Special and general types
Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
A cofiber sequence is the dual notion to a fiber sequence.
For an (infinity,1)-category with (infinity,1)-pushouts, a sequence of morphisms is a cofiber sequence if there is an (infinity,1)-pushout square
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. See there for more details.
In a stable (infinity,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 .
Revised on August 23, 2014 01:50:38
by Adeel Khan