nLab cofiber sequence

Idea

A cofiber sequence is the dual notion to a fiber sequence.

Definition

Abstractly

For $\mathcal{C}$ an (∞,1)-category with (∞,1)-pushouts, a sequence of morphisms $A \stackrel{f}{\to} B \to C$ is a cofiber sequence if there is an (∞,1)-pushout square of the form

$$\array{ A &\stackrel{f}{\to}& B \\ \downarrow &\swArrow& \downarrow \\ * &\to& C }$$

in $\mathcal{C}$. We say that $C$ is the homotopy cofiber of $f$.

Presentation

Under mild conditions on a category with weak equivalences presenting $\mathcal{C}$ (such as a model category), homotopy cofibers are presented by mapping cones.

Specifically for cofiber sequences of topological spaces see at topological cofiber sequence.

Examples

In a stable (∞,1)-category, every fiber sequence is also a cofiber sequence and conversely.

Non-examples

In the unstable case, most fiber sequences are not cofiber sequences or conversely. For instance, if $0\to K\to G\to H \to 0$ is a short exact sequence of groups, then the corresponding maps of classifying spaces $\mathbf{B}K \to \mathbf{B}G \to \mathbf{B}H$ always form a fiber sequence, but not generally a cofiber sequence.

For a concrete counterexample, consider the short exact squence $0 \to \mathbb{Z}\xrightarrow{2} \mathbb{Z}\to \mathbb{Z}/2 \to 0$. Upon taking classifying spaces this becomes $S^1 \to S^1 \to RP^{\infty}$, in which the first map is a double cover whose cofiber is $RP^2$.

Related concepts

• long exact sequence in homology

  • long exact sequence in generalized homology

• stable (∞,1)-category