nLab homological functor

Definition

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

A homological functor

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

is an (∞,1)-functor that sends homotopy cofiber sequences

$$\cdots \longrightarrow X \longrightarrow Y \longrightarrow Z \longrightarrow \Sigma X \longrightarrow \cdots$$

in $\mathcal{C}$ to long exact sequence

$$\cdots \longrightarrow F(X) \longrightarrow F(Y) \longrightarrow F(Z) \longrightarrow F(\Sigma X) \longrightarrow \cdots$$

in $\mathcal{A}$. One writes

$$F_n \coloneqq F \circ \Sigma^{-n}$$

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 $F$ 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 $F$ is the canonical functor.

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

• 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