Category theory glossary Free glossaries at translation jobs
Home Free Glossaries Free Dictionaries Post Your Translation Job! Free Articles Jobs for Translators

Category theory glossary

By Wikipedia,
the free encyclopedia,

Become a member of at just $12 per month (paid per year)


Use the search bar to look for terms in all glossaries, dictionaries, articles and other resources simultaneously

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



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[1]. (NB other authors use the term "quasicategory" with a different meaning [2].)
  • 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. [3]


A morphism f in a category is called:

  • an epimorphism provided that g = h whenever g\circ f=h\circ 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, g\circ f=g and f\circ h=h.
  • an inverse to a morphism g if g\circ f is defined and is equal to the identity morphism on the domain of f, and f\circ 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 f\circ g=f\circ h; e.g., an injection in Set. In other words, f is the dual of an epimorphism.


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.


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.


  1. ^ Adámek, Jiří; Herrlich, Horst, and Strecker, George E (2004) [1990] (PDF). Abstract and Concrete Categories (The Joy of Cats). New York: Wiley & Sons. p. 40. ISBN 0-471-60922-6. 
  2. ^ Joyal, A. (2002). "Quasi-categories and Kan complexes". Journal of Pure and Applied Algebra 175: 207–222. 
  3. ^


Published - March 2009

This glossary is available under the terms
of the GNU Free Documentation

Find free glossaries at

Find free dictionaries at

Subscribe to free newsletter

Need more translation jobs from translation agencies? Click here!

Translation agencies are welcome to register here - Free!

Freelance translators are welcome to register here - Free!

Submit your glossary or dictionary for publishing at

Free Newsletter

Subscribe to our free newsletter to receive news from us:



Use More Glossaries
Use Free Dictionaries
Use Free Translators
Submit Your Glossary
Read Translation Articles
Register Translation Agency
Submit Your Resume
Obtain Translation Jobs
Subscribe to Free Newsletter
Buy Database of Translators
Obtain Blacklisted Agencies
Vote in Polls for Translators
Read News for Translators
Advertise Here
Read our FAQ
Read Testimonials
Use Site Map


translation directory

christianity portal
translation jobs


Copyright © 2003-2024 by
Legal Disclaimer
Site Map