Surely this must be known, but where is there a reference? If one considers any abelian category A, one can from a 2-category as follows: objects: differential graded objects in A (i.e. chain complexes) 1-arrows: chain maps 2-arrows: equivalence classes of chain homotopies where two chain homotopies a,b:F--->G are equivalent if there exits a map x of degree -2 from the common source of F and G to the common target of F and G such that a - b = xd - dx. The two dimensional composition of chain homotopies is addition, the one dimensional is given by if a:F--->G, b:H--->K (source(H)=target(G)) then a *1 b = aH + Gb (composites in diagrammatic order). Also, can one play the game again, and get a 3-category, etc.? A related question: does anyone know of a notion of "weak n-category" in which associativities are strict, but the middle-four-interchange holds only up to higher dimensional data? (That's what happens here if one uses chain homotopies instead of equivalence classes.) --David Yetter ==========================================================================
Addendum to my reply to Yetter's letter: There were other things I meant to say but was short of time yesterday. 1) The work of Brown, Porter, Higgins deals with the non-additive groupoid case. Now I see that they have replied by referring to SLN 72. 2) Andre Joyal has a beautiful proof of the Dold-Kan theorem showing that it is a qualitative version of Isaac Newton's result that a list of data can be recaptured from the first column of iterated differences. I told Dold this while he was visiting Macquarie; he was delighted. 3) The generalisation of Gray's tensor product of 2-categories is performed as follows. Let me write G for the monoidal category 2-Cat with Gray's tensor product. Then G-Cat is the category of categories with homs enriched in G. The category 3-Cat of 3-categories is contained in G-Cat. Let O(I^n) denote the free n-category on the n-cube I^n (as per the work of Iain Aitchison, Mike Johnson, Richard Steiner and myself; in fact, "Parity complexes" has appeared in Cahiers XXXII-4 1991 315-343 and provides a simple model for O(I^n)). Let Q^n denote the 3-category obtained from O(I^n) by forcing all cells of dimension higher than 3 to be identities. Thus we have a full subcategory T of G-Cat consisting of the Q^n; furthermore, T is dense in G-Cat. We certainly know how to multiply cubes: Q^m (tensor) Q^n = Q^(m+n). This tensor product extends from T to G-Cat by Kan extension along the inclusion (Brian Day studied the general process of extending promonoidal structures along dense functors; his convolution structures are about extension along Yoneda). This is the third stage in a clear recursive construction. Regards, Ross ==============================================================================
participants (2)
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dyetter＠math.ksu.edu -
street＠macadam.mpce.mq.edu.au