nLab
cofiber sequence

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Limits and colimits

Contents

Idea

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

Definition

Abstractly

For 𝒞\mathcal{C} an (infinity,1)-category with (infinity,1)-pushouts, a sequence of morphisms AfBCA \stackrel{f}{\to} B \to C is a cofiber sequence if there is an (infinity,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 (infinity,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 0KGH00\to K\to G\to H \to 0 is a short exact sequence of groups, then the corresponding maps of classifying spaces BKBGBH\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 02/200 \to \mathbb{Z}\xrightarrow{2} \mathbb{Z}\to \mathbb{Z}/2 \to 0 . Upon taking classifying spaces this becomes S 1S 1RP S^1 \to S^1 \to RP^{\infty}, in which the first map is a double cover whose cofiber is RP 2RP^2.

Revised on June 14, 2017 05:52:33 by Urs Schreiber (131.220.184.222)