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54
The Sreplete construction
 In CTCS 55, pages 96  116. Springer Lecture Notes in Computer Science 953
, 1995
"... this paper: (internal version) if C 1 is a quasitopos, then S ..."
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this paper: (internal version) if C 1 is a quasitopos, then S
Cofibrantly generated natural weak factorisation systems
, 2007
"... There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with r ..."
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There is an “algebraisation ” of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of mapswithstructure, where the extra structure on a map now encodes a choice of liftings with respect to the other class. This extra structure has pleasant consequences: for example, a natural w.f.s. on C induces a canonical natural w.f.s. structure on any functor category [A, C]. In this paper, we define cofibrantly generated natural weak factorisation systems by analogy with cofibrantly generated w.f.s.’s. We then construct them by a method which is reminiscent of Quillen’s small object argument but produces factorisations which are much smaller and easier to handle, and show that the resultant natural w.f.s. is, in a suitable sense, freely generated by its generating cofibrations. Finally, we show that the two categories of mapswithstructure for a natural w.f.s. are closed under all the constructions we would expect of them: (co)limits, pushouts / pullbacks, transfinite composition, and so on. 1
Lax Factorization Algebras
"... It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the "squaring monad" on CAT. This structure largely facili ..."
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It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the "squaring monad" on CAT. This structure largely facilitates natural choices for left or right liftings once certain basic natural choices have been made. The use of homomorphisms of such lax algebras is also discussed, with focus on "lax freeness". Mathematics Subject Classification: 18A32, 18C20, 18D05, 55P05. Key words: weak factorization system, cofibrantly generated system, (symmetric) lax factorization algebra, lax homomorphism. Supported by the Ministry of Education of the Czech Republic under project MSM 143100009. y Partial financial assistance by NSERC is acknowledged. 1 1. Introduction Weak factorization systems appear prominently in the definition of Quillen model category: for C, W, F the classes of cofibrations, weak equiva...
General reversibility
 In EXPRESS’06, ENTCS. Elsevier
, 2006
"... The first and the second author introduced reversible ccs (rccs) in order to model concurrent computations where certain actions are allowed to be reversed. Here we show that the core of the construction can be analysed at an abstract level, yielding a theorem of pure category theory which underlies ..."
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The first and the second author introduced reversible ccs (rccs) in order to model concurrent computations where certain actions are allowed to be reversed. Here we show that the core of the construction can be analysed at an abstract level, yielding a theorem of pure category theory which underlies the previous results. This opens the way to several new examples; in particular we demonstrate an application to Petri nets. 1
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
Unique Factorisation Lifting Functors and Categories of LinearlyControlled Processes
 Mathematical Structures in Computer Science
, 1999
"... We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce ..."
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We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearlycontrolled ones. To this end, we introduce and study a notion of pathlinearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearlycontrolled processes (viz., processes controlled by pathlinearisable categories) are sheaf models. Introduction This work is an investigation into the mathematical structure of processes. The processes to be considered embody a notion of state space varying according to a control. This we formalise as a category of states (and their interrelations) Xequipped with a control functor X C f . There are different ways in which the control category C may be required to control t...
MONADS OF EFFECTIVE DESCENT TYPE AND COMONADICITY
"... Abstract. We show, for an arbitrary adjunction F ⊣ U: B→Awith B Cauchy complete, that the functor F is comonadic if and only if the monad T on A induced by the adjunction is of effective descent type, meaning that the free Talgebra functor F T: A→A T is comonadic. This result is applied to several ..."
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Abstract. We show, for an arbitrary adjunction F ⊣ U: B→Awith B Cauchy complete, that the functor F is comonadic if and only if the monad T on A induced by the adjunction is of effective descent type, meaning that the free Talgebra functor F T: A→A T is comonadic. This result is applied to several situations: In Section 4 to give a sufficient condition for an exponential functor on a cartesian closed category to be monadic, in Sections 5 and 6 to settle the question of the comonadicity of those functors whose domain is Set, orSet⋆, or the category of modules over a semisimple ring, in Section 7 to study the effectiveness of (co)monads on module categories. Our final application is a descent theorem for noncommutative rings from which we deduce an important result of A. Joyal and M. Tierney and of J.P. Olivier, asserting that the effective descent morphisms in the opposite of the category of commutative unital rings are precisely the pure monomorphisms. 1.
...Cat Is Locally Presentable Or Locally Bounded If ... Is So
, 2001
"... We show, for a monoidal closed category V = (V 0 , #, I), that the category V Cat of small V categories is locally #presentable if V 0 is so, and that it is locally #bounded if the closed category V is so, meaning that V 0 is locally #bounded and that a side condition involving the monoidal ..."
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We show, for a monoidal closed category V = (V 0 , #, I), that the category V Cat of small V categories is locally #presentable if V 0 is so, and that it is locally #bounded if the closed category V is so, meaning that V 0 is locally #bounded and that a side condition involving the monoidal structure is satisfied.
KPurity and orthogonality
 Theory Appl. Categ
"... This paper was inspired by the hardtobeleive fact that Jiˇrí Rosick´y is getting sixty. We are happy to dedicate our paper to his birthday. Abstract. A logic of orthogonality characterizes all “orthogonality consequences” of a given class Σ of morphisms, i.e. those morphisms s such that every obje ..."
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This paper was inspired by the hardtobeleive fact that Jiˇrí Rosick´y is getting sixty. We are happy to dedicate our paper to his birthday. Abstract. A logic of orthogonality characterizes all “orthogonality consequences” of a given class Σ of morphisms, i.e. those morphisms s such that every object orthogonal to Σ is also orthogonal to s. A simple fourrule deduction system is formulated which is sound in every cocomplete category. In locally presentable categories we prove that the deduction system is also complete (a) for all classes Σ of morphisms such that all members except a set are regular epimorphisms and (b) for all classes Σ, without restriction, under the settheoretical assumption that Vopěnka’s Principle holds. For finitary morphisms, i.e. morphisms with finitely presentable domains and codomains, an appropriate finitary logic is presented, and proved to be sound and complete; here the proof follows immediately from previous joint results of Jiˇrí Rosick´y and the first two authors. 1.
Elementary Axioms for Local Maps of Toposes
 Manuscript, submitted for 1999 Category Theory Conference in Coimbra
, 2001
"... We present a complete elementary axiomatization of local maps of toposes. 1 ..."
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We present a complete elementary axiomatization of local maps of toposes. 1