nLab
homological functor

Context

Homotopy theory

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

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Definition

Let throughout 𝒞\mathcal{C} be a stable (∞,1)-category, 𝒜\mathcal{A} an abelian category,

A homological functor

π:𝒞𝒜 \pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A}

is an (∞,1)-functor that transforms every homotopy cofiber sequence

XYZΣX X\to Y\to Z\to \Sigma X

in 𝒞\mathcal{C} into a long exact sequence

π(X)π(Y)π(Z)π(ΣX) \dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots

in 𝒜\mathcal{A}. One writes

π nπΣ n \pi_n \coloneqq \pi\circ \Sigma^{-n}

for nn \in \mathbb{N} and for Σ\Sigma the suspension functor.

Examples

Example
  • 𝒞\mathcal{C} is arbitrary, 𝒜\mathcal{A} is the category of abelian groups and π\pi is π 0𝒞(S,)\pi_0 \mathcal{C}(S,-) for some object S𝒞S\in\mathcal{C}

  • 𝒞\mathcal{C} is equipped with a t-structure, 𝒜\mathcal{A} is the heart of the t-structure, and π\pi is the canonical functor.

  • 𝒞=D(𝒜)\mathcal{C} = D(\mathcal{A}) is the derived category of the abelian category 𝒜\mathcal{A} and π=H 0\pi=H_0.

  • Any of the above with 𝒞\mathcal{C} and 𝒜\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.