ISSN 1201-561X THEORY AND APPLICATIONS OF CATEGORIES Volume 1, 1995 Oriented singular homology Michael Barr 1 Functorial and algebraic properties of Brown's P functor Luis-Javier Hern\'andez-Paricio 10 On finite induced crossed modules, and the homotopy 2-type of mapping cones Ronald Brown and Christopher D. Wensley 54 Kan extensions along promonoidal functors Brian Day and Ross Street 72 Symmetric monoidal categories model all connective spectra R. W. Thomason 78 Distributive adjoint strings Robert Rosebrugh and R. J. Wood 119 A forbidden-suborder characterization binarily-composable diagrams in double categories Robert Dawson 146 Categorical data-specifications Frank Piessens and Eric Steegmans 156 On the size of categories Peter Freyd and Ross Street 174 Copyright: The Editors of Theory and Applications of Categories 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. The method of distribution of the journal is via the Internet tools WWW/gopher/ftp. The journal is archived electronically and in printed paper format. Subscription information: Individual subscribers receive (by e-mail) abstracts of articles as they are published. Full text of published articles is available in .dvi and Postscript format. Details will be e-mailed to new subscribers and are available by WWW/gopher/ftp. 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, Robert Rosebrugh, rrosebrugh@mta.ca. Information for authors: The typesetting language of the journal is TeX, and LaTeX is the preferred flavour. TeX source of articles for publication should be submitted by e-mail directly to an appropriate Editor. They are listed below. Please obtain detailed information on submission format and style files from the journal's WWW server at URL http://www.tac.mta.ca/tac/ or by anonymous ftp from ftp.tac.mta.ca in the directory pub/tac/info. You may also write to tac@mta.ca to receive details by e-mail. Editorial board: John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@triples.math.mcgill.ca Lawrence Breen, Universit\'e de Paris 13: breen@dmi.ens.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, University of Genoa: carboni@vmimat.mat.unimi.it P. T. Johnstone, University of Cambridge: ptj@pmms.cam.ac.uk G. Max Kelly, University of Sydney: kelly\_m@maths.su.oz.au Anders Kock, University of Aarhus: kock@mi.aau.dk F. William Lawvere, State University of New York at Buffalo: mthfwl@ubvms.cc.buffalo.edu Jean-Louis Loday, Universit\'e de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.ruu.nl Susan Niefield, Union College: niefiels@gar.union.edu Robert Par\'e, Dalhousie University: pare@cs.dal.ca Andrew Pitts, University of Cambridge: ap@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@charlie.math.unc.edu Ross Street, Macquarie University: street@macadam.mpce.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca R. W. Thomason, Universit\'e de Paris 7: thomason@mathp7.jussieu.fr Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Sydney: walters\_b@maths.su.oz.au R. J. Wood, Dalhousie University: rjwood@cs.da.ca } %Note that this is a redefinition. Its purpose is to compile the %label lastpage so that we can get the page numbers right. %It also adds the authors' address \def\enddocument{ \label{lastpage} \@endstuff \@checkend{document}\clearpage\begingroup \if@filesw \immediate\closeout\@mainaux \def\global\@namedef##1##2{}\def\newlabel{\@testdef r}% \def\bibcite{\@testdef b}\@tempswafalse \makeatletter\input \jobname.aux \if@tempswa \@warning{Label(s) may have changed. Rerun to get cross-references right}\fi\fi\endgroup\deadcycles\z@\@@end} % Below are the codes for the two reference styles. They can be called by % \begin{references} ... \end{references} and \begin{references*} ... % \end{references*} . In addition, the first can be called by \references % ... \endreferences . \newdimen\refindent \refindent 8mm \def\references{% \fontdimen2\twlrm=4.3pt% space instead of 3.91663 \fontdimen3\twlrm=4.2pt%stretch instead of 1.95831 \fontdimen4\twlrm=1.7pt%shrink instead of 1.30554 \section*{References} \sloppy\footnotesize \begin{list}{} {\leftmargin\refindent \itemindent-\refindent \itemsep2pt \topsep0pt}} \def\endreferences{\end{list}} \newcounter {reflister} \expandafter\def\csname references*\endcsname {% \fontdimen2\twlrm=4.3pt% space instead of 3.91663 \fontdimen3\twlrm=4.2pt%stretch instead of 1.95831 \fontdimen4\twlrm=1.7pt%shrink instead of 1.30554 \section*{References} \sloppy\footnotesize \begin{list}{{\rm[\arabic{reflister}]}}{\usecounter{reflister}} {\leftmargin\refindent \itemindent-\refindent \itemsep2pt \topsep0pt}} \expandafter\let\csname endreferences*\endcsname=\endreferences % Modification of the `plain' definition, taken over by latex. Note the % failure to parametrize the definition, one of the many such that testify % to Knuth's conception of plain as a provisional package. \def\footnoterule{%\kern-3\p@ \hrule width 1truein \kern 2.6\p@} % the \hrule is .4\p@ high % Follows are a small number of procedures I have personally found very % useful. The first three define new math operations like Hom. In the % first, the word is spaced like a mathop and is used if there are to be % arguments. It is better not used if, like Hom, it is followed by an % opening paranthesis. The next two are without the spacing and are used % for boldface, roman face, resp. The usage is: \mathrm{Hom} that defines % a new macro \Hom that puts the word Hom in ordinary type. \def\mathopdef#1{\expandafter\def\csname#1\endcsname{{\mathop{\rm#1}}}} \def\mathbfdef#1{\expandafter\def\csname#1\endcsname{{\hbox{\rm\bf#1}}}} \def\mathrmdef#1{\expandafter\def\csname#1\endcsname{{\hbox{\rm#1}}}} \def\mld#1$${\null\,\vcenter\bgroup\def\\{\cr&}\openup9pt\m@th \ialign\bgroup\strut\hfil$\displaystyle{##}$% &$\displaystyle{{}##}$\hfil\crcr #1 \crcr\egroup\egroup\,$$} \let\emld\relax % for compatibility with old code %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Here is a typical usage of \mld (multi-line display) %% %% %% %% The pattern is %% %% $$\mld formula 1 & formula 2 \\ %% %% formula 3 \\ %% %% ..........\\ %% %% formula n $$ %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \endinput