Differential categories by R.F. Blute, J.R.B. Cockett and R.A.G. Seely This paper is available at http://www.math.mcgill.ca/rags/difftl/difftl.ps.gz Abstract Following work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a "coalgebra modality") and a differential combinator, satisfying a number of coherence conditions. In such a category, one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth (infinitely differentiable). Although such categories do not necessarily arise from models of linear logic, one should think of this as replacing the usual dichotomy of linear vs. stable maps established for coherence spaces. After establishing the basic axioms, we give a number of examples. The most important example arises from a general construction, a comonad on the category of vector spaces. This comonad and associated differential operators fully capture the usual notion of derivatives of smooth maps. Finally, we derive additional properties of differential categories in certain special cases, especially when the differential comonad is a storage modality, as in linear logic. In particular, we introduce the notion of categorical model of the differential calculus, and show that it captures the not-necessarily-closed fragment of Ehrhard-Regnier differential lambda-calculus. It is important to note that differential categories in our sense are strictly more general than the Ehrhard-Regnier structures. For example, by developing them without monoidal closed structure, we capture various "standard models" of differentiation which do not have these properties (principly our examples over vector spaces). Type theorists should notice that although the emphasis of the paper is on the categorical semantics, the category theory is presented in a way that makes the underlying type theory very transparent; the analogy with linear logic is naturally striking, as it provides one of the intuitions, although our models are not necessarily models of linear logic. -- <rags@math.mcgill.ca> <www.math.mcgill.ca/rags> 24-Jul-2005 17:01:30 -0300,6237;000000000000-00000008