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categories: preprint + CT99 photos
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
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
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
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
email@example.com \ and don't be late!
http://www.iti.cs.tu-bs/~koslowj \ Jimi Hendrix (Voodoo Child, SR)