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. (NB other authors use the term "quasicategory" with a different meaning.)
- 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 an identity morphism (of some object).
- 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 given 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.
- balanced if every bimorphism is an isomorphism.
- R-linear (R is a commutative ring) if A is locally small, each hom set is an R-module, and composition of morphisms is R-bilinear. The category A is also said to be over R.
Read more about this topic: Glossary Of Category Theory
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