In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. (Def: http://en.m.wikipedia.org/wiki/Higher_category_theory)

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Relationship between differential cohesion and synthetic differential geometry

I was wondering what is the relationship between differential cohesion and synthetic differential geometry? I know the basics of synthetic differential geometry from Kock's text, but I am not ...
2
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1answer
100 views

Yoneda Lemma for 2-categories - lax version

Is there a sort of lax Yoneda Lemma for 2-categories? Here is what I seem to have proven (although I have not checked all the details): If $\mathcal{C}$ is a (weak) 2-category, $A$ is an object of ...
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47 views

Why are morphisms of monads lax and not oplax natural transformations

Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them. A natural ...
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160 views

Textbooks on higher category theory

What textbooks on higher category theory are there? What books do you recommend? I am looking for self-contained introductions, no research reports. There are lots of informal summaries and arXiv ...
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42 views

Homotopy groups of mapping spaces

If I have an $\infty$-category $\mathcal{C}$ (AKA quasi-category), can I say anything about the homotopy groups of the mapping spaces $\mathrm{Hom}_\mathcal{C}(X,Y)$ for two objects $X$ and $Y$? ...
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104 views

Should an object in the category always be a formal mathematical structure?

Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So ...
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1answer
57 views

List of universal properties

At the moment I am looking into category theory and I am wondering if there exists a list of universal properties? I couldn't seem to find one.
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1answer
46 views

Coherence results

As far a I know, there are two kinds of results which are called 'coherence theorems'. For bicategories they take approximately the following forms: Every diagram of a certain form commutes A ...
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175 views

How to intrinsically think about simplicial objects.

It seems that a simplicial set should be thought as a space/thing, with encoded building information. The set of $n$-simplicies is the set of n-dimensional pieces and the boundary and degeneracy maps ...
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149 views

Why is it difficult to define n-category?

Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally. If we have a definition for 1-categories (category) and a definition for ...
2
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82 views

Higher transformations between natural transformations and so on

In category theory, arrows between categories are functors, arrows between functors are natural transformations, so a natural question is to ask what are arrows between natural transformations ? I've ...
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1answer
49 views

Example of endofunctor in Cat that is not a 2-functor.

Is there a good example of an endofunctor $\def\Cat{\operatorname{Cat}}\Cat \to \Cat$ (seeing $\Cat$ just as category) that is not a 2-functor?
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1answer
90 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 ...
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1answer
46 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 ...
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1answer
74 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 ...
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1answer
54 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 ...
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35 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 ...
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171 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 ...
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1answer
85 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 ...
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96 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
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1answer
87 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 ...
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2answers
81 views

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

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?
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2answers
103 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 ...
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26 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 ...
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1answer
60 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 ...
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1answer
66 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 ...
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81 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 ...
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1answer
135 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 ...
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3answers
178 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 ...
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1answer
25 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 ...
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52 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
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2answers
53 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 ...
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24 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 ...
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79 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?
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65 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
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1answer
74 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
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1answer
142 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
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1answer
79 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
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1answer
112 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 ...
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2answers
101 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 ...
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2answers
225 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 ...
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1answer
75 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 ...
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1answer
60 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 ...
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122 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 ...
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1answer
61 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 ...
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3answers
112 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 ...
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1answer
43 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 ...
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96 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 ...
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1answer
180 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 ...
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1answer
83 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 ...