categories

With this observation in mind how far are we from defining $\forall \mathcal{C} \ \text{category}\ \pi_1(\mathcal{C})$? Let me be more clear. Let be $n$ the following category $0 \rightarrow 1 ...
1
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0answers
31 views

are equivalences in an $(\infty,1)$-category preserved under colimits

Let $C$ be an $(\infty,1)$-category (e.g. quasicategory) having all small colimits. If $f_i:x_i \to y_i$ are equivalences in $C$ indexed by a small set $I$, is $f:=\mathrm{colim}_{i \in I} f_i$ an ...
0
votes
1answer
54 views

Morphism of morphism

I have some ideas about how to define morphism of morphisms in category theory, but I don't know anything about higher category theory. The idea is quite simple, and can be iterated easily. I'll only ...
0
votes
1answer
41 views

Strong (trivial) cofibration in Lurie's HTT

in Lurie's book HTT in annexe A, proof of Proposition A.2.8.2 page 824, he mentions that a map is a "strong (trivial) cofibration" but I didn't succeed to find the definition of this notion that seems ...
1
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0answers
30 views

presentable vs inductive categories

Let $\kappa$ be a regular cardinal, and let $D$ be a small $\infty$-category which admits $\kappa$-small colimits. Then is the canonical map $D \to Ind_\kappa(D)$ an equivalence, and if not why not? I ...
4
votes
2answers
94 views

question about monoidal structure of a 2-category

Consider an extension of the 1-category of vector spaces and linear maps down to a 2-category $\mathcal{C}$ whose objects are $k$-linear categories. What is the symmetric monoidal structure on the ...
0
votes
1answer
63 views

2-category in HoTT: chapter 9 from the HoTT book

After reading Awodey's book, and the HoTT book (and numerous other entries on these topics), I am (ambitiously) trying to do the exercises after the category theory chapter. This concernes the ...
2
votes
0answers
78 views

Transfinite composition and $\kappa$-categories

I wonder whether the following ideas make sense, and whether something in that direction has been written down somewhere; do you know a reference? The last definition is supposed to be a ...
2
votes
1answer
83 views

How does a section of a stack give a sheaf?

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a ...
0
votes
2answers
68 views

Are there any stable $(\infty,1)$-topoi?

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?
2
votes
2answers
78 views

Homotopy direct limit versus direct limit

Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and ...
2
votes
0answers
21 views

Does the lax Gray tensor product preserve fully faithful 2-functors?

The question is in the title. By fully faithful 2-functor, I mean 2-functors such that the maps on the hom categories are isomorphism, and by preserve, I mean in each variable. I have an argument ...
3
votes
1answer
37 views

Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to ...
3
votes
1answer
55 views

Defining a monoidal category without elements

I am trying to generalize the notion of monoid object internal to a (not necessarily strict) monoidal category, by weakening the associativity and unitarity diagrams (see this nlab entry.) Of course ...
1
vote
0answers
55 views

n-globular sets and n-categories

Is the forgetfull functor between the category of n-categories and the category of n-globular sets always monadic? It's seems so, but, in nLab, they are talking only about "2-globular sets and ...
8
votes
1answer
108 views

How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a ...
6
votes
3answers
139 views

Why is the 'mapping space' between two objects in a quasi-category a Kan complex?

Let $X$ be a quasi-category. (i.e. a simplicial set satisfying the weak Kan extension condition). Given two 'objects' $a,b\in X_0$ in $X$, one defines the mapping space $X(a,b)$ to be the pullback of ...
1
vote
1answer
20 views

Let $f : C \rightarrow D$ denote an arrow of a 2-category. Does “cone to $f$” have a commonly accepted meaning?

Let $f : C \rightarrow D$ denote an arrow in a $2$-category. Is there a commonly accepted meaning for the term "cone to $f$," generalizing the case where $f$ is a functor and $C$ and $D$ are ...
0
votes
0answers
20 views

Bicategorical Limits with parameters

Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a pseudofunctor (other type of 2-functors may be considered as well) between bicategories. Suppose that $\mathcal{C}$ has all limits. ...
1
vote
0answers
45 views

2-category as a 2-monad?

