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Here are some rough analogies: Model Category :: $(\infty, 1)$-category Basis :: Vector space Local coordinates :: Manifold I especially like the last one. When you do, say, differential geometry …

One thing that might interest you is my result with Clark Barwick which gives an axiomatiation + uniqueness result for the homotopy theory of higher categories: arXiv:1112.0040 (i.e. $(\infty,n)$-cat …

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 us …

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 …

Yes, this is true. It follows form the work of Barwick and Kan on relative categories as a model for $\infty$-categories. The idea is similar to how Thomason's work shows that every homotopy type can …

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 exi …

As per Todd's suggestion I am posting this as an answer. The $(\infty, n)$-category of bordisms is an important example for many reasons, the most imporant of which is its role in the Baez-Dolan cob …

Yes, it is the same as $\mathbb{F}$. As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints …

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 t …

The answer is yes, fully-faithful functors are stable under co-base change. This is a model independent statement and so we can in particular take $\infty$-category to mean Segal categories. Then t …

This is not true. Here is a counter example. We let $\mathcal{C}_*$ be the following simplicial category. It has two objects 0 and 1. Their only endomorphisms are the identity. There are no morphisms …

First, as Rune pointed out in the comments, his paper with David Gepner gives a very general approach to your wish list. However to make it so general that it applies to arbitrary monoidal $(\infty,1) …

I might be confused about your question. Are you asking... How is trivializing the $O(n)$-action the same as giving an $O(n)$-equivariant non-degenerate trace? (as per Lurie's theorem 3.1.8). How ca …

Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can consid …

So I am trying to learn a bit more about Dominic Verity's model of higher categories, namely weak complicial sets. The underlying object is a stratified simplicial set which satisfies a sort of inner …