The higher-category-theory tag has no usage guidance, but it has a tag wiki.

**60**

votes

**7**answers

18k 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 ...

**58**

votes

**6**answers

18k views

### What are Jacob Lurie's key insights?

This question is inspired by this Tim Gowers blogpost.
I have some familiarity with the work of Jacob Lurie, which contains ideas which are simply astounding; but what I don't understand is which key ...

**51**

votes

**5**answers

6k views

### Why higher category theory?

This is a soft question.
I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...

**48**

votes

**2**answers

2k views

### What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...

**45**

votes

**7**answers

3k views

### Are higher categories useful?

Of course, personally, I think the answer is a big Yes!
However once, a while ago, while giving a talk about higher category theory, I was asked a question about whether higher category theory was ...

**44**

votes

**4**answers

2k views

### Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...

**42**

votes

**4**answers

7k views

### Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...

**37**

votes

**6**answers

4k views

### Non-examples of model structures, that fail for subtle/surprising reasons?

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...

**28**

votes

**4**answers

3k views

### Invertible matrices of natural numbers are permutations… why?

I have heard the following statement several times and I suspect that there is an easy and elegant proof of this fact which I am just not seeing.
Question: Why is it true that an invertible nxn ...

**26**

votes

**1**answer

853 views

### n-categorical description of Chern classes

The Chern classes of a rank $n$ vector bundle on $X$ are obtained from composing the associated classifying map in $[X, BU(n)]$ with the maps $BU(n) \to B^{2i} \mathbb{Z}$ corresponding to the ...

**25**

votes

**2**answers

1k views

### How do you define (infinity,1) categories in Homotopy Type Theory?

One of the major motivations of Homotopy Type Theory is that it naturally builds in higher coherences from the beginning. One important setting where higher coherence requirements get annoying is ...

**24**

votes

**2**answers

2k views

### generalisations of the Seifert-van Kampen Theorem?

I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.
My attention was ...

**23**

votes

**4**answers

2k views

### What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

In this post on the n-Category Café, Urs Schreiber says that, "The theory of G-principal bundles makes sense in any $(\infty,1)$-topos." I followed the link to the nLab and tried to chase definitions, ...

**23**

votes

**2**answers

3k views

### What's the current state of the classification of not-fully-extended TQFTs?

Recall that a $(k,k+1,\dots,k+n)$-TQFT is (supposed to be) a functor from the $n$-category whose $j$-morphisms are (isomorphism classes of) compact $(k+j)$-dimensional manifolds with boundary to some ...

**23**

votes

**4**answers

1k views

### Homotopy groups of spheres in a $(\infty, 1)$-topos

Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces).
You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, ...

**22**

votes

**3**answers

1k views

### t-structures and higher categories?

I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer).
Given a triangulated category, one ...

**21**

votes

**0**answers

722 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 ...

**20**

votes

**6**answers

2k views

### Concrete example of $\infty$-categories.

I've seen many different notion of $\infty$-categories, actual I've seen the operadic-globular ones of Batanin and Leinster and the opetopic too and eventually I'll see the simplicial ones too. ...

**20**

votes

**1**answer

3k views

### Is Lemma A.1.5.7 in Higher Topos Theory correct?

Hello to everyone,
I am studying the properties of combinatorial model categories, following the exposition given by Jacob Lurie in Higher Topos Theory ([HTT] from now on), in section A.2.6.
At some ...

**20**

votes

**2**answers

3k views

### Derived Algebraic Geometry and Chow Rings/Chow Motives

I recently heard a talk about Chow motives and also read Milne's exposition on motives. If I understand it correctly, the naive definition of the Chow ring would be that it simply consists of all ...

**20**

votes

**1**answer

748 views

### Can equivalences be strictified to isomorphisms?

In category theory there are lots of examples of isomorphisms that cannot be strictified to become identities. For instance, every monoidal category is equivalent to a strict monoidal category, where ...

**19**

votes

**2**answers

629 views

### Homotopy-theoretic derived Morita equivalences

Recall that two $k$-algebras $A, B$ are Morita equivalent iff their categories of left modules are equivalent. However, this relation turns out to be rather fine and one introduces a coarser ...

**19**

votes

**3**answers

2k views

### Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...

**18**

votes

**3**answers

1k views

### What do whitehead towers have to do with physics?

First let me say something that I don't completely understand, since I do not know enough physics. If I say anything wrong, someone please tell me:
For the spinning particle, there is a sigma-model, ...

**18**

votes

**2**answers

592 views

### Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...

**18**

votes

**1**answer

636 views

### How to stop worrying about enriched categories?

Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...

**17**

votes

**6**answers

2k 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 ...

**17**

votes

**3**answers

1k views

### What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a ...

**16**

votes

**2**answers

884 views

### Internal categories in simplicial sets

Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories?
Note that this ...

