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 id_{X} in Hom(*X*, *X*),
such that id_{X} *f* = *f*
and *f* id_{X} = *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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