Category Theory  

In General
* History: Discovered by Eilenberg and MacLane in connection with the naturality of the universal coefficient theorem.
$ Concrete category: A class of objects Obj(A), together with arrows (morphisms or mappings)

Mor(A) = {Hom(X, Y) | X, Y in Obj(A), Hom(X, Y) is a set of arrows XY} ,

and a composition law Hom(X, Y) × Hom(Y, Z) → Hom(X, Z), for all X, Y, Z in Obj(A), such that (1) The composition is associative, h(gf) = (hg)f; and (2) for all X in Obj(A), there exists idX in Hom(X, X), such that idX f = f and f idX = f.
$ Equivalently: A category can be defined as a directed network with an associative composition and identities; Hom(X,Y) are the edges from X to Y.
* Remark: The arrows don't have to be necessarily interpreted as maps, but could for example be relationships.
@ References: Eilenberg & MacLane TAMS(45); Muller BJPS(01) [re foundations].
> Online resources: see MathWorld page; SEP page; Wikipedia page.

Morphism > s.a. Equivalence; Homomorphism.
$ Def: An element f in Hom(X, Y), where X and Y are some elements of a category A.
* Automorphism: A bijective endomorphism of an object in a category (onto itself) f : GG; An inner automorphism is one that is generated by a g in G, and the map f : GG can be written a \(\mapsto\) gag–1.
* Epimorphism: One such that gf = hf implies g = h; It is called endomorphism if it is from an object onto itself.
* Isomorphism: A morphism of an object onto itself.
* Monomorphism: One such that fd = fe implies d = e; Or, an f in Hom(A, B) such that for all g, g' in Hom(X, A), (fg = fg') implies (g = g'); The composition of monomorphisms is a monomorphism; For some categories (e.g., sets) it is the same as a 1-1 morphism, and can be seen as an embedding.
* Retraction: One such that ff = f.
> Online resources: see MathWorld page; Wikipedia page.

Constructions and Operations > s.a. limit [inductive], projective [limit]; Nerve; tensor [product].
$ Dual category: A* is defined by Obj(A*) = Obj(A) and, if the morphisms of A are {Hom(X, Y)}, Mor(A*) = {Hom*(X, Y) | X, Y in A}, where Hom*(X, Y):= Hom(Y, X).
$ Direct product: AB is an object C together with f in Hom(C, A), g in Hom(C, B), such that for all C', f ' in Hom(C', A), and g' in Hom(C', B), there exists h in Hom(C', C) such that f ' = hf and g' = hg; If it exists it is unique.
* Examples: In the usual cases it is the Cartesian product of the underlying sets, with some natural structure, and f, g are "projections"; E.g., the Cartesian product in the category of sets or the direct product of graphs; > s.a. manifolds.
$ Direct sum: AB is an object C together with f in Hom(A, C), g in Hom(B, C), such that for all C', f ' in Hom(A, C'), and g' in Hom(B, C'), there exists h in Hom(C, C') such that f ' = hf and g' = hg; If it exists it is unique.
* Example: Disjoint union for sets; > s.a. modules.

Related Concepts and Generalizations > s.a. categories in physics; Diagram; functor; Moduli Space; Subobject.
* Categorical logic: A branch of category theory, similar to mathematical logic but with a stronger connection to theoretical computer science; It represents both syntax and semantics by a category, and an interpretation by a functor; > see Wikipedia page.
@ n-categories: Baez & Dolan JMP(95); Baez qa/97-LNCS [intro].
@ Braided categories: Majid qa/95 [introduction, algebras and Hopf algebras].
@ Categorical logic: Cho et al a1512 [effectus theory].
> Higher category theory: see nLab page.

General References > s.a. types and examples; categories in physics; group theory; set theory.
@ Books: Mitchell 65; MacLane 71; Higgins 71; Schubert 72; Herrlich & Strecker 73; Strooker 78; Kelly 82 [enriched]; Lawvere & Schanuel 09 [I]; Awodey 10; Simmons 11 [intro]; Leinster 14 [intro, many exercises].
@ Short: see foreword of Isbell 64.
@ Other: Dodson 80 [and spacetime topology]; Oxtoby 87 [and measure theory]; Freyd & Scedrov 90; Landry PhSc(99)sep; Kaufmann & Ward a1312 [Feynman categories, categorical formulation for operations and relations].

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