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**11 - 15**of**15**### The Syntax of Coherence

, 1999

"... This article tackles categorical coherence within a two-dimensional generalization of Lawvere’s functorial semantics. 2-theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded, many coherence results become simple statements about the qu ..."

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This article tackles categorical coherence within a two-dimensional generalization of Lawvere’s functorial semantics. 2-theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded, many coherence results become simple statements about the quasi-Yoneda lemma and 2-theory-morphisms. Given two 2-theories and a 2-theory-morphism between them, we explore the induced relationship between the corresponding 2-categories of algebras. The strength of the induced quasi-adjoints are classified by the strength of the 2-theorymorphism. These quasi-adjoints reflect the extent to which one structure can be replaced by another. A two-dimensional analogue of the Kronecker product is defined and constructed. This operation allows one to generate new coherence laws from old ones. 1

### CLONE PROPERTIES OF TOPOLOGICAL SPACES

"... Abstract. Clone properties are the properties expressible by the first order sentence of the clone language. The present paper is a contribution to the field of problems asking when distinct sentences of the language determine distinct topological properties. We fully clarify the relations among the ..."

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Abstract. Clone properties are the properties expressible by the first order sentence of the clone language. The present paper is a contribution to the field of problems asking when distinct sentences of the language determine distinct topological properties. We fully clarify the relations among the rigidity, the fix-point property, the image-determining property and the coconnectedness. 1.

### and

, 2000

"... We show that although the algebraic semantics of place/transition Petri nets under the collective token philosophy can be fully explained in terms of strictly symmetric monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory, because it la ..."

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We show that although the algebraic semantics of place/transition Petri nets under the collective token philosophy can be fully explained in terms of strictly symmetric monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory, because it lacks universality and also functoriality. We introduce the notion of pre-nets to overcome this, obtaining a fully satisfactory categorical treatment, where the operational semantics of nets yields an adjunction. This allows us to present a uniform logical description of net behaviors under both the collective and the individual token philosophies in terms of theories and theory morphisms in partial membership equational logic. Moreover, since the universal property of adjunctions guarantees that colimit constructions on nets are preserved in our algebraic models, the resulting semantic framework has good compositional properties. C ○ 2001 Academic Press Key Words: PT Petri nets; pre-nets; collective/individual token philosophy; monoidal categories; partial membership equational logic; configuration structures; concurrent transition systems.

### Algebraic Theories in Quantum Field Theory and Quantum Algebra

, 1999

"... 1 Introduction Over the past few years, there have been several major innovations in quantum field theory (QFT) and quantum algebra (QA). Following Graeme Segal's seminal definition of conformal field theory, many researchers studying QFT, have now begun using categories and functors in a new a ..."

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1 Introduction Over the past few years, there have been several major innovations in quantum field theory (QFT) and quantum algebra (QA). Following Graeme Segal's seminal definition of conformal field theory, many researchers studying QFT, have now begun using categories and functors in a new and fundamental way. The categories (loosely) correspond to segments of space-time while the functors take segments of space-time to linear maps of Hilbert spaces. In QA, researchers are working with many new structures that are radically different from standard algebraic structures. Many of these structures are neither commutative nor associative and have relations that hold only &quot;up to homotopy.&quot; Both QFT and QA have created many new and diverse mathematical structures that need to be studied from a level of generality that encompasses them all. We propose to study these two fields of mathematical physics from the point of view of a category theoretic version of universal algebra called functorial semantics. Functorial semantics deals with algebraic theories that describe algebraic structures. Our goal is to generalize functorial semantics enough to place both of these fields into a single formalism. Placing them within one formalism will help us understand the deep connections within each area and between both areas. We will use the powerful tools available in functorial semantics to better understand and prove theorems about the structures in QFT and QA.

### Talk given at the Conference on Topics in Geometry and Physics

, 1992

"... hep-th/9304061 Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli spaces ..."

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hep-th/9304061 Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli spaces