From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1431 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: Weak algebraic structures Date: Mon, 28 Feb 2000 14:20:46 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017826 31222 80.91.229.2 (29 Apr 2009 15:10:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:10:26 +0000 (UTC) Cc: T.Leinster@dpmms.cam.ac.uk (Tom Leinster) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Mar 4 02:38:20 2000 -0400 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id BAA21033 for categories-list; Sat, 4 Mar 2000 01:55:00 -0400 (AST) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: ELM [version 2.5 PL1] Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 56 Xref: news.gmane.org gmane.science.mathematics.categories:1431 Archived-At: The following two papers on homotopy-algebraic structures - or "weakened" algebraic structures, if you prefer - are now available. The first one, "Up-to-Homotopy Monoids", is 8 pages long and is essentially a set of notes for the talk I gave at the Louvain-la-Neuve PSSL in October. It can serve as an introduction to the second one, "Homotopy Algebras for Operads" (100 pages, but don't let that scare you: it should be easy for category theorists). Abstracts are below. Tom Leinster * * * "Up-to-Homotopy Monoids" Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold `up to homotopy' in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. It is a much-abbreviated version of the paper `Homotopy Algebras for Operads', and does not assume any knowledge of operads. Available on the mathematics archive: http://xxx.lanl.gov/abs/math.QA/9912084 * * * "Homotopy Algebras for Operads" We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy P-algebra in M is, provided only that some of the morphisms in M have been marked out as `homotopy equivalences'. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any n-fold loop space is an n-fold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A_infinity-spaces, A_infinity-algebras and non-strict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on `change of base', e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we reflect on the advantages and disadvantages of our definition, and on how the definition really ought to be replaced by a more subtle infinity-categorical version. Available on the mathematics archive: http://xxx.lanl.gov/abs/math.QA/0002180