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52
THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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We take another look at the Chu construction and show how to simplify it by looking at
Higher fundamental functors for simplicial sets, Cahiers Topologie Géom
 Diff. Catég
"... Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how ..."
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Cited by 11 (8 self)
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Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (nonreversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy ncategory functor ↑Πn: Smp = nCat, left adjoint to a nerve Nn = nCat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.
A Classification of Accessible Categories
 Max Kelly volume, J. Pure Appl. Alg
, 1996
"... For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the ..."
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For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the
Categorybased Constraint Logic
, 1999
"... This paper presents an (abstract) model theoretic semantics for ECLP, without directly addressing the computational aspect. This is a rather novel approach on the area of constraints where almost all efforts have been devoted to computational and operational issues; it is important the reader unders ..."
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Cited by 7 (3 self)
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This paper presents an (abstract) model theoretic semantics for ECLP, without directly addressing the computational aspect. This is a rather novel approach on the area of constraints where almost all efforts have been devoted to computational and operational issues; it is important the reader understands the modeltheoretic and foundational orientation of this paper. However, we plan to gradually develop the computational side based on these foundations as further research (Section 7.2 sketches some of the directions of such further research). Some computational aspects of this theory can already be found in (Diaconescu, 1996c). This semantics is
On Tree Coalgebras and Coalgebra Presentations
, 2002
"... For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra A t . The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a uniqu ..."
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For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra A t . The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique A t which takes the root of t to s. Moreover, the tree coalgebras are finitely presentable and form a strong generator. Thus, these categories of coalgebras are locally finitely presentable; in particular every system is a filtered colimit of finitely presentable systems.
The Reflectiveness of Covering Morphisms in Algebra And Geometry
, 1997
"... . Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lie ..."
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Cited by 7 (5 self)
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. Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies in M precisely when it is the pullback along the unit jB : B ! IB of its reflexion If : IA ! IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A ! B is said to be a covering of B if, for some effective descent morphism p : E ! B, the pullback p f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass  or rather a subcategory  of the category C #B ae C=B whose objects are those f : A ! B with f 2 \Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is re...
Representations, Hierarchies, and Graphs of Institutions
, 1996
"... For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em ..."
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For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of reusing proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various wellknown institutions of total, ordersorted and partial algebras and firstorder structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, ordersorted and partial algebras and firstorder structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of socalled parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd KriegBr\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.
SUPPORT VARIETIES – AN IDEAL APPROACH
, 2005
"... Abstract. We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applic ..."
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Abstract. We define support varieties in an axiomatic setting using the prime spectrum of a lattice of ideals. A key observation is the functoriality of the spectrum and that this functor admits an adjoint. We assign to each ideal its support and can classify ideals in terms of their support. Applications arise from studying abelian or triangulated tensor categories. Specific examples from algebraic geometry and modular representation theory are discussed, illustrating the power of this approach which is inspired by recent work of Balmer. Contents
Algebraic theories of quasivarieties
 Journal of Algebra
, 1998
"... Analogously to the fact that Lawvere’s algebraic theories of (finitary) varieties are precisely the small categories with finite products, we prove that (i) algebraic theories of many–sorted quasivarieties are precisely the small, left exact categories with enough regular injectives and (ii) algebra ..."
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Analogously to the fact that Lawvere’s algebraic theories of (finitary) varieties are precisely the small categories with finite products, we prove that (i) algebraic theories of many–sorted quasivarieties are precisely the small, left exact categories with enough regular injectives and (ii) algebraic theories of many–sorted Horn classes are precisely the small left exact categories with enough M–injectives, where M is a class of monomorphisms closed under finite products and containing all regular monomorphisms. We also present a Gabriel–Ulmer–type duality theory for quasivarieties and Horn classes. 1 Quasivarieties and Horn Classes The aim of the present paper is to describe, via algebraic theories, classes of finitary algebras, or finitary structures, which are presentable by implications. We work with finitary many–sorted algebras and structures, but we also mention the restricted version to the one–sorted case on the one hand, and the generalization to infinitary structures on the other hand. Recall that Lawvere’s thesis [11] states that Lawvere–theories of varieties, i.e., classes of algebras presented by equations, are precisely the small categories with finite products, (in the one sorted case moreover product–generated by a single object; for many–sorted varieties the analogous statement can be found in [4, 3.16, 3.17]). More in detail: If we denote, for small categories A, by P rodωA the full subcategory of Set A formed by all functors preserving finite products, we obtain the following: (i) If K is a variety, then its Lawvere–theory L(K), which is the full subcategory of K op of all finitely generated free K–algebras, is essentially small, and has finite products. The variety K is equivalent to P rodωL(K).