Setting in Category Theory
In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings.
Read more about this topic: Algebraic Topology
Famous quotes containing the words setting in, setting, category and/or theory:
“A happy marriage perhaps represents the ideal of human relationship—a setting in which each partner, while acknowledging the need of the other, feels free to be what he or she by nature is: a relationship in which instinct as well as intellect can find expression; in which giving and taking are equal; in which each accepts the other, and I confronts Thou.”
—Anthony Storr (b. 1920)
“The setting was really perfect for a brisk bubbling murder....”
—Vladimir Nabokov (1899–1977)
“I see no reason for calling my work poetry except that there is no other category in which to put it.”
—Marianne Moore (1887–1972)
“The theory of rights enables us to rise and overthrow obstacles, but not to found a strong and lasting accord between all the elements which compose the nation.”
—Giuseppe Mazzini (1805–1872)