. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...

Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capabilities. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are †-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras. An earlier version of this paper has been in circulation since November 2005 with the somewhat different title Quantum measurements as coalgebras. 1 1

. The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P(N ). However, this is only a partial axiomatization, since it is based on a concrete, ad hoc chosen, category of symmetries Sym N . In this paper we give a completely abstract characterization of the category of concatenable processes of N , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets. Introduction Concatenable processes of Petri nets have been introduced in [3] to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. [14, 5, 2, 12, 3]. Generally speaking, Petri net processes---whose standard version is given by the Goltz-Reisig non-sequential processes [5]---are structures needed to acc...

... L are simplicial sets, then there is a strong deformation retraction of the fundamental crossed complex of the cartesian product K \Theta L onto the tensor product of the fundamental crossed complexes of K and L. This satisfies various side-conditions and associativity/interchange laws, as for the chain complex version. Given simplicial sets K 0 ; : : : ; K r , we discuss the r-cube of homotopies induced on (K 0 \Theta : : : \Theta K r ) and show these form a coherent system. We introduce a definition of a double crossed complex, and of the associated total (or codiagonal) crossed complex. We introduce a definition of homotopy colimits of diagrams of crossed complexes. We show that the homotopy colimit of crossed complexes can be expressed as the

Abstract. We introduce the notion of strongly concatenable process as a refinement of concatenable processes [3] which can be expressed axiomatically via a

We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and -autonomous categories, in the same sense that the simply-typed -calculus with surjective pairing is the internal language for cartesian closed categories. We show that the eight equality and three commutation congruence axioms of the -autonomous type theory characterise -autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek [12]: the equality of maps in any -autonomous category freely generated from a discrete graph is decidable. 1 Introduction In this paper we introduce a family of type theories which can be regarded as internal languages for autonomous (or symmetric monoidal closed) and -autonomous categories, in the same sense that the standard simply-typed -calculus with surjective pairing is...

We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and -autonomous categories, in the same sense that simply-typed -calculus (augmented by appropriate constructs for products and the terminal object) is the internal language for cartesian closed categories. The rules are presented in the style of Gentzen's Sequent Calculus. A key feature is the systematic treatment of naturality conditions by explicitly representing the categorical composition, or cut in the type theory, by explicit substitution, and the introduction of new let-constructs, one for each of the three type constructors ?;\Omega and (, and a Parigot-style ¯-abstraction to give expression to the involutive negation. The commutation congruences of these theories are precisely those imposed by the naturality conditions. In particular the type theory for -autonomous categories may be regarded as a term assignment system for the multiplicative (\Omega ; (;?;?)-fragmen...

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. The concatenable processes of a Petri net N can be characterized abstractly as the arrows of a symmetric monoidal category P[N ]. Yet, this is only a partial axiomatization, since P[N ] is built on a concrete, ad hoc chosen, category of symmetries. In this paper we give a fully equational description of the category of concatenable processes of N , thus yielding an axiomatic theory of the noninterleaving behaviour of Petri nets. Introduction C oncatenable processes of Petri nets have been introduced in [3] to account, as their name indicates, for the issue of process concatenation. Let us briefly reconsider the ideas which led to their definition. The development of theory Petri nets, focusing on the noninterleaving aspects of concurrency, brought to the foreground various notions of process, e.g. [14, 5, 2, 12, 3]. Generally speaking, Petri net processes---whose standard version is given by the Goltz-Reisig non-sequential processes [5]---are structures needed to account for th...

Abstract. In this paper, we present new concepts of Ann-categories, Ann-functors, and a transmission of the structure of categories based on Ann-equivalences. We build Anncategory