Category of topological spacesIn mathematics, the category of topological spaces, often denoted Top, is the whose s are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds, with compactly generated spaces as objects and continuous maps as morphisms or with the .
Diagram (category theory)In , a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.
Complete categoryIn mathematics, a complete category is a in which all small s exist. That is, a category C is complete if every F : J → C (where J is ) has a limit in C. , a cocomplete category is one in which all small colimits exist. A bicomplete category is a category which is both complete and cocomplete. The existence of all limits (even when J is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a : for any two objects there can be at most one morphism from one object to the other.
Category of groupsIn mathematics, the Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a . The study of this category is known as group theory. There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to . M has two adjoints: one right, I: Mon→Grp, and one left, K: Mon→Grp. I: Mon→Grp is the functor sending every monoid to the submonoid of invertible elements and K: Mon→Grp the functor sending every monoid to the Grothendieck group of that monoid.
Category of ringsIn mathematics, the category of rings, denoted by Ring, is the whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is , meaning that the class of all rings is proper. The category Ring is a meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure.
ToposIn mathematics, a topos (USˈtɒpɒs, UKˈtoʊpoʊs,_ˈtoʊpɒs; plural topoi ˈtɒpɔɪ or ˈtoʊpɔɪ, or toposes) is a that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory.
Preadditive categoryIn mathematics, specifically in , a preadditive category is another name for an Ab-category, i.e., a that is over the , Ab. That is, an Ab-category C is a such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: and where + is the group operation. Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this term for certain special preadditive categories (see below).
EpimorphismIn , an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism f : X → Y that is right-cancellative in the sense that, for all objects Z and all morphisms , Epimorphisms are categorical analogues of onto or surjective functions (and in the the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion is a ring epimorphism. The of an epimorphism is a monomorphism (i.e. an epimorphism in a C is a monomorphism in the Cop).
Equaliser (mathematics)In mathematics, an equaliser is a set of arguments where two or more functions have equal values. An equaliser is the solution set of an equation. In certain contexts, a difference kernel is the equaliser of exactly two functions. Let X and Y be sets. Let f and g be functions, both from X to Y. Then the equaliser of f and g is the set of elements x of X such that f(x) equals g(x) in Y. Symbolically: The equaliser may be denoted Eq(f, g) or a variation on that theme (such as with lowercase letters "eq").
Adjoint functorsIn mathematics, specifically , adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e.
