Glossary of category theory

This is a glossary of properties and concepts in category theory in mathematics.

Categories

A category A is said to be:
* small provided that the class of all morphisms is a set (i.e., not a proper class); otherwise large.
* locally small provided that the morphisms between every pair of objects "A" and "B" form a set.
* Some authors assume a foundation in which the collection of all classes forms a "conglomerate", in which case a quasicategory is a category whose objects and morphisms merely form a conglomerate [cite book |last=Adámek |first=Jiří |coauthors=Herrlich, Horst, and Strecker, George E |title=Abstract and Concrete Categories (The Joy of Cats) |origyear=1990 |url=http://katmat.math.uni-bremen.de/acc/ |format=PDF |year=2004 |publisher= Wiley & Sons |location=New York |isbn=0-471-60922-6 |pages=40] . (NB other authors use the term "quasicategory" with a different meaning [cite journal|last=Joyal|first=A.|title=Quasi-categories and Kan complexes|journal=Journal of Pure and Applied Algebra|volume=175|year=2002|pages=207–222] .)
* isomorphic to a category B provided that there is an isomorphism between them.
* equivalent to a category B provided that there is an equivalence between them.
* concrete provided that there is a faithful functor from A to Set; e.g., Vec, Grp and Top.
* discrete provided that each morphism is the identity morphism.
* thin category provided that there is at most one morphism between any pair of objects.
* a subcategory of a category B provided that there is an inclusion functor from A to B.
* a full subcategory of a category B provided that the inclusion functor is full.
* wellpowered provided for each object "A" there is only a set of pairwise non-isomorphic subobjects.
* complete provided that all small limits exist.
* cartesian closed provided that it has a terminal object and that any two objects have a product and exponential.
* abelian provided that it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.
* normal provided that every monic is normal. [http://planetmath.org/encyclopedia/NormalCategory.html]

Morphisms

A morphism "f" in a category is called:
* an epimorphism provided that $g=h$ whenever $gcirc\; f=hcirc\; f$. In other words, "f" is the dual of a monomorphism.
* an identity provided that "f" maps an object "A" to "A" and for any morphisms "g" with domain "A" and "h" with codomain "A", $gcirc\; f=g$ and $fcirc\; h=h$.
* an inverse to a morphism "g" if $gcirc\; f$ is defined and is equal to the identity morphism on the domain of "f", and $fcirc\; g$ is defined and equal to the identity morphism on the codomain of "g". The inverse of "g" is unique and is denoted by "g" ^-1
* an isomorphism provided that there exists an "inverse" of "f".
* a monomorphism (also called monic) provided that $g=h$ whenever $fcirc\; g=fcirc\; h$; e.g., an injection in Set. In other words, "f" is the dual of an epimorphism.

Functors

A functor "F" is said to be:
* a constant provided that "F" maps every object in a category to the same object "A" and every morphism to the identity on "A".
* faithful provided that "F" is injective when restricted to each hom-set.
* full provided that "F" is surjective when restricted to each hom-set.
* isomorphism-dense (sometimes called essentially surjective) provided that for every "B" there exists "A" such that "F"("A") is isomorphic to "B".
* an equivalence provided that "F" is faithful, full and isomorphism-dense.
* amnestic provided that if "k" is an isomorphism and "F"("k") is an identity, then "k" is an identity.
* reflect identities provided that if "F"("k") is an identity then "k" is an identity as well.
* reflect isomorphisms provided that if "F"("k") is an isomorphism then "k" is an isomorphism as well.

Objects

An object "A" in a category is said to be:
* isomorphic to an object B provided that there is an isomorphism between "A" and "B".
* initial provided that there is exactly one morphism from "A" to each object B; e.g., empty set in Set.
* terminal provided that there is exactly one morphism from each object B to "A"; e.g., singletons in Set.
* zero object if it is both initial and terminal, such as a trivial group in Grp.

References