9 votes
Accepted

The relation between t-structures and derived category

Assume $\mathcal D$ is a presentable stable $\infty$-category with a $\mathrm t$-structure (which is accessible and compatible with filtered colimits), and let $\mathcal A$ be its heart, $\mathcal{D(A)...
Maxime Ramzi's user avatar
  • 13.3k
7 votes

Heart of a bounded $t$-structure on the derived category of coherent sheaves

One can construct t-structures on the bounded derived category of coherent sheaves on a smooth projective curve (or higher-dimensional variety) by tilting, see Bayer's notes, Prop. 3.6.1, and the ...
Evgeny Shinder's user avatar
6 votes
Accepted

Chromatic t-structures?

To expand on Tim's answer, the arguments generalize to show that $Sp_{K(n)}$ admits no non-trivial t-structures in general. The crucial ingredient is that $Sp_{K(n)}$ has no non-trivial localising or ...
Piotr PstrÄ…gowski's user avatar
6 votes

Chromatic t-structures?

The second question turns out to have a surprisingly easy negative answer. This is depressing on two counts: both that the answer is negative and that it's so easy. Suppose that $Sp_{K(n)}$ has a $t$-...
6 votes

The relation between t-structures and derived category

I had reason to think about this a few years ago. When $\mathcal D$ arises as the derived category of an abelian category (with a possibly exotic $t$-structure), a construction of a realization ...
Dan Petersen's user avatar
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5 votes
Accepted

On various relations between "additional axioms" for AB4 and Grothendieck abelian categories

I don't think (3) implies (1). For example, the opposite category of the category of abelian groups satisfies (3), but is not AB5.
Jeremy Rickard's user avatar
4 votes

On various relations between "additional axioms" for AB4 and Grothendieck abelian categories

Obviously, (2) implies (1). Indeed, if directed colimits are preserved by a conservative exact functor taking values in a category where they are exact, then they are exact in the source category. ...
Leonid Positselski's user avatar
2 votes

Monoidality of truncation of spectra

A number of statements equivalent to preservation of $E_n$-algebras under truncation are given in a paper I wrote with Michael Batanin called "Bousfield Localization and Eilenberg-Moore Categories". ...
David White's user avatar
  • 29.4k

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