From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/178 Path: news.gmane.org!not-for-mail From: Andrej Bauer Newsgroups: gmane.science.mathematics.categories Subject: Assemblies without coproducts? Date: Thu, 19 Mar 2009 12:40:18 +0100 Message-ID: Reply-To: Andrej Bauer NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1237470510 12119 80.91.229.12 (19 Mar 2009 13:48:30 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 19 Mar 2009 13:48:30 +0000 (UTC) To: categories list Original-X-From: categories@mta.ca Thu Mar 19 14:49:47 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LkId1-0005n1-LA for gsmc-categories@m.gmane.org; Thu, 19 Mar 2009 14:49:47 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LkHy3-00043a-Cc for categories-list@mta.ca; Thu, 19 Mar 2009 10:07:27 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:178 Archived-At: This question is mostly for the realizabilitologists on the list. Let A be a PCA. The category of assemblies (or pers) over A has finite coproducts because any PCA contains true, false, and if-then-else. Let now A be a typed PCA (TPCA), according to John Longley. This means we have a non-empty set of types, operations * and -> on types (not necessarily freely generating the types). For each type t we have a set of values A_t. We require the K and S combinators to exist, as well as pairing and projections. We do NOT require that there be a boolean type, or a type of natural numbers. Some examples of TPCAs: - finite sets, with * and -> interpreted as cartesian product and exponential - Goedel's T - countably-based algebraic lattices - any PCA A where the type structure is then trivial and A_t = A. Assemblies over a TPCA are formed like the usual assemblies, except we have to specify underlying types. An assembly (S,t,|=) is a set S with a type t and a realizability relation |= between S and A_t. Now, do assemblies over a tpca A have binary coproducts? If A contains a type which resembles the booleans, we can do it. But I don't see how to do it in general. It's probably a trick involving higher-order functions. Andrej