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I think you prove this pretty much as one does in an ordinary abelian category. First note that the coequalizer $f,g: X \rightarrow Y$ is also given by the pushout of $(f, 1)$ and $(g, 1)$, both maps …

If $p$ is an inner fibration then the answer is yes (HTT.2.4.1.7). Lurie doesn't even define p-cocartesian unless $p$ is an inner fibration, but it's not hard to redefine the notion slightly more gen …

Though I don't know of an example off-hand, I don't believe it's the case that every cellular model category presents a presentable $\infty$-category. In practice, we can usually find a combinatorial …

Consider the diagram where the top left horizontal arrow is given by $\eta$. Observe that homotopy classes of lifts of the map $X_s \times \Delta^1 \rightarrow S$ are the same as homotopy classes o …

Here's a way to cook up lots of examples. Notice that $\Lambda^n_0 = \Delta^0 \star \partial \Delta^{n-1}$ so that a functor $\Lambda^n_0 \to \mathcal{C}$ is the same as a functor $\partial\Delta^{n-1 …

Here are a few observations... I think there exist stable infinity categories that are not the dg-nerve (resp. $A_\infty$-nerve) of a dg-category (resp. $A_\infty$ category). In particular, the cate …

Maybe I should record these as answers: (1) If you add "left proper" to the requirement then the answer is that there are lots of examples. A theorem of Dugger says that any left proper, cofibrantly …

The answer is 'no' (sorry I didn't realize this before!) I remembered someone showing me a weird counterexample in model categories a couple weeks ago (elaborated on from this paper by Rosicky: http: …

The lemma is true as stated. The Cartesian equivalence in question does not imply an equivalence in the Joyal model structure after forgetting the markings; rather, it implies an equivalence after for …

The way I always remember this stuff is as follows: Given a map $J \to \mathsf{Cat}$ form the associated cocartesian fibration $E \to J$. By assumption, $I$ is the actual colimit (as opposed to the l …

Here's a proof which is certainly overkill, but it has the merit of using references so you can read the proofs in detail. We have $i: \mathcal{C} \subset \mathcal{D}$ a fully faithful subcategory wi …

I like Maxime's argument better, but here's another. As you say, the case when $C=\bullet$ is well-known. But we can reduce to that case! The map $D_{/f} \to D$ is pulled back from $\mathsf{Fun}(C, D) …

By Corollary HA.4.8.5.20, the functor from $\mathbb{E}_{n+1}$-algebras to $\mathbb{E}_n$-monoidal categories and colimit-preserving, $\mathbb{E}_n$-monoidal functors is fully faithful. (Notice that th …

Let me propose an answer to the question which isn't quite what you ask for. In fact, for the most part I agree with Denis that the correct object really is $\mathsf{Fil}(\mathcal{C})$. Also, I should …

Since this example is kinda fun, let me spell it out. (The intuition should be clear though: the simplicial category I defined is really the result of taking a not-so-exciting (2,2)-category and apply …