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Zbl 0564.18005
Grandis, Marco
On distributive homological algebra. I: $RE$-categories. (English)
[J] Cah. Topol. Géom. Différ. 25, 259-301 (1984). ISSN 1245-530X

This is the first of three papers representing work whose original goal was the understanding of {\it E. C. Zeeman}'s diagrams associated with filtered complexes [Ann. Math. (2) 66, 557--585 (1957; Zbl 0097.38703)]. That no such diagrams occur for bifiltered complexes owes itself to a lack of distributivity; hence the title of the trilogy. \par The paper studies the 2-category of $RE$-categories. An $RE$-category is a category $A$ together with an order on the homsets (making $A$ into a 2-category) and an involution $\sim: A\sp{\text{op}}\to A$ (which is the identity on objects and is regular: $aaa=a$), all subject to certain axioms. The category of relations $\mathrm{Rel}\ E$ in an exact category $E$ is an $RE$-category, and those arising in this way are characterized. Here ``exact" is in the sense of {\it B. Mitchell} [Theory of categories (1965; Zbl 0136.00604), p. 18]; such a category is pointed, but not necessarily additive. While 1-cells between $RE$-categories are what one expects, the 2-cells are lax natural transformations; even so, the 2-category of $RE$-categories is shown to be complete.
[Ross H. Street (North Ryde)]

MSC 2000:: *18B10 Category of relations
18G40 Spectral sequences (homological algebra)

Keywords: Zeeman diagrams; filtered differential group; bifiltered complexes; distributivity; category of relations; exact category; 2-category; RE-categories

Citations: Zbl 0564.18006; Zbl 0097.38703; Zbl 0136.00604

Cited in: Zbl 0634.18009 Zbl 0577.18003 Zbl 0564.18006

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