equivalences in/of $(\infty,1)$-categories
In (∞,1)-category theory the typical notion of filtered object in category theory simplifies: instead of a sequential diagram of monomorphisms, it is just any tower diagram.
Let $\mathcal{C}$ be an (∞,1)-category.
A filtering on an object $X \in \mathcal{C}$ is a sequential diagram $X_\bullet \colon (\mathbb{Z}, \lt) \to \mathcal{C}$
such that
is the sequential homotopy colimit of the tower.
Dually a cofiltering of $X$ is a tower $X_\bullet$ such that
is the homotopy limit.
This appears as (Higher Algebra, def. 1.2.2.9).
If $\mathcal{C}$ is a stable (∞,1)-category with sequential limits/sequential colimits and with a t-structure, then every filtering/cofiltering on $X$ induces a spectral sequence of a filtered stable homotopy type which converges to the homotopy groups of $X$.
The spectral sequence of a filtered stable homotopy type associated with the chromatic tower (regarded as a filtered object in an (infinity,1)-category) is the chromatic spectral sequence (Wilson 13, section 2.1.2)
Last revised on May 22, 2022 at 02:04:10. See the history of this page for a list of all contributions to it.