id: 05123039
dt: j
an: 05123039
au: Blute, R.F.; Cockett, J.R.B.; Seely, R.A.G.
ti: Differential categories.
so: Math. Struct. Comput. Sci. 16, No. 6, 1049-1083 (2006).
py: 2006
pu: Cambridge University Press, Cambridge
la: EN
cc:
ut: differential category; additive symmetric monoidal category; comonad;
differential combinator; coKleisli category; linear logic; coherence
spaces; storage modality; categorical model; differential $λ$-calculus
ci:
li: doi:10.1017/S0960129506005676
ab: Summary: 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
$S_\infty$ 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
comonad is a storage modality, as in linear logic. In particular, we
introduce the notion of a categorical model of the differential
calculus, and show that it captures the not-necessarily-closed fragment
of Ehrhard-Regnier differential $λ$-calculus.
rv: