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Your calculation is correct. For every two objects $X, Y \in \mathcal{C}$, the hom space $L^H\mathcal{C}(X,Y)$ has the right lifting property against $\partial \Delta^n \to \Delta^n$ for $n \geq 3$. F …

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 …

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 …

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 …

We will use the fact that $M$ is invertible. Let ${}_BN_A$ be an inverse to $M$. Thus we have isomorphisms $${}_AM \otimes_B N_A \cong {}_AA_A$$ and $${}_BN \otimes_A M_B \cong {}_BB_B$$ If we make th …

Alexander Campbell's guess is correct. Here is a reference. Lemma 10.2 of this paper Clark Barwick, Christopher Schommer-Pries, On the Unicity of the Homotopy Theory of Higher Categories, arXiv:1112. …

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 …

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 …

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 …

Suppose that $C$ is a complete closed monoidal category and $I$ is any small category. Then the functor category $Fun(I,C)$ is again a closed monoidal category with the pointwise tensor product $F \ot …

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 …

I assume you meant "symmetric monoidal functors". Yes, this seems to hold. By your description you have constructed maps: $$ \eta: \mathcal{C} \cong \Omega B \mathcal{C}$$ $$ \varepsilon: B \Omega \ …

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) …

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 …

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 …