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The consequence is that we have given a functor C : Mon − → Ooc in a way which required arbitrary choices. The arbitrary choice of one object for C(M ) means that if we begin with a one-object category C , construct M = U (C ), and then construct C(M ), the result will not be the same as C unless it happens that the one object of C is M . Thus C ◦ U = idOoc , so that U is not the inverse of C. ) C is not surjective on objects, since not every small category with one object is in the image of C; in fact a category D is C(M ) for some monoid M only if the single object of D is actually a monoid and the arrows of D are actually the arrows of that monoid.

Thus (G ◦ F )i = Gi ◦ Fi for i = 0, 1. We note that the composition circle is usually omitted when composing functors so that we write GF (C) = G(F (C)). It is sometimes convenient to refer to a category CAT which has all small categories and ordinary large categories as objects, and functors between them. Since trying to have CAT be an object of itself would raise delicate foundational questions, we do not attempt here a formal definition of CAT. 5 Properties of Cat We note some properties without proof.

Another example is the functor which forgets all the structure of a semigroup. This is a functor U : Sem − → Set. There are lots of semigroups with the same set of elements; for example, the set {0, 1, 2} is a semigroup on addition (mod 3) and also a different semigroup on multiplication (mod 3). The functor U applied to these two different semigroups gives the same set, so U is not injective on objects, in contrast to the forgetful functor from monoids to semigroups. We will not give a formal definition of underlying functor.

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Category theory by Barr M.

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