1
vote
0answers
38 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 ...
2
votes
2answers
35 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 ...
3
votes
0answers
30 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
69 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?
3
votes
1answer
40 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
82 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
68 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
73 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
37 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
28 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. ...
3
votes
2answers
157 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
58 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
37 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
100 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
43 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
0answers
23 views

Two properties of the category $Pr^L$

This is a continuation of another question about the symmetric monoidal category of locally presentable categories; now for the (∞,1)-analogue. Googling a bit I found two astonishing (albeit quite ...
0
votes
3answers
71 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 ...
0
votes
0answers
75 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
161 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
66 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
198 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
213 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: ...
4
votes
1answer
86 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
125 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 ...
2
votes
0answers
26 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
180 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 ...
3
votes
1answer
76 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
107 views

Reasons for coherence for bi/monoidal categories

Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this ...
2
votes
1answer
102 views

Deligne tensor product

I would like to know something about a tensor product of categories and it seems Deligne tensor product is what I am looking for. But the paper "Categories tannakiennes" by Deligne is not available ...
1
vote
0answers
62 views

2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
1
vote
0answers
52 views

Basics of Lie 2-algebras?

Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ? For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie ...
4
votes
0answers
101 views

What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
3
votes
1answer
58 views

Dual of a 2-category

Let $C$ be a $2$-category. There are two ways of dualizing $C$: The first one is well-known and also generalizes to arbitrary $(\infty,1)$-categories: We dualize "at each stage". The second one only ...
1
vote
1answer
220 views

Do the terms “quiver” and “meta graph” refer to the same concept?

Do the terms "quiver" and "metagraph" refer to the same concept? Or is there a distinction I am missing. My sources are Quiver - http://ncatlab.org/nlab/show/quiver Metagraph - ...
9
votes
1answer
357 views

Pullbacks of categories

Let $\mathfrak{Cat}$ be the 2-category of small categories, functors, and natural transformations. Consider the following diagram in $\mathfrak{Cat}$: $$\mathbb{D} \stackrel{F}{\longrightarrow} ...
2
votes
0answers
85 views

Morphisms with an arbitrary number of objects

Is this structure familiar for you? It consists of a category $C$ a set $M$ a function ``$\operatorname{arity}$'' defined on $M$ a function $\operatorname{Obj}_m$ defined for every ...
13
votes
5answers
652 views

Concrete examples of 2-categories

I've been reading some of John Baez's work on 2-categories (eg here) and have been trying to visualize some of the constructions he gives. I'm interested in coming up with 'concrete' examples of ...
0
votes
1answer
115 views

How to define a 2-category

I found this definition in http://arxiv.org/abs/hep-th/0304074 (pp.13) for 2-categories: It is a collection $\mathcal{C}_0$ of objects, $\mathcal{C}_1$ of morphisms, and $\mathcal{C}_2$ of 2-morphisms ...
5
votes
1answer
89 views

Automorphisms and bicategories

I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are determined up to unique isomorphism, algebraic closures (and by extension, their ...
6
votes
1answer
360 views

Mathematics needed for higher dimensional category theory?

I'm a undergrad(third year, Manchester uni) that is thinking of doing a PhD in this area or category theory in general. Just wondering, what branches of Maths should I focus on? As I've been told ...
7
votes
0answers
162 views

$(\mbox{Sh,Sh-map})$ represents the category of sheaves on a stack

I'm trying to understand the following theorem, and I think I don't understand the definitions. Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid ...
6
votes
1answer
189 views

Introductory texts for weak $\omega$-categories

As I'm constantly running across higher categories these days, I'm wondering what is a good starting point to get into the theory? While I am aware of nLab and the n-Category Café, I am having a real ...