I used recently the following hierarchy of notions (one of them coinciding with L.D.'s one) as abstract settings for homotopical algebra. A synthesis is given in Cahiers Top. Geom. Diff. Cat. 33 (1992, 135-175 (ch. 5-6); a longer 1991 preprint on the subject is no longer available, but a new version will soon be ready. - "h-category", or "category with homotopies". (This notion goes back to K.H. Kamps (Manuscripta Math. 3 (1970), 237-255), who used it in a slightly different, equivalent form under the name of "generalized homotopy system"; it is an extension of a Kan "homotopy system", or category with cylinder endofunctor) Formally, an h-category is a category enriched over Reflexive Graphs, with a suitable monoidal closed structure. Concretely, an h-category has objects, maps f: A -> B and "cells" (or homotopies, or transformations) alpha: f -> g (for f: A -> B). Objects and maps form a category; further, there is a "vertical identity" id f: f -> f for every map and a "reduced horizontal composition" of cells with maps: k alpha h (for h: A' -> A, k: B -> B') under the obvious axioms for identities and associativity (id B) alpha (id A) = alpha, k (id f) h = id (kfh), (k'k) alpha (hh') = k' (k alpha h) h'. Of course one may separate the left and right composition of cells with maps: k alpha, alpha h. Note that there is no vertical composition. - h1-category = h-category + vertical involution (under some weak axioms) - h2-category = h-category + vertical composition (under some weak axioms; for instance the vertical comosition is not required to be associative) (strict h2-category = h2-category with axioms for vertical identities and vertical associativity = sesquicategory in the sense of R. Street's reply = (probably) "category with transformations" in the sense of L.D.'s message - h3-category = h-category + vertical involution and composition (under some weak axioms) (strict h3-category = h3-category with axioms for vertical identities, vertical inverses and vertical associativity = category enriched over groupoids, with the "funny" structure = sesquigroupoid? - h4-category = h3-category + second-order homotopy relation making it a sort of relaxed 2-category. To write down the complete definitions would be too long here. But the following examples should be sufficient to make them clear, and also to motivate them a) Top = Topological spaces, continuous maps and homotopies. It is h4, not strict - the vertical identities, involution and composition just "work well" up to second-order homotopy. b) C*A = Chain complexes (over a preadditive category A), chain maps, homotopies. It is h4 and strictly h3 (the vertical composition of homotopies is obtained from the sum, and works well in the strict sense). This sesquicategory could be of interest for L.D., as an algebraic model of his situation. The presence of a vertical involution (strictly well behaved) should also be of interest. c) Dga = Differential graded algebras (over some unital ring), their homomorphisms and homotopies. It is just an h-category - the multiplicativity conditions on homotopies prevent to reverse or add them. Best regards, Marco Grandis ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++