might not be injective. Consequently, theorems or techniques (such as homology, cohomology or homotopy theory) which are only dependent on the homotopy class of spaces and maps involved may be applied to with the assumption that and that is actually the inclusion of a subspace. Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of to points of and hence of embedding within despite the fact that the function need not be one-to-one. Categorical application and interpretation One can use the mapping cylinder to construct homotopy colimits: this follows from the general statement that any category with all pushouts and coequalizers has all colimits. That is, given a diagram, replace the maps by cofibrations (using the mapping cylinder) and then take the ordinary pointwise limit (one must take a bit more care, but mapping cylinders are a component). Conversely, the mapping cylinder is the homotopy pushout of the diagram where and . Mapping telescope Given a sequence of maps the mapping telescope is the homotopical direct limit. If the maps are all already cofibrations (such as for the orthogonal groups ), then the direct limit is the union, but in general one must use the mapping telescope. The mapping telescope is a sequence of mapping cylinders, joined end-to-end. The picture of the construction looks like a stack of increasingly large cylinders, like a telescope. Formally, one defines it as See also Cofibration Mapping cylinder (homological algebra) Homotopy colimit Mapping path space, which can be viewed as the mapping cocylinder References Algebraic topology