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, category and/or theory:
“May we two stand,
When we are dead, beyond the setting suns,
A little from other shades apart,
With mingling hair, and play upon one lute.”
—William Butler Yeats (1865–1939)
“The truth is, no matter how trying they become, babies two and under don’t have the ability to make moral choices, so they can’t be “bad.” That category only exists in the adult mind.”
—Anne Cassidy (20th century)
“Could Shakespeare give a theory of Shakespeare?”
—Ralph Waldo Emerson (1803–1882)