Dear categorists, just to announce that my preprint, Weak identity arrows in higher categories is now available from http://arxiv.org/abs/math.CT/0507116. I look forward to all sorts of comments. Cheers, Joachim. ---------------------------------------------------------------- Joachim Kock <kock@mat.uab.es> Departament de Matemà tiques -- Universitat Autònoma de Barcelona Edifici C -- 08193 Bellaterra (Barcelona) -- ESPANYA Phone: +34 93 581 25 34 Fax: +34 93 581 27 90 <A HREF="http://mat.uab.es/~kock/">http://mat.uab.es/~kock/</A> ---------------------------------------------------------------- Abstract: There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened --- these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category $\Delta$ is replaced by a certain `fat' delta of `coloured ordinals', where the degeneracy maps are only up to homotopy. The first part of this exposition is aimed at a broad mathematical readership and contains also a brief introduction to simplicial viewpoints on higher categories in general. It is explained how the definition of fair $n$-category is almost forced upon us by three standard ideas. The second part states some basic results about fair categories, and give examples. The category of fair $2$-categories is shown to be equivalent to the category of bicategories with strict composition law. Fair $3$-categories correspond to tricategories with strict composition laws. The main motivation for the theory is Simpson's weak-unit conjecture according to which $n$-groupoids with strict composition laws and weak units should model all homotopy $n$-types. A proof of a version of this conjecture in dimension $3$ is announced, obtained in joint work with A.~Joyal. Technical details and a fuller treatment of the applications will appear elsewhere. 12-Jul-2005 02:18:33 -0300,2509;000000000001-00000005