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On the construction of free algebras for equational systems
 IN: SPECIAL ISSUE FOR AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2007). VOLUME 410 OF THEORETICAL COMPUTER SCIENCE
, 2009
"... The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applica ..."
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The purpose of this paper is threefold: to present a general abstract, yet practical, notion of equational system; to investigate and develop the finitary and transfinite construction of free algebras for equational systems; and to illustrate the use of equational systems as needed in modern applications.
Term Equational Systems and Logics (Extended Abstract)
"... We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an intern ..."
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We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an internal completeness result may be used to synthesise complete equational logics. Indeed, as an application, we synthesise a sound and complete nominal equational logic, called Synthetic Nominal Equational Logic, based on the category of Nominal Sets.
DESCENT FOR MONADS
"... Abstract. Motivated by a desire to gain a better understanding of the “dimensionbydimension” decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describ ..."
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Abstract. Motivated by a desire to gain a better understanding of the “dimensionbydimension” decompositions of certain prominent monads in higher category theory, we investigate descent theory for endofunctors and monads. After setting up a basic framework of indexed monoidal categories, we describe a suitable subcategory of Cat over which we can view the assignment C ↦ → Mnd(C) as an indexed category; on this base category, there is a natural topology. Then we single out a class of monads which are wellbehaved with respect to reindexing. The main result is now, that such monads form a stack. Using this, we can shed some light on the free strict ωcategory monad on globular sets and the free operadwithcontraction monad on the category of collections.
Traces for Coalgebraic Components
 MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of sta ..."
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This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.
Term Equational Rewrite Systems and Logics
"... We introduce an abstract general notion of system of equations and rewrites between terms, called Term Equational Rewrite System (TERS), and develop a sound logical deduction system, called Term Equational Rewrite Logic (TERL), to reason about equality and rewriting. Further, we give an analysis of ..."
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We introduce an abstract general notion of system of equations and rewrites between terms, called Term Equational Rewrite System (TERS), and develop a sound logical deduction system, called Term Equational Rewrite Logic (TERL), to reason about equality and rewriting. Further, we give an analysis of algebraic free constructions which together with an internal completeness result may be used to synthetise a complete TERL. Indeed, as an application, we derive a sound and complete equational rewrite