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I think Theo's "no go" is exactly right. Here is an example which might make things easier to understand: Let X be any category. I am going to construct an interesting 2-category with one object which …

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 …

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 \ …

You can always do this. Take any $b$ and define $d = p_2 b$. Then $b' = (p_1, d)$ is equivalent to the original $b$. To see this note that $$(p_1, m) \circ b = (p_1b, m \circ (p_1 b, d)) \simeq id = …

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 …

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 …

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 …

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 …

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 …

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. …

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 …

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 …

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 …

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 …