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Let me try to give some intuition by examining two important examples. One should start from the definition: the suspension $ΣX$ is the universal choice of $Y$ filling of a square $$\require{AMScd} \ …

I think you made a sign mistake, and asked for a left adjoint to $\Omega^\infty$ since the monoid ring is a left along to the forgetful functor. If so then the answer is yes. $\Sigma^\infty_+:\mathr …

Since the homotopy category depends only on the 2-skeleton of the simplicial set, the easiest thing to do is to take the 2-skeleton of an ∞-groupoid. For example, let $E$ be the nerve of the contracti …

A significant technical improvement has been found by J. Shah in the theory of Kan extensions. Unfortunately I do not know of an exposition that does only the classical case, but reading the proof of …

A(n ∞-)category with $G$-action is just a functor $BG\to \mathrm{Cat}_∞$. Then, if $\mathcal{C},\mathcal{D}$ are (∞-)categories with $G$-action, we can get another (∞-)category with $G$ action $\mathr …

I already answered some version of this question in this answer, but let me try to expand a bit on the concrete advantages in mathematical practice. For understanding the following you need to take on …

TL DR: That is not enough. If you let $\psi_i:G(i)\to F(i)$ be the left adjoint of $\phi_i$ you also need the condition that for every $f:i\to j$ the canonical morphism $$\psi_jG(f)\to F(f)\psi_i$$ ad …

Yes, you can compute the mapping spaces in ∞-categories by taking the biggest Kan subcomplex of the internal hom. The trick is not to use the Joyal model structure, but instead the model structure on …

In the case where $\mathcal{C}$ is presentable, this is constructed in proposition 2.9 of Robalo, Marco, $K$-theory and the bridge from motives to noncommutative motives, Adv. Math. 269, 399-550 …

In fact what you need is that your ∞-category is additive (i.e. that it has direct sums and that the canonical commutative monoid structure on the mapping spaces is group-like). All stable categories …

I assume that with ``monoidal ∞-groupoid'' you mean an $E_1$-space. In this case the answer is yes. It is well known that $E_1$-spaces can be modeled by functors $$X:\Delta^{\mathrm{op}}\to \operatorn …

I'm not aware of anyone writing the proof down, but I think we can patch it together as an easy consequence of several facts in Lurie's Higher Topos Theory (henceforth HTT). The statement, as I under …

Let me add a short observation to Dylan's fantastic answer. There is indeed a more concrete construction of the symmetric monoidal structure on the $\infty$-category of spectra: it is the localized Da …

Let $\mathcal{C},\mathcal{D}$ two $\infty$-categories and $f:\mathcal{C}\to\mathcal{D}$ and $g:\mathcal{D}\to \mathcal{C}$ two functors. Recall (HTT.5.2.2.7) that a natural transformation $u:1_{\math …

As discussed in the comments, I'm writing here the proof of the following fact: Let $\mathscr{X}_∞$ be the ∞-topos of sheaves on $\mathrm{FinTop}^{op}$ under the atomic topology (the topology wher …