In this paper we elaborate on a 2-categorical construction of the homotopy category of a Quillen ... more In this paper we elaborate on a 2-categorical construction of the homotopy category of a Quillen model category. Given any category A and a class of morphisms Σ ⊂ A containing the identities, we construct a 2-category Ho(A) obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in [1]. The inclusion 2-functor A −→ Ho(A) has a universal property which yields the 2-localization of A at Σ provided that the arrows of Σ become equivalences in Ho(A). This result together with a fibrant-cofibrant replacement is then used to obtain the 2-localization of a model category C at the weak equivalences W. The set of connected components of the hom categories yields a novel proof of Quillen's results. We follow the general lines established in [1], [2] for model bicategories.
In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates be... more In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A F −→ B is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similary we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A −→ Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi. Acknowledgement We are grateful to the referees of AiM for their comments on this paper, and in particular for the suggestion to give proper emphasis to our results on conical σ-limits.
In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates be... more In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A F −→ B is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similary we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A −→ Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi. Acknowledgement We are grateful to the referees of AiM for their comments on this paper, and in particular for the suggestion to give proper emphasis to our results on conical σ-limits.
A locally connected topos is a Galois topos if the Galois objects generate the topos. We show tha... more A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bilimits of topoi, we show that every Galois topos has a point.
Replacing {0} by the whole ideal of infinitesimals yields a weaker notion of archimedean element ... more Replacing {0} by the whole ideal of infinitesimals yields a weaker notion of archimedean element that we call quasiarchimedean. It is known that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky topology) are exactly the hyperarchimedean algebras. We characterise all the algebras with compact maximal spectrum as being quasihyperarchimedean MV-algebras, which in a sense are non semisimple hyperarchimedean algebras. We develop some basic facts in the theory of MV-algebras along the lines of algebraic geometry, where infinitesimals play the role of nilpotent elements, and prove a MV-algebra version of Hilbert's Nullstellensatz. Finally we consider the relations (some inedited) between several elementary classes of MValgebras in terms of the ideals that characterise them, and present elementary (first order with denumerable disjunctions) proofs in place of the set-theoretical usually found in the literature. Contents 1. Background on MV-algebras of continuos functions 1 2. Quasihyperarchimedean algebras 3 3. The MV-Nullstellensatz. 4 4. The relations between some classes of MV-algebras 6 References 8
We give the definitions of model bicategory and q-homotopy, which are natural generalizations of ... more We give the definitions of model bicategory and q-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory C, denote by C f c the full sub-bicategory of the fibrant-cofibrant objects. We prove that the 2-dimensional localization of C at the weak equivalences can be computed as a bicategory Ho(C) whose objects and arrows are those of C f c and whose 2-cells are classes of q-homotopies up to an equivalence relation. When considered for a model category, q-homotopies coincide with the homotopies as considered by Quillen. The pseudofunctor C q −→ Ho(C) which yields the localization is constructed by using a notion of fibrant-cofibrant replacement in this context. We include an appendix with a general result of independent interest on a transfer of structure for lax functors, that we apply to obtain a pseudofunctor structure for the fibrant-cofibrant replacement. is a biequivalence of bicategories for every bicategory D; where Hom W,Θ (C, D) stands for the full sub-bicategory of Hom(C, D) given by those pseudofunctors that send weak equivalences into equivalences. We also show that the homotopy bicategory Ho(C) is locally small (resp. a 2-category) when C is so. We define q-homotopies as w-homotopies which satisfy