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For example, if : N2 ! B is de ned by ( (m; n) = 1 m < n 0 else then 9n : N ! B is valid, and so is 8m 9n : 1 ! B . 10 To sum the previous discussion, classical rst order logic involves characteristic arrows and subobjects in the topos Set, and can thus be modeled in that topos. It is natural to conjecture that any topos can model rst order logic in an analogous fashion. Let T be a topos. We already know that and the various Sub(X) are lattices. To model rst order logic, we must do two things. First, we must provide with additional structure modeling logical implication; speci cally, we must de ne a material implication arrow ): !

52 CHRIS HILLMAN 3. Every mathematicalstatement, or proposition, mentioning variables x1; x2; : : :xn which are understood to belong to sets (or other objects) X1 ; X2 ; : : :Xn can be regarded as a function from X1 X2 : : :Xn to B . For example, consider ; : R ! B where ( (x) = 1 x > 2 0 else ( (x) = 1 x > 5 0 else and ' : N4 ! B where ( m m m '(j; k; `; m) = 1 j + k = ` 0 else In short, propositions are the characteristic functions of Set. 4. Every such proposition is associated with a truth set, namely the subset of the domain on which the proposition holds true.

Given two monic maps dom ! X dom take their pullback. ^ ^ dom de nes a subobject of A, whereas X^ ! dom de nes a subobject of B. ^ represent the same subobject A u B of X, which is a Moreover, ^= common subobject of A; B; moreover, by the de ning property of pullback squares it is the greatest common subobject, so we have found the desired greatest lower bound. Next, construct the sum dom + dom , giving the diagram X ???? dom x? ?? y dom ????! dom + dom Use the UMP of a sum to nd a unique arrow dom + dom !

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A Categorical Primer by Chris Hillman


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