It is well known that a category is the same as a monad in the 2-category of spans. So I am wondering if there is a similar statement hold for higher categories: can a bicategory be given as a weak ...
3
votes
2answers
45 views

Distributivity in linear monoidal categories

Let $\mathcal{C}$ be a linear monoidal category, that is a monoidal category (with tensor product $\otimes$) enriched over $\mathbf{Vect}$. Now as far as I can tell the axioms for a linear category ...
1
vote
0answers
21 views

Is it possible to express the pentagon condition by associativity?

Let me start by considering associativity. Let $C$ be a set with a binary operation (multiplication) $C\times C\to C$. The associativity law is $(ab)c=a(bc)$. It can be expressed by a commutativity ...
3
votes
0answers
38 views

Derived pseudo-functor

Let $ \mathfrak {X}\to \mathfrak{Y} $ be a pseudofunctor (in which $\mathfrak{X} $ is a model category and $\mathfrak{Y} $ is a bicategory). I would like to understand when there is a derived functor ...
0
votes
2answers
74 views

What is the name and properties of a category C with hom(C) is subclass of ob(C)?

Suppose we have a category $C$ such as $hom(C) \subset ob(C)$ (so every arrow $f: a \rightarrow b$ of $C$ is also an object of $C$). What is the name and the main properties of such category?
1
vote
0answers
59 views

Torsors for 2-groups

Let $\mathbb{G}$ be a 2-group, by which I mean a strict monoidal category in which all objects are invertible (up to coherent isomorphisms) and all morphisms are invertible (strictly). What is the ...
3
votes
1answer
54 views

Difference between the simplicial nerve and the nerve of a simplicial category

In Jacob Lurie's Higher Topos Theory book, he defines the following notion of a simplicial nerve: Definition 1.1.5.5. Let $\mathcal{C}$ be a simplicial category. The simplicial nerve ...
5
votes
1answer
114 views

$2$-Morphisms in the Fundamental $2$-Groupoid

I'm trying to write down a clean definition of the fundamental $2$-groupoid $\pi_{\leq 2}(X)$ of a topological space $X$. Specifically, I'm concerned with how to properly define $2$-morphisms. Here is ...
2
votes
1answer
70 views

Totally ordered set Category

If 2 is the totally ordered set, and C is any category, this is given F is a functor from $ 2 \to C $ then what type of objects and arrows of the functor between them. As far as I understand as 2 is ...
6
votes
1answer
88 views

Higher categorical Yoneda lemma

One of the most powerful results in ordinary category theory is the Yoneda lemma, and so the following question seems natural: Is there an analogue of the Yoneda lemma for (weak) $n$-categories? I ...
0
votes
2answers
54 views

Can the equivalence between principle bundles and maps to classifying spaces be turned into an adjunction.

We have that $G-PBun(X)$, the category of topological principal bundles for a structure group $G$ is equivalent to $Top[X,BG]$ where $BG$ is the classifying space of $G$. This almost looks like an ...
0
votes
0answers
34 views

Has any NNO been indentified in an infinity topos?

The Natural Numbers Object ($NNO$) in $Set$ is just the Integers. Have any been identified or conjectured in $(\infty,1)$-topos that isn't an ordinary topos, in particular it could be a $(2,1)$-topos. ...
4
votes
2answers
205 views

How to understand the definition of sets in homotopy type theory and the role of univalence?

