inclusion I is an exact functor. This occurs if and only if C is closed under taking kernels and cokernels. Note that there are examples of full subcategories of an abelian category that are themselves abelian but where the inclusion functor is not exact, so they are not abelian subcategories (see below). C is a thick subcategory if it is closed under taking direct summands and satisfies the 2-out-of-3 property on short exact sequences; that is, if is a short exact sequence in A such that two of lie in C, then so does the third. In other words, C is closed under kernels of epimorphisms, cokernels of monomorphisms, and extensions. Note that P. Gabriel used the term thick subcategory to describe what we here call a Serre subcategory. C is a topologizing subcategory if it is closed under subquotients. C is a Serre subcategory if, for all short exact sequences in A we have M in C if and only if both are in C. In other words, C is closed under extensions and subquotients. These subcategories are precisely the kernels of exact functors from A to another abelian category. C is a localizing subcategory if it is a Serre subcategory such that the quotient functor admits a right adjoint. There are two competing notions of a wide subcategory. One version is that C contains every object of A (up to isomorphism); for a full subcategory this is obviously not interesting. (This is also called a lluf subcategory.) The other version is that C is closed under extensions. Here is an explicit example of a full, additive subcategory of an abelian category that is itself abelian but the inclusion functor is not exact. Let k be a field, the algebra of upper-triangular matrices over k, and the category of finite-dimensional -modules. Then each is an abelian category and we have an inclusion functor identifying the simple projective, simple injective and indecomposable projective-injective modules. The essential image of I is a full, additive subcategory, but I is not exact. History Abelian categories were introduced by (under the name of "exact category") and in order to unify various cohomology theories. At the time, there was a cohomology theory for sheaves, and a cohomology theory for groups. The two were defined differently, but they had similar properties. In fact, much of category theory was developed as a language to study these similarities.