R. Rosebrugh and R.J. Wood. An adjoint characterization of the category of sets. Proc. Amer. Math. Soc., 122(2):409--413, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... characterization of Chu(V; k) as the cofreely generated self dual closed category on V [Pav93] and Rosebrugh and Wood s appealing one axiom characterization of Set as the unique choice of V (any category) for which the Yoneda embedding Y V : V Set V op has a string of four left adjoints [RW94]. Another rationale for choosing V = Set is its accessibility as a universally understood category, with sets and functions nowadays being taught early on. Whereas category theory is unavoidable for the general V enriched case, the ordinary case can be treated quite comprehensively from the ....

R. Rosebrugh and R.J. Wood. An adjoint characterization of the category of sets. Proc. Amer. Math. Soc., 122(2):409--413, 1994.

....no language at all, and no equational theory beyond the equational tautologies x = x. There is therefore no mental plane to speak of in sets, making them the most physical of all the objects of traditional concrete (set based) mathematics, if not of all category theory (and perhaps even there, cf. [RW94]) Set op is equivalent to the category of complete atomic Boolean algebras (CABA s) But the free CABA generated by the set X is the power set 2 2 X . Hence the Boolean operations of each arity X , X empty, finite, or infinite, consist of all functions from 2 X to 2. This is the maximum ....

R. Rosebrugh and R.J. Wood. An adjoint characterization of the category of sets. Proc. Amer. Math. Soc., 122(2):409--413, 1994.

....at various times that it would appear as an exercise in some textbook. In 1979, a longer, but related, proof appeared in [F1] We advised the author of this history and sent him our proof. This was reported in [F2] but our proof was still not published. Now that there is actually an application [RW], we decided publication was in order. We have expressed the construction in a form we believe begs generalization to, for example, parametrized categories [SS] Note throughout that small can mean finite . For an object A of a category A, we let Idem(A) fe : A Gamma A : ee = eg denote the ....

Robert Rosebrugh and Richard Wood, An adjoint characterization of the category of sets, Proceedings AMS 122 (1994) 409-413.

....is often known as the Interpolation Lemma. 7) Entirely analogous to Example 6) is the idempotence of the wavy hom for a continuous category as in [4] 8) Also related to Example 5) is the string U a V a W a X a Y : set Gamma set set op , with Y the Yoneda embedding, which was shown in [13] to characterize set among categories with set valued homs. Here V Y has constant value 1 and XU has constant value ; 9) In [17] cofibrations were studied in the context of proarrow equipment. It was observed there that the defining adjoint strings for both left cofibrations and right ....

....Vol. 1, No. 6 140 = Z B C( Phi; Y B) Theta C(Y B;UV Psi) Z B B(X Phi; B) Theta C(Y B;UV Psi) and LG( Phi; Psi) UV Y )Z ( Phi; Psi) Z B UV Y ( Phi; B) Theta Z(B; Psi) Z B C( Phi; UV Y B) Theta C(Y B; Psi) In the case at hand, we recall from [13] that X Phi = Phi(1) V Psi = Psi( and US = S Deltaset( Gamma; where S Delta Gamma denotes S fold multiple. Now, in the last coend expression for GL( Phi; Psi) taking account of the Yoneda lemma, we have GL( Phi; Psi) R B set( Phi(1) B) Theta Psi( Deltaset(B; ....

Robert Rosebrugh and R. J. Wood. An adjoint characterization of the category of sets. Proc. Amer. Math. Soc., 122(2):409--413, 1994.

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