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(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 …

Provided at least one of $M$ and $N$ is locally compact, the $\infty$-topos $\mathrm{Sh}(M \times N)$ is the product of $\mathrm{Sh}(M)$ and $\mathrm{Sh}(N)$ in $\mathrm{RTop}$. This is HTT 7.3.1.11. …

I was surprised (and delighted) to discover that the answer is no. Here is an example; in fact it's a special case of the example Mike referred to, but worked out in more detail. Let S be a set and …

Question 1: The model category $\mathcal{C}$ should be left proper, i.e. the pushout of a weak equivalence along a cofibration is again a weak equivalence. (Dually, there is a notion of right proper. …

Mostly I refer you to my answer here and also this question. To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a m …

Let A be a bicategory. Can I construct in some "natural" way a bicategory L(A) such that for every bicategory B, the bicategory of lax functors Lax(A, B) is (bi)equivalent to the bicategory of pseudo …

I would like to know the standard usage of "lax colimit" and "oplax colimit" in the 2-categorical literature. The nLab does not give an explicit definition of "lax colimit", as far as I can see, and …

In Higher Topos Theory, Lurie says that "Postnikov towers are convergent" in a presentable $\infty$-category $\mathcal{C}$ if $\mathcal{C}$ is equivalent to the $\infty$-category $\mathrm{Post}(\mathc …

The inclusion $\partial \Delta^n \times \Delta^1 \subseteq X(n+1)$ isn't any kind of anodyne extension, though. It's formed by attaching an n-simplex to $\partial \Delta^n \times \Delta^1$ with bound …

Certainly unbiased definitions are the norm in modern homotopy theory. I guess an example of a biased definition is the (original?) definition by Stasheff of an $A_\infty$ space—the homotopy theorist …

You might want to take a look at the responses to How to think about model categories? Even if you only cared about presentable (∞,1)-categories, you still might prefer the general (∞,1)-categorical …

$\newcommand{\SSet}{\mathsf{SSet}}\DeclareMathOperator{\Map}{Map}$First, let's record the fact that for any $A$ in $\SSet_{/S}$ and any right fibration $p : X \to S$, the simplicial set $\Map_{\SSet_{ …

Since the join of simplicial sets is associative and $\Delta^m = \Delta^0 \star \cdots \star \Delta^0$ ($m+1$ times), we should start by trying to understand things like $\Lambda^n_j \star \Delta^0$, …

Here is a counterexample for your next-to-last question. Let S be a set with more than one element and consider the two full subcategories of Cat on, respectively, the single category which is the di …

I'll denote your category of 2-vector spaces by 2Vect. By your preliminary remarks, 2Vect is actually the category of Vect-valued presheaves on Δ≤1 where Δ≤1 denotes the full subcategory of Δ on the …