some extra conditions related to fibrations and cofibrations. The notion of w-homotopy was introduced in [8] and is taken with respect only to the class W. Note that in op. cit. w-homotopies are called homotopies, but here we chose to use the prefix w-to explicitly distinguish them from q-homotopies. This definition allows to use previous results of [8], where for an arbitrary pair (A, W) given by a family of arrows of a bicategory we construct a bicategory Ho(A, W) whose objects and arrows are those of A and whose 2-cells are constructed with w-homotopies. In this general case there is no vertical composition of w-homotopies, and thus the 2-cells of Ho(A, W) are given by classes of finite sequences of w-homotopies by an appropriate equivalence relation. Associated with Ho(A, W) there is a canonical 2-functor A i −→ Ho(A, W). A natural way in which a proof of Theorem 6.4 can be attempted is to consider Quillen's proof of the localization theorem for model categories in [38], and to try to generalize it to this context using q-homotopies. In doing so, the first obstacle one faces is that there is no canonical way to compose (whisker) arbitrary q-homotopies with arrows. This is already so for the 1-dimensional case in [38], as is well known: for arbitrary arrows A f G G g G G B in a model category, f ℓ ∼ g doesn't imply uf ℓ ∼ ug (where ℓ ∼ is the left homotopy relation). The other composition poses no problem, f u ℓ ∼ gu is easily seen to hold. The way in which this obstacle is avoided in [38] is by considering A to be a cofibrant object, and B fibrant, so that there is a left homotopy between two arrows if and only if there is a right homotopy. An important feature of (left) w-homotopies is that both compositions (whiskerings) are canonically defined, and so this obstacle disappears. There is a rich interplay between wand q-homotopies, and that was the key we have found to overcoming the obstacle described above for the q-homotopies and furthermore proving Theorem 6.4. On the one hand, w-homotopies allow us to define a homotopy bicategory. On the other hand, q-homotopies satisfy several results analogous to the ones that can be found in [38], that we establish and use in this article. More explicitly, to prove Theorem 6.4 we will use the following results: A. Ho(C) inherits the bicategory structure of Ho(C, W) constructed in [8], in such a way that Ho(C) = Ho f c (C, W) ⊂ Ho(C, W), that is the full sub-bicategory of Ho(C, W) given by the fibrant-cofibrant objects. We will deduce this from the following two facts: A1. q-homotopies can be composed vertically: for any pair of composable q-homotopies there is a single q-homotopy representing the composition. A2. For any w-homotopy in C f c there is a fibrant w-homotopy in the same class. For any fibrant w-homotopy there is a fibrant q-homotopy in the same class (we say that a homotopy is fibrant if its cylinder is so; this is made precise in Definition 4.1). In fact, we show more generally that, when X is cofibrant and Y is fibrant, all possible B. There is a fibrant-cofibrant replacement for model bicategories, that is an assignment from C to C, defined on objects and arrows (but not on 2-cells) X f G G g G G Y ❀ RQX RQf G G RQg G G RQY , satisfying the expected conditions for a replacement: all the objects QX are cofibrant, all the objects RQX are fibrant-cofibrant, and there are trivial fibrations QX ρ X −→ X, trivial cofibrations QX λ QX −−→ RQX, and invertible 2-cells RQX o o λ QX RQf ⇓λ Qf QX ⇓ρ f Qf ρ X G G X f RQY o o λ QY QY ρ Y G G Y Note that Q and RQ are just assignments on objects and arrows and not necessarily pseudofunctors, because we are not assuming that there is a (pseudo)functorial factorization, just as in Quillen's original axioms in [38]. Nevertheless, we will show that there are structural 2-cells (homotopies) which make Q and RQ, suitably defined, into pseudofunctors Ho f (C, W) −→ Ho f (C, W), and ρ and λ into pseudonatural transformations. In the above, Ho f (C, W) denotes the bicategory determined by the fibrant left homotopies (indistinctly w-or q-, see A2 above). It is this bicategory that plays a fundamental role in our proof of Theorem 6.4. Note that, even though its 2-cells can be given by qhomotopies, the composition (whiskering) in the bicategory Ho f (C, W) of a homotopy with an arrow on the left has to be defined by using the general composition of w-homotopies, which as mentioned above does not need the consideration of right homotopies. Although ultimately for the fibrant-cofibrant replacement both left and right homotopies are necessary, the latter become implicit in our construction. This