Bear with me, I'm a physicist. In homotopy type theory, as I understand it, a type $X$ is a set if all the morphisms over its terms $x:X$ are identies. When I say "morphisms", then I view the term as ...
0
votes
1answer
67 views

What is homotopy in $(\infty,1)$-categories (as weak Kan complexes)

One of the peculiar (and somewhat appealing) features of quasi-categories is that many properties from ordinary category theory characterized equality are characterized by some form of homotopy ...
2
votes
1answer
50 views

Reference request for 2-category theory

In the theory of 2-categories, we have the following theorem. A pseudofunctor is an equivalence of 2-categories if it is essentially surjective on objects, and the induced functor on Hom categories ...
1
vote
2answers
117 views

What does the following notation means in the context of 2-categories? $\bullet$

I know the elementary definition of an adjunction and the compact version (i.e. in terms of naturality of two hom-sets) but I am reading a definition which I cant understand its notation, namely the ...
5
votes
1answer
51 views

How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
0
votes
3answers
98 views

From where comes the horizonal composition in a 2-category?

Viewing a 2-category as a category enriched in $\mathsf{Cat}$, I can see from where comes the vertical composition: morphisms of a 2-category are objects of $\mathsf{Cat}$ and 2-morphisms of this ...
1
vote
1answer
41 views

Is a $0$-category best viewed as a set, or a setoid?

I'm way, WAY out of my depth with this question, but curiosity got the better of me. Anyway. I've heard that $0$-categories are basically just sets. This seems wrong to me, since sets are kind of ...
0
votes
0answers
87 views

Is it true in a presentable infinity category that algebras are homotopy colimits of free algebras?

In 1-categorical algebra one knows that, in a locally presentable category, every algebra for a finite product theory is a colimit of free algebras. Is the same true for algebras of finite product ...
6
votes
1answer
172 views

Is it possible to formalize a “universe” of categories as a one-sorted first-order theory with one binary relation and no functions?

This is a modification of a question I asked earlier. In that question, I hadn't placed any limits on the number of binary relations allowed, so my question had an affirmative answer, but a trivial ...
1
vote
1answer
77 views

What is an object classifier and how does give a natural numbers object?

According to a history of topos theory by McLarty, Blass (1989) showed that the existence of an object classifier over a given topos implies that the topos has a natural number object. What is an ...
6
votes
1answer
229 views

Is it possible to formalize (higher) category theory as a one-sorted theory, just like we did with set theory?

Set theory is typically formalized as a one-sorted theory without urelements. Is it possible to do the same with category theory or higher category theory, formalizing the whole affair as a theory ...
6
votes
1answer
247 views

What was the Lawveres explanation of adjoint functors in terms of Hegelian Philosophy?

I was contemplating asking this question on Philsophy.SE but felt it was better directed here as there are a dearth of category theorists there. According to the wikipedia entry on Categorical Logic: ...
5
votes
1answer
103 views

What is a (-1)-morphism?

So, I read the John Baez essay "Lectures on n-categories and cohomology" and I understand the notion of a (-1)-category" and a (-2)-category" and how to derive them. However, I'm not totally clear on ...
8
votes
1answer
137 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 ...
3
votes
1answer
141 views

What is curvature, in terms of holonomy functors?

It is well known and understood that linear connections, as holonomy functors, are composition-preserving mappings from the path groupoid to a structure group $G$. This extends the idea of a 1-form ...
2
votes
0answers
28 views

Are lax kernel pairs stable under change of base?

The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps ...
3
votes
2answers
214 views

What is a monad in a $2$-category?

The wikipedia article on monads somewhat mysteriously notes that Monads can be defined in any 2-category ${\mathfrak C}$. The monads defined above are for ${\mathfrak C}$ = Cat. where Cat is the ...
4
votes
1answer
85 views

Are bicategories and lax 2-categories the same?

My question is that whether the definition of bicategories is the same as the definition of lax 2-categories. I heard that they are both week versions of 2-categories. Are they the same? If not, how ...
4
votes
1answer
121 views

Lax algebras as lax morphisms

Wondering for the ncatlab I've encountered the following pages: one about lax-morphisms and the other about lax-algebras for $2$-monads. For what I could get it seems that lax algebras can be ...