Category theory

In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.

In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category.

In mathematics, a direct limit (also called inductive limit) is a colimit of a "directed family of objects".

Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms).

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive.

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse.

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor ⊗ : C × C → C which is associative up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism.

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved.

In mathematics, a ringed space can be equivalently thought of either (a) a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space.

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations.

In mathematics, categorical algebra is a subfield of algebra that approaches algebra from the categorical point of view.

In category theory, a field of mathematics, a category algebra is an associative algebra, defined for any locally finite category and commutative ring with unity.