allows for a concise, clear proof of the localization theorem, where many of the technical difficulties of working with bicategories are avoided. C. The 2-functor C i −→ Ho(C, W) factors through Ho f (C, W), and we define q = RQ i (more precisely, as its corestriction to Ho(C) = Ho f c (C, W)). Its universal property is deduced from the properties that RQ and i have separately. D. The pseudofunctor structures for Q and R above are obtained by a transfer of pseudofunctor structure, a novel technique introduced in this paper. A natural context in which to state and prove the transfer results is the more general one of lax functors between bicategories, and this is done in Appendix A. E. The definition of Ho f (C, W) Q −→ Ho f (C, W) on 2-cells and its pseudofunctor structure are obtained as the unique ones for which ρ X , ρ f form a pseudonatural transformation Q ρ ⇒ id (we transfer to Q the pseudofunctor structure of the identity pseudofuntor). A
Cahiers de Topologie et Géométrie Différentielle Catégoriques, 2005
La theorie des points proches des varietes differentielles reelles d'Andre Weil generalise la... more La theorie des points proches des varietes differentielles reelles d'Andre Weil generalise la notion fondamentale de jet d'Ehresmann, et comme celui-ci, comprend tout le calcul differentiel des derivees d'ordre superieur. Dans cet article nous generalisons et developpons cette theorie pour le cas des varietes banachiques complexes. Etant donnes une algebre de Weil W et un ouvert B d'un espace de Banach, l'analyticite et la dimension infinie nous imposent des modifications dans la definition de B[W], le prolongement d'espece W de B, pour que ce dernier ait les proprietes souhaitees (Definition 2.8). Pour une fonction holomorphe f, nous demontrons une formule explicite en termes des derivees d'ordre superieur pour la fonction f[W] induite entre les prolongements d'espece W. Dans une seconde partie, nous considerons un modele analytique de la GDS muni d'un plongement j de la categorie des ouverts d'espaces de Banach, et nous montrons que le calcu...
In this paper we develop a general representation theory for mv-algebras. We furnish the appropri... more In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any mv-algebra is isomorphic to the mv-algebra of all global sections of a sheaf of mv-chains on a compact topological space. This result is intimately related to McNaughton's theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton's. On spite of the language utilized in this abstract, we wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory. preface We wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory. Our basic reference on mv-algebras is the book 'Algebraic Foundations of Many-valued Reasoning' [1], and we refer to this book and not to the original sources for the known specific results we utilize. Of sheaf theory we need only some basic general facts, the reader may consult the book 'Sheaves in Geometry and Logic' [9]. The basic reference for category theory is of course the classical 'Categories for the working mathematician' [8]. Contents 1. Background, terminology and notation 4 2. The prime spectrum of mv-algebras 8 3. Duality between mv-algebras and sheaves of mv-chains 9 4. The maximal and the [0, 1]-valued morphisms spectra 12 5. Hyperachimedian algebras 16 6. Finitely presented mv-algebras 20 7. The pushout-pullback lemma 24 8. The compacity lemma. 25 9. The theorem of McNaughton 27 10. Appendix. Sheaf theory of posets 29 References 34
In [1], Grothendieck develops the theory of pro-objects over a category C. The fundamental proper... more In [1], Grothendieck develops the theory of pro-objects over a category C. The fundamental property of the category Pro(C) is that there is an embedding C c −→ Pro(C), the category Pro(C) is closed under small cofiltered limits, and these limits are free in the sense that for any category E closed under small cofiltered limits, pre-composition with c determines an equivalence of categories Cat(Pro(C), E)+ ≃ Cat(C, E), (where the "+" indicates the full subcategory of the functors preserving cofiltered limits). In this paper we develop a 2-dimensional theory of pro-objects. Given a 2-category C, we define the 2-category 2-Pro(C) whose objects we call 2-pro-objects. We prove that 2-Pro(C) has all the expected basic properties adequately relativized to the 2-categorical setting, including the universal property corresponding to the one described above. We have at hand the results of Cat-enriched category theory, but our theory goes beyond the Cat-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.
