Category Theory| 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 X → Y} ,
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 : G
→ G; An inner automorphism is one that is generated by a g
in G, and the map f : G → G 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: A
⊗ B 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: A ⊕
B 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.
@ References: Tull a1801 [quotient categories].
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]; Cho a1910-PhD [effectus theory and quantum mechanics].
> 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];
Leinster a1612 [intro];
Grandis 18 [and applications, for beginners];
Fong & Spivak 19 [applications].
@ 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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