ISSN 1201-561X

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THEORY AND APPLICATIONS OF CATEGORIES

Volume 21, 2008-2009

The Tholen Festscrift: A special volume dedicated to Walter Tholen on the occasion of his 60’th birthday.

Foreword The guest editors 1

A convenient category for directed homotopy L. Fajstrup and J. Rosicky 7

Limit preserving full embeddings V. Trnkova and J. Sichler 21

Protolocalisations of exact Mal’cev categories Francis Borceux, Marino Gran, Sandra Mantovani 37

Doubles for monoidal categories Craig Pastro and Ross Street 61

Completions in biaffine sets Elisabetta Felaco and Eraldo Giuli 76 Weakly Mal’cev categories N. Martins-Ferreira 91

The Editors of TAC wish to thank Maria Manuel Clementino, George Janelidze, Jiri Rosicky and Robert Rosebrugh who acted as guest editors for this special volume.

\begin{matrix}c \\ ◯ \end{matrix} The Editors of Theory and Applications of Categories 2009. Permission to copy for private use

granted.

Extensions in the theory of lax algebras Christoph Schubert and Gavin J. Seal 118

Approximate Mal’tsev operations Dominique Bourn and Zurab Janelidze 152

Notes on effective descent and projectivity in quasivarieties of universal algebras Ana Helena Roque 172

Descent for compact 0-dimensional spaces George Janelidze and Manuela Sobral 182

Analytic functors and weak pullbacks J. Adamek and J. Velebil 191

Relative injectivity as cocompleteness for a class of distributors Maria Manuel Clementino and Dirk Hofmann 210

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods.

Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication.

Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information. Individual subscribers receive by e-mail abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor.

Information for authors. The typesetting language of the journal is TEX, and L^{A}TEX 2ε is strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files from http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor. Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor. Gavin Seal, McGill University: gavin seal@fastmail.fm

Transmitting editors.

Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown (at) btinternet.com Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Cuill Inc.: valeria@cuill.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, University of Western Sydney: s.lack@uws.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca