Diagonal functorIn , a branch of mathematics, the diagonal functor is given by , which maps as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the : a product is a universal arrow from to . The arrow comprises the projection maps. More generally, given a , one may construct the , the objects of which are called . For each object in , there is a constant diagram that maps every object in to and every morphism in to .
Pointed spaceIn mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as that remains unchanged during subsequent discussion, and is kept track of during all operations. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e.
Discrete categoryIn mathematics, in the field of , a discrete category is a category whose only morphisms are the identity morphisms: homC(X, X) = {idX} for all objects X homC(X, Y) = ∅ for all objects X ≠ Y Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set | homC(X, Y) | is 1 when X = Y and 0 when X is not equal to Y. Some authors prefer a weaker notion, where a discrete category merely needs to be equivalent to such a category.
Cone (category theory)In , a branch of mathematics, the cone of a functor is an abstract notion used to define the of that functor. Cones make other appearances in category theory as well. Let F : J → C be a in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of and morphisms in C. The J is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory.
Natural transformationIn , 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 involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called .
Functor categoryIn , a branch of mathematics, a functor category is a category where the objects are the functors and the morphisms are natural transformations between the functors (here, is another object in the category). Functor categories are of interest for two main reasons: many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable; every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.
Initial and terminal objectsIn , a branch of mathematics, an initial object of a C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists exactly one morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object.
