Morphisms in a sentence

Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.

Any class can be viewed as a category whose only morphisms are the identity morphisms.

Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph (augmented with loops as needed) where composition of morphisms is concatenation of paths.

For any given set I, the discrete category on I is the small category that has the elements of I as objects and only the identity morphisms as morphisms.

Hence, considering complete lattices with complete semilattice morphisms boils down to considering Galois connections as morphisms.

In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense.

On the other hand, some authors have no use for this distinction of morphisms (especially since the emerging concepts of "complete semilattice morphisms" can as well be specified in general terms).

The existence of identity morphisms and the composability of the morphisms are guaranteed by the reflexivity and the transitivity of the preorder.

Any preordered set (P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y. Between any two objects there can be at most one morphism.

A regular monomorphism equalizes some parallel pair of morphisms.

A ring object in C is an object R equipped with morphisms (addition), (multiplication), (additive identity), (additive inverse), and (multiplicative identity) satisfying the usual ring axioms.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor.

A small category with a single object is the same thing as a monoid : the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation.

Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.

By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them; we are studying the relationships between various classes of mathematical structures.

Category theory deals with abstract objects and morphisms between those objects.

Consider the category Grp of all groups with group homomorphisms as morphisms.

Definition If F and G are functors between the categories C and D, then a natural transformation η from F to G is a family of morphisms that satisfy two requirements.

Fibrations, Coverings Particular kinds of morphisms of groupoids are of interest.

For example, in game semantics, the category of games has games as objects and strategies as morphisms: we can interpret types as games, and programs as strategies.