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6. Conclude that the objects + 1X ; are isomorphic in C=X. 7. Show that isomorphism classes of objects in C=X form a commutative monoid. Exercise: de ne a category S=C, called the coslice category, which is \dual" to the slice category C=S. The categories S=C and C=S are often called comma categories. Exercise: suppose C is a category in which products always exist. 1. Show that we can \multiply" the objects of the coslice category X=C. More precisely, given X ! M and X ! N de ne an arrow X !

Given an object (X; x) of TopZ, let H(X; x) = 1(X; x) be the homotopy group consisting of the homotopy' classes of paths in X which begin and end at x. Similarly, given an arrow (X; x) ! (Y; y) of TopZ let H' : hX ! hY be the group homomorphism de ned by ] 7! ) Verify that this de nes a functor from TopZ to Grp. (See, for instance, 8] for more about the homotopy functor). Exercise: Let X be a topological space. If T is the collection of open sets of X, considered as a poset under inclusion, let OX be the corresponding preorder category.

Exercise: verify that for any x; y 2 L, x ^ y and x _ y de ne a product and sum (respectively) in L. If L has a top element 1 (satisfying 1 ^ x = x for all x) and bottom element 0 (satisfying 0 _ x = x for all x), verify that these are nal and initial objects of L (respectively). If L is a distributive lattice with top and bottom elements 0; 1 (respectively), and if L has an exponential, L is said to be a Heyting algebra; in this context yx is written x ) y and is called the psuedocomplement of x relative to y.

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