A la mémoire de Charles Ehresmann Résumé. Nous définissons la notion de 2-catégorie 2-filtrante e... more A la mémoire de Charles Ehresmann Résumé. Nous définissons la notion de 2-catégorie 2-filtrante et donnons une construction explicite du bicolimite d'un 2-foncteurà valeurs dans les catégories. Une catégorie considérée une 2-catégorie triviale est 2-filtrante si et seulement si c'est une catégorie filtrante, et notre construction conduità une catégorieéquivalenteà la catégorie qui s'obtient par la construction usuelle de colimites filtrantes de catégories. Pour cette construction des axiomes plus faibles suffisent, et nous appelons la notion correspondante 2-catégorie pré 2-filtrante. L'ensemble complet des axiomes est nécessaire pour montrer que les bicolimites 2-filtrantes ont les propriétés correspondantes aux propriétés essentielles des colimites filtrantes.
Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C, W) that al... more Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C, W) that allow us to construct the bicategorical localization with respect to W by dealing only with the 2-cells, that is without adding objects or arrows to C. We show that in this case, the 2-cells of the localization can be given by the homotopies with respect to W, a notion defined in this article which is closely related to Quillen's notion of homotopy for model categories but depends only on a single family of arrows. This localization result has a natural application to the construction of the homotopy bicategory of a model bicategory, which we develop elsewhere, as the pair (C f c , W) given by the weak equivalences between fibrant-cofibrant objects satisfies the conditions given in the present article.
We consider locales B as algebras in the tensor category sℓ of sup-lattices. We show the equivale... more We consider locales B as algebras in the tensor category sℓ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections sh(B) q −→ E in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over sℓ for the sℓ-functor Rel(E) Rel(q *) −→ Rel(sh(B)) (B-Mod) 0 into the sℓ-category of discrete B-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those known so far. This equivalence follows from two independent results. We develop an explicit construction of the localic groupoid G : G
In this paper we elaborate on a 2-categorical construction of the homotopy category of a Quillen ... more In this paper we elaborate on a 2-categorical construction of the homotopy category of a Quillen model category. Given any category A and a class of morphisms Σ ⊂ A containing the identities, we construct a 2-category Ho(A) obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in [1]. The inclusion 2-functor A −→ Ho(A) has a universal property which yields the 2-localization of A at Σ provided that the arrows of Σ become equivalences in Ho(A). This result together with a fibrant-cofibrant replacement is then used to obtain the 2-localization of a model category C at the weak equivalences W. The set of connected components of the hom categories yields a novel proof of Quillen's results. We follow the general lines established in [1], [2] for model bicategories.
In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates be... more In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A F −→ B is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similary we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A −→ Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi. Acknowledgement We are grateful to the referees of AiM for their comments on this paper, and in particular for the suggestion to give proper emphasis to our results on conical σ-limits.
In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates be... more In this paper we introduce sigma limits (which we write σ-limits), a concept that interpolates between lax and pseudolimits: for a fixed family Σ of arrows of a 2-category A, a σ-cone for a 2-functor A F −→ B is a lax cone such that the structural 2-cells corresponding to the arrows of Σ are invertible. The conical σ-limit of F is the universal σ-cone. Similary we define σ-natural transformations and weighted σ-limits. We consider also the case of bilimits. We develop the theory of σ-limits and σ-bilimits, whose importance relies on the following key fact: any weighted σ-limit (or σ-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary Cat-valued 2-functor as a conical σ-bicolimit of representable 2-functors, for a suitable choice of Σ, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a Cat-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class Σ, which we call σ-filtered. Our main result is: A pseudofunctor A −→ Cat is flat if and only if it is a σ-filtered σ-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi. Acknowledgement We are grateful to the referees of AiM for their comments on this paper, and in particular for the suggestion to give proper emphasis to our results on conical σ-limits.