**16**

votes

**2**answers

1k views

### Why do my quantum group books avoid homotopical language?

I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.
Many notes closer to "Kontsevich ...

**16**

votes

**1**answer

688 views

### Lurie's approach to the bar-cobar adjunction

I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...

**15**

votes

**2**answers

1k views

### Lax Functors and Equivalence of Bicategories?

Lax functors of bicategories were introduced at the very inception of bicategories, and I'm trying to get a better feel for them. They are the same as ordinary 2-functors, but you only require the ...

**15**

votes

**1**answer

635 views

### Skeleton category of the category of skeleton categories?

A category is a skeleton if, roughly speaking, no two distinct objects within the category are isomorphic. To every category is associated a skeleton, and two categories are categorically "equivalent" ...

**15**

votes

**2**answers

1k views

### What's an example of an “adjunction up to adjunction”?

(There are several dialects of 2-categorical language. Mine is the one where everything is weak by default. You might need to insert the word "weak" or "pseudo" (or even "strong", if your default is ...

**14**

votes

**1**answer

930 views

### Are there non-categorical notions in topos theory?

Suppose that $\mathcal{T}$ is an abstract $2$-category we know is equivalent to the $2$-category of Grothendieck topoi via some equivalence $$\phi:\mathcal{T} \to \mathfrak{Top},$$ and let $E$ be an ...

**14**

votes

**2**answers

686 views

### Are the axioms for higher category-theory effectively computable?

I ask this, although I don't conduct any research in the area, or even plan to. -- There seems to be general agreement that the axioms for higher categories grow very rapidly in complexity as the ...

**14**

votes

**1**answer

480 views

### Is there a good notion of “induction” for representations of 2-categories?

One of the most important observations in the representation theory of algebras is that if one has a subalgebra $A\subset B$, then these is a functor $B\otimes_A -\colon A\operatorname{-pmod}\to ...

**14**

votes

**2**answers

524 views

### Infinite dimensional 2-Hilbert spaces

Is there a definition of an infinite dimensional 2-Hilbert space?
Finite dimensional 2-Hilbert spaces have been discussed by Baez in
http://arxiv.org/abs/q-alg/9609018
In the more recent paper by ...

**14**

votes

**0**answers

121 views

### Specific cases of the tangle hypothesis in terms of “classical” n-categories

As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal ...

**13**

votes

**1**answer

1k views

### How much do universes matter in topos theory?

Suppose we fix two Grothendieck universes $\mathcal{U} \in \mathcal{V}.$ Then one has that $\mathcal{U}$-$\mathbf{Set},$ the category of $\mathcal{U}$-small sets, is a locally $\mathcal{U}$-small, ...

**13**

votes

**1**answer

679 views

### compact objects in model categories and $(\infty,1)$-categories

In an ordinary category $C$, one says that an object $X$ is $\kappa$-compact if the representable functor $Hom(X,-)\colon C \to Set$ preserves $\kappa$-filtered colimits. We say $C$ is locally ...

**13**

votes

**3**answers

618 views

### Is there a tricategory of bicategories and biprofunctors?

Background
There is a bicategory where the objects are categories, the 1-morphisms are profunctors, and the 2-morphisms are morphisms of profunctors. The non-obvious part of this assertion is that ...

**13**

votes

**2**answers

474 views

### Model for the (infinity,1)-category of (homotopy-)limit preserving functors

I've got a simplicial model category $M.$ I'd like to get my hands on the (infinity,1) category of homotopy limit preserving functors from M to Spaces in order to compare it to another simplicial ...

**13**

votes

**2**answers

651 views

### Homotopy Fixed Points of SO(2) on Fully Dualizable Algebras

Note: by fixed points, I always mean homotopy fixed points.
As explained in Jacob Lurie's paper on the cobordism hypothesis, we have an action of O(2) on the $\infty $-groupoid $X$ given by ...

**13**

votes

**1**answer

560 views

### The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...

**13**

votes

**2**answers

432 views

### A model category of abelian categories?

Let $\mathcal{M}$ be the following category:
The objects are small abelian categories with chosen zero object, biproducts, kernels, and cokernels.
The morphisms are functors that preserve the ...

**13**

votes

**1**answer

926 views

### A Model Category of Segal Spaces?

So in Julie Bergner's work on (infty, 1)-categories arXiv:0610239, she considers several model categories which model (infty, 1)-categories, which are known to be equivalent. I'm guessing that there ...

**13**

votes

**1**answer

370 views

### Unobstructedness of braided deformations of symmetric monoidal categories in higher category theory

Let $k$ be a field of characteristic zero, and $\mathcal{C}$ be a $k$-linear additive symmetric monoidal category. A braided deformation of $\mathcal{C}$ over a local artin ring $R$ with residue ...

**13**

votes

**2**answers

768 views

### Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...

**13**

votes

**0**answers

1k views

### Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...