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1 vote

TR2 for homotopy category of stable $\infty$-category

I’d like to give some more details to complement Maxime’s answer. $$\require{AMScd} \begin{CD} 1 @>>> 2 @>>> 3 \\ @VVV @VVV @VVV \\ 4 @>>> 5 @>>> 6 \\ @. @VVV @VVV \...
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2 votes

Composition map in $\infty$-categories

I had also been wondering about this for a while. I think I've figured this out. First we observe that $\operatorname{St}_{\mathcal{C}}(\{x\})=\mathfrak{C}[\mathcal{C}](-,x)$; this follows by directly ...
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  • 223
4 votes
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TR2 for homotopy category of stable $\infty$-category

"This is equivalent to the assertion that the construction of the large diagram computes the suspension functor $\Sigma$" My previous answer was based on me misreading this quote :) You want ...
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10 votes
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Proposition A.2.6.15 in HTT

Retracts of weak equivalences are weak equivalences. Now if $f'$ is a retract of $f$ and you start with such a diagram with $f'$ on the left, you can create a new diagram with $f$, the same $X', X''$ ...
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2 votes
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Strictness of two operations on proarrow equipments

I believe the answer to (2) is yes. First, apply the strictification theorem for bicategories twice, to make composition of arrows and proarrows both strictly associative. Thus, when our equipment is ...
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5 votes
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Is every folk cofibration of strict $\omega$-categories a monomorphism?

I just thought (or maybe remember) a neat proof of this fact. It involve ideas I worked on a few years ago but never published - but that's short enough so that I can explain the key ideas on MO. Let ...
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9 votes
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Is there a Dold-Kan theorem for circle actions?

No, they are not equivalent, even for $C = Sp$. Indeed, the category of spectra with $S^1$-action is also the category of $\mathbb S[S^1]$-modules, and is compactly generated by a single object. On ...
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  • 8,310
3 votes
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Is there a "duality involution" on presentable categories?

The answer is no, even if you restrict to the full subcategory of $Pr^L$ spanned by the $Psh(C)$'s. I'll answer in the $1$-categorical case but : a- the $\infty$-categorical case follows because ...
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5 votes

Is there a "duality involution" on presentable categories?

$\newcommand\Pr{\mathit{Pr}}\newcommand\Pres{\mathit{Pres}}$Not an answer, but too long for a comment. Gabriel–Ulmer duality lends some intuition here. For simplicity I consider the finitely ...
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  • 5,045
5 votes
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Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?

Regarding monadicity (rather than comonadicity), the (2-categorical variant of the) question is answered in Bunge–Carboni's The symmetric topos. In their paper, $\mathbf A$ denotes the 2-category of ...
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