Accessible category

Let \kappa be an infinite regular cardinal, i.e. a cardinal number that is not the sum of a smaller number of smaller cardinals; examples are \aleph _{0} (aleph-0), the first infinite cardinal number, and \aleph_{1} , the first uncountable cardinal). A partially ordered set (I, \leq) is called \kappa-directed if every subset J of I of cardinality less than \kappa has an upper bound in I . In particular, the ordinary directed sets are precisely the \aleph_0-directed sets. Now let C be a category. A direct limit (also known as a directed colimit) over a \kappa-directed set (I, \leq) is called a \kappa-directed colimit. An object X of C is called \kappa-presentable if the Hom functor \operatorname{Hom}(X,-) preserves all \kappa-directed colimits in C. It is clear that every \kappa-presentable object is also \kappa'-presentable whenever \kappa\leq\kappa', since every \kappa'-directed colimit is also a \kappa-directed colimit in that case. A \aleph_0-presentable object is called finitely presentable. Examples • In the category Set of all sets, the finitely presentable objects coincide with the finite sets. The \kappa-presentable objects are the sets of cardinality smaller than \kappa. • In the category of all groups, an object is finitely presentable if and only if it is a finitely presented group, i.e. if it has a presentation with finitely many generators and finitely many relations. For uncountable regular \kappa, the \kappa-presentable objects are precisely the groups with cardinality smaller than \kappa. • In the category of left R-modules over some (unitary, associative) ring R, the finitely presentable objects are precisely the finitely presented modules. == -accessible and locally presentable categories==

The category C is called \kappa-accessible provided that: • C has all \kappa-directed colimits • C contains a set P of \kappa-presentable objects such that every object of C is a \kappa-directed colimit of objects of P. An \aleph_0-accessible category is called finitely accessible. A category is called accessible if it is \kappa-accessible for some infinite regular cardinal \kappa. When an accessible category is also cocomplete, it is called locally presentable. A functor F : C \to D between \kappa-accessible categories is called \kappa-accessible provided that F preserves \kappa-directed colimits. Examples • The category Set of all sets and functions is locally finitely presentable, since every set is the direct limit of its finite subsets, and finite sets are finitely presentable. • The category R-Mod of (left) R-modules is locally finitely presentable for any ring R. • The category of simplicial sets is finitely accessible. • The category Mod(T) of models of some first-order theory T with countable signature is \aleph_1 -accessible. \aleph_1 -presentable objects are models with a countable number of elements. • Further examples of locally presentable categories are finitary algebraic categories (i.e. the categories corresponding to varieties of algebras in universal algebra) and Grothendieck categories. ==Theorems==

One can show that every locally presentable category is also complete. Furthermore, a category is locally presentable if and only if it is equivalent to the category of models of a limit sketch. Adjoint functors between locally presentable categories have a particularly simple characterization. A functor F : C \to D between locally presentable categories: • is a left adjoint if and only if it preserves small colimits, • is a right adjoint if and only if it preserves small limits and is accessible. ==Notes==