# Applications of Categories in Computer Science: Proceedings by M. P. Fourman, P. T. Johnstone, A. M. Pitts

By M. P. Fourman, P. T. Johnstone, A. M. Pitts

Purposes of class conception and comparable subject matters of arithmetic to computing device technological know-how were a transforming into sector in recent times. This e-book includes chosen papers at the topic from the London Mathematical Society Symposium held on the college of Durham in July 1991.

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5 Examples 1. The Kleisli category based on the increasing path comonad T\ is cartesian closed, with exponentiation structure built from the standard splitmerge combination, which form an isomorphism. 34 BROOKES AND GEVA : COMPUTATIONAL COMONADS Using the computational pairing Imerge and split2, we obtain intensional forms of currying, uncurrying, and application which we will call cuTrV/, unaTrry2 and appj. ry / (a) x id) = e a. Since split2 o Imerge = id, it follows that cu7ry/(unaTrry2 h) = h.

The functor fun maps each algorithm from Two x Two to Two into a function from Two x Two to Two. In particular, fun(pb) = fun(lb) = fun(rb) = fun(db) = b. Similarly, fun(pbn) = b for all n > 0, and fun(b*) = b. In fact, b* = alg(b). Figure 12 illustrates the result of taking the extensional quotient of Figure 6. Boxes enclose equivalence classes of algorithms, arcs between boxes represent the quotient ordering, and within each box we retain the pointwise order to ease comparison with Figure 6. As expected, the quotient figure is isomorphic to Figure 4 when we identify each equivalence class with the function computed.

COCKETT & SEELY: WEAKLY DISTRIBUTIVE CATEGORIES 55 Next, given a two-tensor-polycategory P , the weakly distributive category W(P) is just the category part of P , viz. those morphisms whose source and target are singletons. The distributivities are essentially given by the (cut) rule and the axioms (<8> R) and (© L). For instance, 6% is given as (note the "exchange"): A,C— And the coherence conditions follow from the equivalences for polycategories. It is clear from the constructions above that WP(W) is isomorphic toW; indeed they are the same category.