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A natural transformation Phi={Phi_C:F(C)->D(C)} between functors F,G:C->D of categories C and D is said to be a natural isomorphism if each of the components is an ...

In a category C, a terminal object is an object T in Ob(C) such that for any other object X in Ob(C), there is one, and only one morphism from X to T.

A category modeled after the properties of the category of sets. A category E is a topos if E has finite limits and every object of E has a power object (Barr and Wells 1985, ...

Abstract Algebra

A functor is said to be faithful if it is injective on maps. This does not necessarily imply injectivity on objects. For example, the forgetful functor from the category of ...

Let F,G:C->D be functors between categories C and D. A natural transformation Phi from F to G consists of a family Phi_C:F(C)->G(C) of morphisms in D which are indexed by the ...

A presheaf C of categories consists of the following data: 1. For every local homeomorphism f:Y->X of topological spaces X, Y, a category C(f:Y->X); 2. For every diagram f ...

A natural transformation Phi_Y:B(AY)->Y is called unital if the leftmost diagram above commutes. Similarly, a natural transformation Psi_Y:Y->A(BY) is called unital if the ...

A technical mathematical object which bears the same resemblance to binary relations as categories do to functions and sets.

The free product G*H of groups G and H is the set of elements of the form g_1h_1g_2h_2...g_rh_r, where g_i in G and h_i in H, with g_1 and h_r possibly equal to e, the ...