S which both have the same (two-sided) unit element e and which satisfy the interchange identity (5). Prove that· and are equal, and that each is commutative. 6. Combine Exercises 4 and 5 to prove that the fundamental group ofa topological group is abelian. 7.

Thus our definition of product category gives an automatic definition of "functor of two variables" ~ just as the definition of the product X x Y of two topological spaces gives an automatic definition of "continuous function of two variables". Fix one argument in a bifunctor S; the result is an ordinary functor of the remaining argument. The whole bifunctor S is determined by these two arrays of one-variable functors in the following elementary way. Proposition 1. Let B, C, and D be categories.

Let C( U) denote the set of all continuous real-valued functions h : U -+ R; the assignment h r-. hi V restricting each h to the subset V is a function C(U)-+C(V) for each V CU. This makes C a contravariant functor on Open (X) to Set. This functor is called the sheaf of germs of continuous functions on X. On a smooth manifold, the sheaf of germs of C'~) -differentiable functions is constructed in similar fashion (cf. Mac Lane-Moerdijk [1992]). Mod-R is a contravariant functor from rings R to categories.