homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Let throughout $\mathcal{C}$ be a stable (∞,1)-category, $\mathcal{A}$ an abelian category,
A homological functor
is an (∞,1)-functor that transforms every homotopy cofiber sequence
in $\mathcal{C}$ into a long exact sequence
in $\mathcal{A}$. One writes
for $n \in \mathbb{N}$ and for $\Sigma$ the suspension functor.
$\mathcal{C}$ is arbitrary, $\mathcal{A}$ is the category of abelian groups and $\pi$ is $\pi_0 \mathcal{C}(S,-)$ for some object $S\in\mathcal{C}$
$\mathcal{C}$ is equipped with a t-structure, $\mathcal{A}$ is the heart of the t-structure, and $\pi$ is the canonical functor.
$\mathcal{C} = D(\mathcal{A})$ is the derived category of the abelian category $\mathcal{A}$ and $\pi=H_0$.
Any of the above with $\mathcal{C}$ and $\mathcal{A}$ replaced by their opposite categories.
Any reduced generalized homology theory.
Last revised on May 10, 2016 at 11:47:38. See the history of this page for a list of all contributions to it.