A locally connected topos is a Galois topos if the Galois objects generate the topos. We show tha... more A locally connected topos is a Galois topos if the Galois objects generate the topos. We show that the full subcategory of Galois objects in any connected locally connected topos is an inversely 2-filtered 2-category, and as an application of the construction of 2-filtered bilimits of topoi, we show that every Galois topos has a point.
Replacing {0} by the whole ideal of infinitesimals yields a weaker notion of archimedean element ... more Replacing {0} by the whole ideal of infinitesimals yields a weaker notion of archimedean element that we call quasiarchimedean. It is known that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky topology) are exactly the hyperarchimedean algebras. We characterise all the algebras with compact maximal spectrum as being quasihyperarchimedean MV-algebras, which in a sense are non semisimple hyperarchimedean algebras. We develop some basic facts in the theory of MV-algebras along the lines of algebraic geometry, where infinitesimals play the role of nilpotent elements, and prove a MV-algebra version of Hilbert's Nullstellensatz. Finally we consider the relations (some inedited) between several elementary classes of MValgebras in terms of the ideals that characterise them, and present elementary (first order with denumerable disjunctions) proofs in place of the set-theoretical usually found in the literature. Contents 1. Background on MV-algebras of continuos functions 1 2. Quasihyperarchimedean algebras 3 3. The MV-Nullstellensatz. 4 4. The relations between some classes of MV-algebras 6 References 8
We give the definitions of model bicategory and q-homotopy, which are natural generalizations of ... more We give the definitions of model bicategory and q-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory C, denote by C f c the full sub-bicategory of the fibrant-cofibrant objects. We prove that the 2-dimensional localization of C at the weak equivalences can be computed as a bicategory Ho(C) whose objects and arrows are those of C f c and whose 2-cells are classes of q-homotopies up to an equivalence relation. When considered for a model category, q-homotopies coincide with the homotopies as considered by Quillen. The pseudofunctor C q −→ Ho(C) which yields the localization is constructed by using a notion of fibrant-cofibrant replacement in this context. We include an appendix with a general result of independent interest on a transfer of structure for lax functors, that we apply to obtain a pseudofunctor structure for the fibrant-cofibrant replacement. is a biequivalence of bicategories for every bicategory D; where Hom W,Θ (C, D) stands for the full sub-bicategory of Hom(C, D) given by those pseudofunctors that send weak equivalences into equivalences. We also show that the homotopy bicategory Ho(C) is locally small (resp. a 2-category) when C is so. We define q-homotopies as w-homotopies which satisfy some extra conditions related to fibrations and cofibrations. The notion of w-homotopy was introduced in [8] and is taken with respect only to the class W. Note that in op. cit. w-homotopies are called homotopies, but here we chose to use the prefix w-to explicitly distinguish them from q-homotopies. This definition allows to use previous results of [8], where for an arbitrary pair (A, W) given by a family of arrows of a bicategory we construct a bicategory Ho(A, W) whose objects and arrows are those of A and whose 2-cells are constructed with w-homotopies. In this general case there is no vertical composition of w-homotopies, and thus the 2-cells of Ho(A, W) are given by classes of finite sequences of w-homotopies by an appropriate equivalence relation. Associated with Ho(A, W) there is a canonical 2-functor A i −→ Ho(A, W). A natural way in which a proof of Theorem 6.4 can be attempted is to consider Quillen's proof of the localization theorem for model categories in [38], and to try to generalize it to this context using q-homotopies. In doing so, the first obstacle one faces is that there is no canonical way to compose (whisker) arbitrary q-homotopies with arrows. This is already so for the 1-dimensional case in [38], as is well known: for arbitrary arrows A f G G g G G B in a model category, f ℓ ∼ g doesn't imply uf ℓ ∼ ug (where ℓ ∼ is the left homotopy relation). The other composition poses no problem, f u ℓ ∼ gu is easily seen to hold. The way