Historical notes Category theory

in 1942–45, samuel eilenberg , saunders mac lane introduced categories, functors, , natural transformations part of work in topology, algebraic topology. work important part of transition intuitive , geometric homology axiomatic homology theory. eilenberg , mac lane later wrote goal understand natural transformations. required defining functors, required categories.

stanislaw ulam, , writing on behalf, have claimed related ideas current in late 1930s in poland. eilenberg polish, , studied mathematics in poland in 1930s. category theory also, in sense, continuation of work of emmy noether (one of mac lane s teachers) in formalizing abstract processes; noether realized understanding type of mathematical structure requires understanding processes preserve structure. achieve understanding, eilenberg , mac lane proposed axiomatic formalization of relation between structures , processes preserve them.

the subsequent development of category theory powered first computational needs of homological algebra, , later axiomatic needs of algebraic geometry. general category theory, extension of universal algebra having many new features allowing semantic flexibility , higher-order logic, came later; applied throughout mathematics.

certain categories called topoi (singular topos) can serve alternative axiomatic set theory foundation of mathematics. topos can considered specific type of category 2 additional topos axioms. these foundational applications of category theory have been worked out in fair detail basis for, , justification of, constructive mathematics. topos theory form of abstract sheaf theory, geometric origins, , leads ideas such pointless topology.

categorical logic well-defined field based on type theory intuitionistic logics, applications in functional programming , domain theory, cartesian closed category taken non-syntactic description of lambda calculus. @ least, category theoretic language clarifies these related areas have in common (in abstract sense).

category theory has been applied in other fields well. example, john baez has shown link between feynman diagrams in physics , monoidal categories. application of category theory, more specifically: topos theory, has been made in mathematical music theory, see example book topos of music, geometric logic of concepts, theory, , performance guerino mazzola.

more recent efforts introduce undergraduates categories foundation mathematics include of william lawvere , rosebrugh (2003) , lawvere , stephen schanuel (1997) , mirroslav yotov (2012).