Special and general types
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
Special notions
Variants
differential cohomology
Extra structure
Operations
Theorems
A cofiber sequence is the dual notion to a fiber sequence.
For an (∞,1)-category with (∞,1)-pushouts, a sequence of morphisms is a cofiber sequence if there is an (∞,1)-pushout square of the form
in . We say that is the homotopy cofiber of .
Under mild conditions on a category with weak equivalences presenting (such as a model category), homotopy cofibers are presented by mapping cones.
Specifically for cofiber sequences of topological spaces see at topological cofiber sequence.
In a stable (∞,1)-category, every fiber sequence is also a cofiber sequence and conversely.
In the unstable case, most fiber sequences are not cofiber sequences or conversely. For instance, if is a short exact sequence of groups, then the corresponding maps of classifying spaces always form a fiber sequence, but not generally a cofiber sequence.
For a concrete counterexample, consider the short exact squence . Upon taking classifying spaces this becomes , in which the first map is a double cover whose cofiber is .
Last revised on January 17, 2021 at 05:58:08. See the history of this page for a list of all contributions to it.