1
vote
0answers
93 views

Retracts of 2-categories

Let $\mathbf{C}$ be a strict $2$-category and let $M = \{(x_\alpha,y_\alpha)\}$ be a collection of object-pairs so that each hom-category $\mathbf{C}(x_\alpha,y_\alpha)$ has an initial element ...
7
votes
1answer
174 views

Localizing 2-categories about a single morphism

This question is a straight-up reference request, but of course I will be grateful for an answer in the event that no references are readily available. Consider a strict $2$-category $\mathbf{C}$ and ...
1
vote
0answers
125 views

Axioms for a symmetric monoidal bicategory

I start reading the axioms for a symmetric monoidal bicategory. The axioms include so many diagrams to be satisfied. I am wondering if people really use these axioms directly to check a given data is ...
1
vote
2answers
166 views

Defining degeneracies for semi-simplicial sets with inner Kan conditions

Suppose we are given a category enriched over semi-simplicial sets, i.e. for the simplices in this category we have well-defined boundary maps, but no degeneracy maps. Suppose also that in this ...
6
votes
2answers
231 views

Weakening simplicial identities

The generators $d_i, s_i$ for morphisms of the simplicial category satisfy simplicial identities: $d_jd_i = d_id_{j−1}$ for $i < j$ $s_jd_i = d_is_{j−1}$ for $i < j$ $s_jd_i = id$ for $i = ...
3
votes
0answers
86 views

Bicategorical limits with parameters

(This question was asked in http://math.stackexchange.com/questions/741334/bicategorical-limits-with-parameters with no answer.) Let $F(-,-)\colon \mathcal{A}\times \mathcal{B}\to \mathcal{C}$ be a ...
4
votes
2answers
268 views

What's an initial object in a poset-enriched category?

I have a functor $F:C \to D$ between poset-enriched categories, and I'd like to show that the induced map on classifying spaces is a homotopy-equivalence. To this end, I am trying to establish the ...
1
vote
1answer
497 views

On the foundations for large categories

There are some basic discussions on the motivations of large categories and small categories [here] (On the large cardinals foundations of categories) and [here] (Large cardinal axioms and ...
5
votes
1answer
190 views

What's the link between topological spaces as locales and topological spaces as infinity-groupoids?

I've seen texts that talk about topological spaces being essentially locales, like Topology via Logic by Vickers, and texts related to homotopy theory that talk about topological spaces being ...
8
votes
1answer
277 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 ...
21
votes
0answers
618 views

categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result? Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and ...
3
votes
2answers
217 views

Localisation in a quasi-category

Let $W$ be a family of arrows in a category $\mathcal{C}$, there is a nature notion of localisation w.r.t. $W$. And if $W$ satisfies some nice properties, we have calculus of fraction. Now consider ...
7
votes
2answers
390 views

Categorification of coends and ends

I describe below a categorified version of the coend construction, "2-coend" for short. It takes as input a collection of 1-categories $\{W_{xy}\}$ which afford left and right representations of a ...
7
votes
0answers
421 views

Where else has Proposition B1.3.17 in the Elephant been proved?

(I asked the same question here and got some helpful comments, but thought I'd re-ask in case I get a more direct response.) This is a sort of reference request. Proposition B1.3.17 in Johnstone's ...
7
votes
1answer
379 views

n-categories enriched in (n+1)-categories

Recall the notion of an $n$-cateogry $C$ enriched in a symmetric monoidal category. Instead of a set of $n$-morphisms $mor(a, b)$ (where $a$ and $b$ are compatible $(n{-}1)$-morphisms), we have an ...
7
votes
1answer
302 views

Are reflective subcategories of complete infinity categories complete?

It is well known that reflective subcategories of complete categories are complete, and that limits in the subcategory are computed by taking the limit in the ambient category and applying the ...
5
votes
1answer
243 views

Pushforwards of stacks of algebras?

This is a refined/sheafified version of this previos question of mine. Let $(X,\mathcal{O}_X)$ be a ringed space or more in general a ringed stack, where the structure sheaf $\mathcal{O}_X$ is a ...
7
votes
0answers
525 views

TQFTs with target category of higher type than the source

In the classical version of the Cobordism Hypothesis, such as, e.g., in Jacob Lurie's On the Classification of Topological Field Theories, one considers the $\infty$-category of symmetric monoidal ...
8
votes
4answers
630 views

Higher categories in logic

I've read somewhere (probably in the nlab) that higher category theory has application in logic. By the way since now the only applications of higher category theory I've seen are in homotopy theory ...
0
votes
1answer
385 views

Weak algebraic structures

The following question can be thought as a sequel of this one. Here I'm looking for a big list of example of weak algebraic structures: here weak means that the structure (i.e. operations) need not ...
9
votes
0answers
428 views

Is there any elementary text unravelling the definitions of 2-category, lax functor and lax transformation, allowing people who do not know in the first place what these things are to really understand the definitions?

The question is in the title. My current research subject is the homotopy theory of $2$-categories. It may be slightly unreasonable to expect my PhD thesis to be read or even looked at by people ...
2
votes
4answers
295 views

Composition of composable 2-cells in a 2-category is unambiguously defined?

I believe the following nice statement is true, but I cannot find a reference or proof it myself. In a 2-category(i.e., bicategory), the composition of composable 2-cells is unambiguously defined. ...
35
votes
7answers
10k views

Is Mac Lane still the best place to learn category theory?

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ... Is Mac Lane still ...
6
votes
0answers
280 views

Rigorous recursive definition of $m$-algebras

Naively, $m$-algebras over a field $\mathbb{k}$ are easily defined recursively: the category of 0-algebras is the symmetric monoidal category of $\mathbb{k}$-vector spaces, and for $m>0$ an ...
6
votes
3answers
1k views

What is higher dimensional algebra?

Could anyone explain what higher dimensional algebra is? I tried to look on the web but I couldn't find a satisfactory definition, the ones that I found are too vague. What I'm looking for is a good ...
4
votes
2answers
452 views

Equivalence in $\infty$-categories

In every $n$-category (weak or strict) can be defined the concept of equivalence via a recursive definition: * an equivalence in a set ($0$-category) is just an identity; * for each $n \in \mathbb N$ ...
5
votes
1answer
245 views

Reference Request: Lax Ends

I've read in a few different places that the standard fact \[ \text{Nat}\,(F,G) \cong \int_x \text{Hom}\,(Fx,Gx) \] can be upgraded to ...
11
votes
4answers
381 views

What properties should a good definition of (weak) $n$-category satisfy?

My (perhaps inaccurate) impression is there are many competing definitions of the notion of a (weak) $n$-category, none of which are generally accepted. I've run across several properties such ...
6
votes
1answer
274 views

The plus construction for stacks of n-types

In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+1\right)$ times, and in general, ...