cofiber sequence in nLab

Contents

Contents

Idea

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

Definition

Abstractly

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

A →f B ↓ ⇙ ↓ * → C \array{ A &\stackrel{f}{\to}& B \\ \downarrow &\swArrow& \downarrow \\ * &\to& C }

in 𝒞\mathcal{C}. We say that CC is the homotopy cofiber of ff.

Presentation

Under mild conditions on a category with weak equivalences presenting 𝒞\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→K→G→H→00\to K\to G\to H \to 0 is a short exact sequence of groups, then the corresponding maps of classifying spaces BK→BG→BH\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→ℤ→2ℤ→ℤ/2→00 \to \mathbb{Z}\xrightarrow{2} \mathbb{Z}\to \mathbb{Z}/2 \to 0 . Upon taking classifying spaces this becomes S 1→S 1→RP ∞S^1 \to S^1 \to RP^{\infty}, in which the first map is a double cover whose cofiber is RP 2RP^2.

• long exact sequence in homology

  • long exact sequence in generalized homology

• stable (∞,1)-category