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.

**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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send feedback and suggestions to bombelli at olemiss.edu – modified
23 jan 2016