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Monads on Composition Graphs
by
Lutz Schröder
Department of Mathematics and Computer Science, University of Bremen
Collections of objects and morphisms that fail to form categories isasmuch as the expected composites of two morphisms need not always be defined have been studied in [2] under the name `composition graphs'. In particular, a notion of adjointness for functors between composition graphs has been introduced which makes use of a naturally arising analogue of Lawvere's comma construction.
We now show that such adjunctions give rise to a notion of monads on composition graphs; such monads can be described by natural transformations fulfilling the usual equations and suitable additional properties (which are invisible on the level of categories) as well as by means of Kleisli triples [1]. Notably, the latter description seems to be better suited for dealing with situations with little or no associativity. Using the Kleisli construction, we show that each monad is associated to an adjunction. On the other hand, the proper generalization of the Eilenberg-Moore construction produces a weaker form of adjunctions (which can, under mild associativity conditions, also be handled by means of the comma construction) and admits comparison functors for such weakly adjoint realizations. As a first nontrivial example, we present a power set monad and a word monad on the composition graph of full unary relation morphisms.
[1] E. Manes: Algebraic Theories, Springer Verlag, Berlin, 1976.
[2] L. Schröder: Composition graphs and free extensions of categories (in German). PhD thesis, University of Bremen, 1999; also: Logos Verlag, Berlin, 1999.
Date received: June 7, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-43.