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*To*: categories@mta.ca (categories list)*Subject*: categories: preprint + CT99 photos*From*: Koslowski <koslowj@iti.cs.tu-bs.de>*Date*: Fri, 18 Feb 2000 14:35:42 +0100 (MET)*Sender*: cat-dist@mta.ca

Dear Colleagues, Together with Robin Cockett and Robert Seely I would like to announce the availablility of a preprint Morphisms and modules for linear bicategories that has been submitted for the CT99 proceedings. You can get it from <http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/RESEARCH/>. Those interested in some of the pictures I took at that meeting may want to look at <http://www.iti.cs.tu-bs.de/TI-INFO/koslowj/PHOTOS/CT99-0/>. If anybody wants his or her picture removed, or wants to order a print or a larger version of a picture, please let me know. -- J"urgen Koslowski Abstract: Linear bicategories are a generalization of bicategories, in which the horizontal composition of 1-cells is replaced by two (coherently linked) horizontal compositions. This notion combines and integrates compositional features typical of bicategories with those arising from linear logic. Linear bicategories, therefore, provide a natural categorical semantics and interpretation for (relational) non-commutative linear logic. In particular, the logical notion of negation (or complementation) turns into a linear notion of adjunction, with involutive negations corresponding to cyclic linear adjunctions. The latter are crucial for the construction of a tricategory of linear bicategories, linear functors, linear transformations and linear modifications. This paper first develops the structure of the afore-mentioned tricategory and describes how the various components have a natural interpretation using the diagrammatic calculus of circuits. Then we transfer the module construction to the linear setting, which leads to a new linear bicategory of linear monads, linear modules, and module transformations over a given linear bicategory. This lives in a tricategory where the 1-cells are given by linear functors that are strict on units. The connections of this construction with the nucleus construction for linear bicategories are indicated at the end. The present paper is primarily expository, setting out the notions necessary for the development of category theory enriched in a linear bicategory. -- J"urgen Koslowski \ If I don't see you no more on this world ITI, TU Braunschweig \ I'll meet you on the next one koslowj@iti.cs.tu-bs.de \ and don't be late! http://www.iti.cs.tu-bs/~koslowj \ Jimi Hendrix (Voodoo Child, SR)

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