By Chris Hillman
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Additional info for A Categorical Primer
J. L. Bell, Toposes and Local Set Theories. Oxford: Clarendon Press, 1988. 3. Garrett Birko and Saunders Mac Lane, A Survey of Modern Algebra. Fourth edition. New York: Macmillan, 1977. 4.
2. If 0 is initial in A, F 0 is initial in B. Dually, if 1 is nal in B, G 1 is nal in A. A CATEGORICAL PRIMER 39 3. F preserves all coproducts; in fact, all coequalizers; indeed, all colimits. Dually, G preserves all limits. Here is another example of a pair of adjoint functors, which is of considerable independent interest. Given an arrow ' : X E ! Y in Set, for each x 2 X we can de ne (x) to be the map taking e 7! '(x; e). This gives an arrow : X ! Y E , where we de ne Y E = f : E ! Y g. Conversely, given we can recover ' by observing that '(x; e) = (x)(e) = ev( (x); e) where setting ev( ; e) = (e) for all : E !
The rst problem is fairly easily solved. Let : ! be the projection onto the rst factor, and take the equalizer of _; : ! This gives a 10 Higher order logics exist (and are handily treated by topos theory), but we will not discuss them. 54 CHRIS HILLMAN subobject of whose characteristic arrow is the desired binary logical operator ): ! Exercise: if Set, verify that the equalizer of ; _ is f(x; y) : x ^ y = xg = f(x; y) : x yg Verify that the operator ): B B ! B agrees with our original de nition.
A Categorical Primer by Chris Hillman