By M. P. Fourman, P. T. Johnstone, A. M. Pitts (Editors)
Purposes of type concept and comparable issues of arithmetic to desktop technological know-how were a growing to be sector in recent times. This publication 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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Extra resources for Applications of Categories in Computer Science: Proceedings of the LMS Symposium, Durham 1991
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. 2. For the strictly increasing paths comonad, we again let 7/5 : D —> T2D be 7£>d = d?.
We plan to explore to what extent (and to what effect) the two approaches can be combined. For instance, given a comonad T and a monad P over the same category C one might obtain (assuming that T and P satisfy certain properties) a category of (T, P)-algorithms, in which a morphism from A to B is a morphism in C from TA to PB. We plan to investigate notions of computation in further domain-theoretic settings. As noted above, we are already working on categories of algorithms on (generalized) concrete data structures .
If T —• A is a tensorfree sequent of -F(C), (for example, is in the image of the unit,) then any derivation of F —• A is equivalent to a derivation in C, since with the cuts restricted to the tensor-free part of -F(C), none of the left or right introduction rules could be used in the derivation (they introduce tensors that could never be eliminated). • Remark: We believe more: viz. that the unit is faithful as well, but as of this writing some details remain to be checked. ) We have not considered the question of units T,± for the tensors ®,0—these can be added if wanted, together with the obvious rules and equations (as done in [Se89]).
Applications of Categories in Computer Science: Proceedings of the LMS Symposium, Durham 1991 by M. P. Fourman, P. T. Johnstone, A. M. Pitts (Editors)