Many important constructions in mathematics can be studied in this context. "Naturality" is a principle, like general covariance in physics, that cuts deeper than is initially apparent. An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions.
These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics. Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.
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Anonymous2013-09-01 2:48
Every functor F: Cightarrow D induces the opposite functor F^\mathrm{op}: C^\mathrm{op}ightarrow D^\mathrm{op}, where C^\mathrm{op} and D^\mathrm{op} are the opposite categories to C and D. By definition, F^\mathrm{op} maps objects and morphisms identically to F. Since C^\mathrm{op} does not coincide with C as a category, and similarly for D, F^\mathrm{op} is distinguished from F. For example, when composing F: C_0ightarrow C_1 with G: C_1^\mathrm{op}ightarrow C_2, one should use either G\circ F^\mathrm{op} or G^\mathrm{op}\circ F. Note that, following the property of opposite category, (F^\mathrm{op})^\mathrm{op} = F.