An example is given by currying, which tells us that for any object , a map is the same as a map , where is the exponential object, given by all maps from to . In the case of topological spaces, if we take to be the unit interval, this leads to a duality between and , which then gives a duality between the reduced suspension, which is a quotient of , and the loop space, which is a subspace of . This then leads to the adjoint relation, which allows the study of spectra, which give rise to cohomology theories.
and a cofibration is defined by having the dual homotopy extension property, represented by dualising the previous diagram:
The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration we get the sequence
and given a cofibration we get the sequence
and more generally, the duality between the exact and coexact Puppe sequences.
This also allows us to relate homotopy and cohomology: we know that homotopy groups are homotopy classes of maps from the n-sphere to our space, written , and we know that the sphere has a single nonzero (reduced) cohomology group. On the other hand, cohomology groups are homotopy classes of maps to spaces with a single nonzero homotopy group. This is given by the Eilenberg–MacLane spaces and the relation
A formalization of the above informal relationships is given by Fuks duality.[1]