Links to publications 2006

Points of affine categories and additivity
A. Carboni and G. Janelidze, 127-131

A category C is additive if and only if, for every object B of C, the category Pt(C,B) of pointed objects in the comma category (C,B) is canonically equivalent to C. We reformulate the proof of this known result in order to obtain a stronger one that uses not all objects of B of C, but only a conveniently defined generating class S. If C is a variety of universal algebras, then one can take S to be the class consisting of any single free algebra on a non-empty set.

Links to publications 2005

Generic commutative separable algebras and cospans of graphs
R. Rosebrugh, N. Sabadini and R.F.C. Walters

We show that the generic symmetric monoidal category with a commutative separable algebra which has a $\Sigma$-family of actions is the category of cospans of finite $\Sigma$-labelled graphs restricted to finite sets as objects, thus providing a syntax for automata on the alphabet $\Sigma$. We use this result to produce semantic functors for $\Sigma$-automata.

Links to publications 2004

Links to publications 2003

Links to Publications 2002

Feedback, trace and fixed point semantics
P. Katis, N. Sabadini, R.F.C. Walters

We introduce a notion of category with feedback-with-delay, closely related to the notion of traced monoidal category, and show that the Circ construction of [JPAA 115: 141--178, 1997] is the free category with feedback on a symmetric monoidal category. Combining with the Int construction of Joyal-Street-Verity [Math. Proc. Camb.Phil. Soc., 119, 447-468, 1996] we obtain a description of the free compact closed category on a symmetric monoidal category. We thus obtain a categorical analogue of the classical localization of a ring with respect to a multiplicative subset. In this context we define a notion of fixed-point semantics of a category with feedback which is seen to include a variety of classical semantics in computer science.