F. W. Lawvere. Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two), pages 134--145. Springer-Verlag, Berlin, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... has been a first important step towards an axiomatic theory of recursive types (see [Sim92] and [Fio94a, Chapters 6 8] Other work on algebraic compactness can be found in [Ad a93, Bar92] Concerning fixed points of endomorphisms, it was noticed by [HP90] after studying the work of [Law64, Law69] that in the presence of cartesian closure they are inconsistent with coproducts (empty or binary) Also algebraic compactness (which yields zero objects) is inconsistent with cartesian closure. This, in principle, precludes a unified treatment of sums, products, exponentials and recursive types ....

F.W. Lawvere. Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications II, volume 92 of Lecture Notes in Mathematics. Springer-Verlag, 1969.

....that the equational theory involving the fixedpoints is a conservative extension of the theory with primitive recursion only. It is well known that adding fixed point operators to the theory of coproducts is from the 1993 Conference on Rewriting Techniques and Applications inconsistent [Law69]; we think the analysis here sheds some additional light on that result (see the remarks at the end of the paper) Normalization implies, as usual, that closed normal form terms are numerals, abstractions, pairs, etc. depending on their type) In retrospect, this makes the use of expansions seem ....

....2 x 2 ) Then C[x:hx] a by induction the hypothesis, and C[x:hx] reduces to b directly. 2 The restrictions on the expansion rules have the unfortunate consequence that the reduction is not closed under substitution. We will have to be careful about this in certain places below. Lawvere [Law69] has pointed that one can do propositional logic in a bicartesian closed category and so cannot postulate fixed points for all maps. It will be useful to sketch the argument here. Fix any type A; the type B = A A will play the role of a boolean type, in which elements of the form oe 1 u and oe 2 v ....

F. W. Lawvere. Diagonal arguments and cartesian closed categories, in Category Theory, Homology Theory, and Their Applications II, LNM 92, SpringerVerlag, 1969.

....of the recursive type constructor . The categorical structure for interpreting the sum type constructor is standard; that is, given by binary coproducts. It then follows that the interpretation of the product type constructor Theta cannot be standard, viz. given by products (cf. [20, 16]) However, the category of games G admits a pretensor constructor Omega (see Section 2, x Type structure) which is a product in the subcategory, G t , of total innocent strategies [15] This situation, which is typical in models of FPC [9, 10] is the one that we axiomatise. The structure for ....

F. Lawvere. Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications II, volume 92 of LNM. SpringerVerlag, 1969.

.... has been a first important step towards an axiomatic theory of recursive types (see [Sim92] and [Fio94a, Chapters 6 8] Other work on algebraic compactness can be found in [Ad a93, Bar92] Concerning fixed points of endomorphisms, it was noticed by [HP90] after studying the work of [Law64, Law69] that in the presence of cartesian closure they are inconsistent with coproducts (empty or binary) Also algebraic compactness (which yields zero objects) is inconsistent with cartesian closure. This, in principle, precludes a unified treatment of sums, products, exponentials and recursive ....

F. W. Lawvere. Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications II, volume 92 of Lecture Notes in Mathematics. Springer Verlag, 1969.

....for domain equations involving the coproduct can be problematic, however. There are categorical impediments to the solution of some equations. For example, the equation D = 1 (D D) where 1 is the terminal object) has no solution in a any non trivial bicartesian closed category (see [12] and [6] Moreover, there are equations which have a non trivial solution in a bicartesian closed category but have no non trivial solution over the profinites. We provide a condition which, in effect, reduces the problem of solving an equation over the profinite domains to one of getting a ....

Lawvere, F. W., Diagonal arguments and cartesian closed categories. In: Category Theory, Homology Theory and their Aplications II. Lecture Notes in Mathematics, vol. 92, Springer-Verlag, 1969. 30 Carl A. Gunter

....a map g : I C in C, then fl(F S A(g) F S A(fl(g) and similarly, given a map f : B C in C, then fl(F S A(f) F S A(fl(f) 2. 3 Fixpoints in Cartesian Categories The main concern of this subsection will be a parametrised version of fixpoints in cartesian categories as introduced in [Law69]. Definition 2.12 Let C be a category with a terminal object 1. A map f ] 1 B is a fixpoint of the map f : B B iff f ] f ] f . A fixpoint operator is an operation on maps ( Gamma) B : C(B; B) Gamma C(1; B) such that f ] is a fixpoint of f for any map f : B B. ....

F. W. Lawvere. Diagonal arguments and cartesian closed categories. In P. Hilton, editor, Category Theory, Homology Theory and their Applications II, LNM, volume 92. Springer-Verlag, 1969.

....the standard axiomatization of its equational theory is sufficient for the coherence theorem. But when we add variants, the standard axiomatization of these features, while sufficient for coherence, clashes with the standard axiomatization of recursive types, yielding an inconsistent theory (see [Law69, HP89a] for variants, that is, coproducts) The solution lies in two observations: 1) the (too) strong axioms are only needed for coercion terms , and (2) in the various models we examined these coercion terms have special interpretations (such as strict, or linear maps) so special in fact, that they ....

....case M of l 1 ) inj l 1 ; Q) l n ) inj l n ; Q) where M : l 1 : t 1 ; l n : t n ] Q: l 1 : t 1 ; l n : t n ] t . More precisely, it is possible to check that the system fVART BETAg fVART COPg is equivalent to fVART BETAg fVART CRNg fVART ETAg. However, it is known [Law69, HP89a] that fVART BETAg fVART COPg is inconsistent with the existence of fixed points. In fact, this may be refined: Proposition 5 The system fVART BETAg fVART CRNg is (equationally) inconsistent with the existence of fixed points. Proof: The categorical equation f VART COP g may be thought of as ....

F. W. Lawvere. Diagonal arguments and cartesian closed categories. In Category theory, homology theory, and their applications II, pages 134--145, Lecture Notes in Mathematics, Vol. 92, Springer-Verlag, 1969.

....taken to be equal to # with a bottom element adjoined together with an element # unrelated to any other element but bottom, and equipped with appropriate extensions of the maps from the standard object of numerals. 2. 3 Fixpoints The main concern of this subsection is fixpoints as introduced in [Law69]. Definition 3. Let C be a category with a terminal object. A map h : 1 # B is a fixpoint of a map f : B # B i# h; f = f . A fixpoint operator for an object B is an operation on maps ( # B : C(B, B) # C(1, B) such that f # is a fixpoint of f for any map f : B # B. Given an ....

F. W. Lawvere. Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications II, LNM, volume 92. Springer-Verlag, 1969.

....X to Kan extensions. The new edition [Mac97] also contains important material on topos theory, 2categories, bicategories, and presheaves. Beyond that, I would recommend categorical logic and fibrations [Jac99, Pho92] enriched category theory [Law73a] and any further writing by Lawvere such as [Law70, Law69, Law91]. Links . http: www.mta.ca #cat dist categories.html . http: www.acsu.buffalo.edu #wlawvere ....

F. W. Lawvere. Diagonal arguments and cartesian closed categories. In Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two), pages 134--145. Springer-Verlag, Berlin, 1969.

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