B. J. Day. An embedding theorem for closed categories. In G.M. Kelly, editor, Category Seminar, Sydney, volume 420 of Lecture Notes in Mathematics, pages 55-64. Springer-Verlag, Berlin-New York, 1974.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for ; 0 free fragment) Suppose there are no occurrences of or 0 in and A. Then A i j= A Proof. The proof essentially follows that of Engberg and Winskel in [17] which is in turn similar to the proof the Yoneda embedding preserves both monoidal closed and Cartesian closed structure [12, 32]. Consider the net whose places are formulae in the ; 0 free fragment, and where there is a transition from fA 1 ; An g to fB 1 ; Bm g just if A 1 : An B 1 : Bm is provable in BI. Suppose interpretation i maps each propositional letter p to the set of markings which ....

B. J. Day. An embedding theorem for closed categories. In G.M. Kelly, editor, Category Seminar, Sydney, volume 420 of Lecture Notes in Mathematics, pages 55-64. Springer-Verlag, Berlin-New York, 1974.

....us to make a remark about full and faithful embeddings. Faithfulness is the semantic counterpart of a syntactic conservativity result, while fullness says that adding such structure does not cause any new maps to added, when we focus on just ccc or smcc types being embedded. Proposition 15 ([12]) Suppose C is a (small) symmetric monoidal closed category. Then the Yoneda embedding takes the symmetric monoidal closed structure of C to that on Set C op , as given by Day s construction. Since Set C op is a dcc, this can be read as showing the conservativity of over multiplicative, ....

....that Day s construction covers essentially all models of the (0; free fragment of : a cartesian dcc embeds into Set C op in a way that preserves all the relevant 26 structure. In this sense, it can truly be said that many of the semantic fundamentals of had already been worked out in [11, 12]. However, one further point is worth making: if we consider coproduct types then there are many models that fall outside the scope of the construction, as explicitly stated. The reason is that Yoneda does not preserve coproducts. But it can be made to preserve them, by replacing Set C op with ....

B. J. Day. An embedding theorem for closed categories. In G.M. Kelly, editor, Category Seminar, Sydney, volume 420 of Lecture Notes in Mathematics, pages 55-64. Springer-Verlag, Berlin-New York, 1974. 46

....fragment) Suppose there are no occurrences of or 0 in Gamma and A. Then Gamma A iff Gamma j= A Proof. The proof essentially follows that of Engberg and Winskel in [17] which is in turn similar to the proof the Yoneda embedding preserves both monoidal closed and Cartesian closed structure [12, 32]. Consider the net whose places are formulae in the ; 0 free fragment, and where there is a transition from fA 1 ; An g to fB 1 ; Bm g just if A 1 : An B 1 : Bm is provable in BI. Suppose interpretation i maps each propositional letter p to the set of markings which ....

B. J. Day. An embedding theorem for closed categories. In G.M. Kelly, editor, Category Seminar, Sydney, volume 420 of Lecture Notes in Mathematics, pages 55--64. Springer-Verlag, Berlin-New York, 1974.

....types being embedded. We can embed a ccc in a dcc in a trivial way, by regarding it as a dcc in which the two closed structures coincide. This implies the conservativity of the equality on ff terms given by cartesian dcc s over that for simply typed calculus. For smcc s we refer to a result of [8], which says that the Yoneda embedding takes symmetric monoidal closed structure on a small category C to that structure just described on Set C op . These embeddings raise the question of a purely functional understanding of ff. For example, we could formulate a model consisting of (certain, ....

B. J. Day. An embedding theorem for closed categories. In G.M. Kelly, editor, Category Seminar, Sydney, volume 420 of Lecture Notes in Mathematics, pages 55--64. Springer-Verlag, Berlin-New York, 1974.

....If C is a symmetric monoidal category, then Set C op is a bicartesian DCC. It is worth remarking that Day describes his results in much greater generality, in the context of enriched categories, so this gives us many more models than those mentioned in the proposition. In a separate paper [11], Day also shows that the Yoneda embedding preserves closed ctructure; this is an analogue of the standard fact that Yoneda preserves CCC structure. From this we may conclude that BI is conservative over MILL, again not only on the level of provability but also on the semantics of proofs. 3.2 ....

B. J. Day. An embedding theorem for closed categories. In G.M. Kelly, editor, Category Seminar, Sydney, volume 420 of Lecture Notes in Mathematics, pages 55--64. Springer-Verlag, Berlin-New York, 1974.

....products; compare [11] Proposition 2.7. For any promonoidal category A, the Yoneda embedding Y : A Gamma [A; V] op is a promonoidal functor (just use the definition and the Yoneda Lemma) The closure in [A; V] op of the representables Y (A) A(A; Gamma) under tensor products and unit (as in [4]) gives a full monoidal subcategory A 0 of [A; V] op , and Y factors through the inclusion via a promonoidal functor N : A Gamma A 0 . This construction has a universal property: to describe it we introduce the ordinary category PMon(A;B) whose objects are promonoidal functors Phi : ....

B.J. Day. An embedding theorem for closed categories. In Lecture Notes in Mathematics 420, pages 55--64, Springer, 1974.

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