From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8310 Path: news.gmane.org!not-for-mail From: Harley Eades III Newsgroups: gmane.science.mathematics.categories Subject: Re: Functional programming and William Lawvere's notion of "variable sets"(vs topos of "static sets") in a Topos ... Date: Thu, 4 Sep 2014 08:07:12 -0400 Message-ID: References: Reply-To: Harley Eades III NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Mac OS X Mail 7.3 \(1878.6\)) Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1409842887 31864 80.91.229.3 (4 Sep 2014 15:01:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 4 Sep 2014 15:01:27 +0000 (UTC) Cc: Categories mailing list To: "Vasili I. Galchin" Original-X-From: majordomo@mlist.mta.ca Thu Sep 04 17:01:21 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XPYXJ-0007AR-6K for gsmc-categories@m.gmane.org; Thu, 04 Sep 2014 17:01:21 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:41124) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XPYWP-0002yB-7G; Thu, 04 Sep 2014 12:00:25 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XPYWN-0003Py-Mo for categories-list@mlist.mta.ca; Thu, 04 Sep 2014 12:00:23 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8310 Archived-At: Hi, Vasya. Something you might find interesting is that stateful computations can = be=20 modeled =97 very nicely in fact =97 using co-algebras. They have been = used to give a formal semantics to object oriented (OO) programming = languages,=20 as well, as embedding OO in purely functions programming languages. I am not aware of a connection between co-algebras and variable sets, = but I am not familiar with the latter. I just thought you might find these interesting. Thanks, .\ Harley Here are a list of references from my DB: @article{Setzer:2006, Author =3D {Anton Setzer}, Journal =3D {Trends in Functional Programming}, Title =3D {Object-Oriented Programming in Dependent Type = Theory}, Year =3D {2006}} @article{Jacobs:2003, Author =3D {Bart Jacobs and Erik Poll}, Journal =3D {Theoretical Computer Science}, Number =3D {3}, Pages =3D {329 - 349}, Title =3D {Coalgebras and monads in the semantics of Java}, Volume =3D {291}, Year =3D {2003}} @article{Jacobs:2002b, Author =3D {Jacobs, Bart}, Journal =3D {Mathematical. Structures in Comp. Sci.}, Month =3D dec, Number =3D {6}, Pages =3D {875--903}, Title =3D {The Temporal Logic of Coalgebras via Galois = Algebras}, Volume =3D {12}, Year =3D {2002}} @incollection{Jacobs:2002, Author =3D {Jacobs, Bart}, Booktitle =3D {Algebraic and Coalgebraic Methods in the = Mathematics of Program Construction}, Editor =3D {Backhouse, Roland and Crole, Roy and Gibbons, = Jeremy}, Pages =3D {237-281}, Publisher =3D {Springer Berlin Heidelberg}, Series =3D {Lecture Notes in Computer Science}, Title =3D {Exercises in Coalgebraic Specification}, Volume =3D {2297}, Year =3D {2002}} @article{Jacobs:2002a, Author =3D {Bart Jacobs}, Journal =3D {Electronic Notes in Theoretical Computer Science}, Number =3D {1}, Pages =3D {112 - 134}, Title =3D {Comprehension for Coalgebras}, Volume =3D {65}, Year =3D {2002}} @article{Rothe:2001, Author =3D {Jan Rothe and Hendrik Tews and Bart Jacobs}, Journal =3D j-jucs, Month =3D {feb}, Number =3D {2}, Pages =3D {175--193}, Title =3D {The Coalgebraic Class Specification Language CCSL}, Volume =3D {7}, Year =3D {2001}} @article{Kurz:20011, Author =3D {Alexander Kurz}, Journal =3D {Theoretical Computer Science}, Number =3D {1--2}, Pages =3D {119 - 138}, Title =3D {Specifying coalgebras with modal logic}, Volume =3D {260}, Year =3D {2001}} @article{Jacobs:2000, Author =3D {Bart Jacobs}, Journal =3D {Electronic Notes in Theoretical Computer Science}, Number =3D {0}, Pages =3D {160 - 195}, Title =3D {Towards a Duality Result in Coalgebraic Modal Logic}, Volume =3D {33}, Year =3D {2000}} @article{Hensel:1999, Author =3D {Hensel, U and Jacobs, B}, Journal =3D {Journal of Logic and Computation}, Number =3D {4}, Pages =3D {463-500}, Title =3D {Coalgebraic theories of sequences in PVS}, Volume =3D {9}, Year =3D {1999}} @article{Jacobs:1999, Author =3D {Bart Jacobs}, Journal =3D {Newsletter 3 of the Dutch Association for = Theoretical Computer Science (NVTI)}, Pages =3D {15-27}, Title =3D {Coalgebras in Specification and Verification for = Object-Oriented Languages}, Year =3D {1999}} @article{Jacobs:1998a, Author =3D {Jacobs, Bart and van den Berg, Joachim and Huisman, = Marieke and van Berkum, Martijn and Hensel, U. and Tews, H.}, Journal =3D {SIGPLAN Not.}, Month =3D oct, Number =3D {10}, Pages =3D {329--340}, Title =3D {Reasoning About Java Classes: Preliminary Report}, Volume =3D {33}, Year =3D {1998}} @incollection{Hensel:1998, Author =3D {Hensel, Ulrich and Huisman, Marieke and Jacobs, Bart = and Tews, Hendrik}, Booktitle =3D {Programming Languages and Systems}, Editor =3D {Hankin, Chris}, Pages =3D {105-121}, Publisher =3D {Springer Berlin Heidelberg}, Series =3D {Lecture Notes in Computer Science}, Title =3D {Reasoning about classes in object-oriented languages: = Logical models and tools}, Volume =3D {1381}, Year =3D {1998}} @article{Jacobs:1998, Author =3D {Bart Jacobs}, Journal =3D {Electronic Notes in Theoretical Computer Science}, Number =3D {0}, Pages =3D {231 - 242}, Title =3D {Coalgebraic Reasoning about Classes in = Object-Oriented Languages}, Volume =3D {11}, Year =3D {1998}} @incollection{Jacobs:1996, Author =3D {Jacobs, Bart}, Booktitle =3D {Object Orientation with Parallelism and = Persistence}, Editor =3D {Freitag, Burkhard and Jones, CliffB. and Lengauer, = Christian and Schek, Hans-J{\"o}rg}, Pages =3D {83-103}, Publisher =3D {Springer US}, Series =3D {The Kluwer International Series in Engineering and = Computer Science}, Title =3D {Objects And Classes, Co-Algebraically}, Volume =3D {370}, Year =3D {1996}} On Aug 4, 2014, at 3:03 AM, Vasili I. Galchin = wrote: > Dear Cat People ... >=20 > I have been re-reading William Lawvere's description of "variable > sets" (a functor category) in "a elementary topoi" on a discrete > poset/preset vs a "abstract sets" in the topos of abstract sets . In > Haskell( and other FPLs .. ) aren't "pure" computations just over the > topos of abstract sets and "mutable" /"time-varying/stateful" > computations aren't just "variable sets" (i.e. a functor over a > discrete preset-poset)?? Please forgive my germ of an idea >=20 > Kind regards, >=20 > Vasya >=20 > P.S. Yes I am familiar with Eugenio Moggi's papers on using monads for > stateful computations .. >=20 >=20 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]