nLab homological functor

Definition

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

A homological functor

$$\pi \;\colon\; \mathcal{C}\longrightarrow \mathcal{A}$$

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

$$X\to Y\to Z\to \Sigma X$$

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

$$\dots \to \pi(X)\to \pi(Y)\to \pi(Z)\to \pi(\Sigma X) \to \dots$$

in $\mathcal{A}$. One writes

$$\pi_n \coloneqq \pi\circ \Sigma^{-n}$$

for $n \in \mathbb{N}$ and for $\Sigma$ the suspension functor.

Examples

Example

• $\mathcal{C}$ 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.

Related concepts

• homotopical functor

• Adams spectral sequence

• Lurie spectral sequence