Luc seems to be describing what, in our work on rewrites etc, Eilenberg wanted to call a one-and-a-half-category, and I changed (and I'm not sure how Sammy feels about it) to sesquicategory when writing the article "Categorical structures" for "Handbook of Algebra" Volume 2 (Elsevier, North Holland). [The preprint is dated Nov 1992 but the volume probably won't appear until late 1994!] There are two symmetric monoidal closed structures on the category Cat of categories. One is the usual cartesian closed structure where the internal hom [A,B] is the usual category of functors A --> B and NATURAL transformations between these. The other closed structure (which I call the "funny" one) has internal hom [[A,B]] the category of functors and transformations between these (which consist of the data for a nat trans without the naturality requirement). This funny structure WAS an example in Eil-Kelly "Closed categories" La Jolla 1965. Carolyn Brown has used the funny structure in her work on Petri nets. Sesquicategories are V-categories where V is Cat with the funny monoidal structure. John G. Stell <john@cs.keele.ac.uk> tells me he has used sesquicategories in computer science, and apparently independently came up with the same name for them. Perhaps of future interest to those interested in sesquicats are the various monoidal structures on 2-Cat. For example, there is John Gray's tensor product of 2-categories, where "natural", instead of being dropped, is replaced by "lax natural". A recent paper of Gordon, Power, Street looks at the case where V = 2-Cat with the monoidal structure obtained by replacing "natural" by "pseudonatural"; we prove that every tricategory is equivalent (in the approp sense) to a V-category. Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
participants (2)
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unknown＠example.com -
street＠macadam.mpce.mq.edu.au