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Here is a way to see it which might constitute "just unfolding the definitions" (at least if one were sitting in on the class at Harvard when it was being taught). Set $Z(T) = Y(T^c)$, (compliment tak …

Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category. Stable $\infty$-categories give …

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 is an argument, which is basically Denis Nardin's comment. To have a model independent proof you need model independent definitions of the hocolim and of the localization. You can define them …

Yes, this follows easily by combining the results of Barwick-Kan and Toen. One way to rephrase your question is the following: Given a relative category $(C,W)$ (i.e. just a category with a subcat …

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 …

The completeness condition is not really about making things invertible which weren't already. It is about where the information about invertible morphisms is stored. We can already see this with $(\i …

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 …

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 …