in which this obstacle is avoided in [38] is by considering A to be a cofibrant object, and B fibrant, so that there is a left homotopy between two arrows if and only if there is a right homotopy. An important feature of (left) w-homotopies is that both compositions (whiskerings) are canonically defined, and so this obstacle disappears. There is a rich interplay between wand q-homotopies, and that was the key we have found to overcoming the obstacle described above for the q-homotopies and furthermore proving Theorem 6.4. On the one hand, w-homotopies allow us to define a homotopy bicategory. On the other hand, q-homotopies satisfy several results analogous to the ones that can be found in [38], that we establish and use in this article. More explicitly, to prove Theorem 6.4 we will use the following results: A. Ho(C) inherits the bicategory structure of Ho(C, W) constructed in [8], in such a way that Ho(C) = Ho f c (C, W) ⊂ Ho(C, W), that is the full sub-bicategory of Ho(C, W) given by the fibrant-cofibrant objects. We will deduce this from the following two facts: A1. q-homotopies can be composed vertically: for any pair of composable q-homotopies there is a single q-homotopy representing the composition. A2. For any w-homotopy in C f c there is a fibrant w-homotopy in the same class. For any fibrant w-homotopy there is a fibrant q-homotopy in the same class (we say that a homotopy is fibrant if its cylinder is so; this is made precise in Definition 4.1). In fact, we show more generally that, when X is cofibrant and Y is fibrant, all possible B. There is a fibrant-cofibrant replacement for model bicategories, that is an assignment from C to C, defined on objects and arrows (but not on 2-cells) X f G G g G G Y ❀ RQX RQf G G RQg G G RQY , satisfying the expected conditions for a replacement: all the objects QX are cofibrant, all the objects RQX are fibrant-cofibrant, and there are trivial fibrations QX ρ X −→ X, trivial cofibrations QX λ QX −−→ RQX, and invertible 2-cells RQX o o λ QX RQf ⇓λ Qf QX ⇓ρ f Qf ρ X G G X f RQY o o λ QY QY ρ Y G G Y Note that Q and RQ are just assignments on objects and arrows and not necessarily pseudofunctors, because we are not assuming that there is a (pseudo)functorial factorization, just as in Quillen's original axioms in [38]. Nevertheless, we will show that there are structural 2-cells (homotopies) which make Q and RQ, suitably defined, into pseudofunctors Ho f (C, W) −→ Ho f (C, W), and ρ and λ into pseudonatural transformations. In the above, Ho f (C, W) denotes the bicategory determined by the fibrant left homotopies (indistinctly w-or q-, see A2 above). It is this bicategory that plays a fundamental role in our proof of Theorem 6.4. Note that, even though its 2-cells can be given by qhomotopies, the composition (whiskering) in the bicategory Ho f (C, W) of a homotopy with an arrow on the left has to be defined by using the general composition of w-homotopies, which as mentioned above does not need the consideration of right homotopies. Although ultimately for the fibrant-cofibrant replacement both left and right homotopies are necessary, the latter become implicit in our construction. This allows for a concise, clear proof of the localization theorem, where many of the technical difficulties of working with bicategories are avoided. C. The 2-functor C i −→ Ho(C, W) factors through Ho f (C, W), and we define q = RQ i (more precisely, as its corestriction to Ho(C) = Ho f c (C, W)). Its universal property is deduced from the properties that RQ and i have separately. D. The pseudofunctor structures for Q and R above are obtained by a transfer of pseudofunctor structure, a novel technique introduced in this paper. A natural context in which to state and prove the transfer results is the more general one of lax functors between bicategories, and this is done in Appendix A. E. The definition of Ho f (C, W) Q −→ Ho f (C, W) on 2-cells and its pseudofunctor structure are obtained as the unique ones for which ρ X , ρ f form a pseudonatural transformation Q ρ ⇒ id (we transfer to Q the pseudofunctor structure of the identity pseudofuntor). A
Cahiers de Topologie et Géométrie Différentielle Catégoriques, 2005
La theorie des points proches des varietes differentielles reelles d'Andre Weil generalise la... more La theorie des points proches des varietes differentielles reelles d'Andre Weil generalise la notion fondamentale de jet d'Ehresmann, et comme celui-ci, comprend tout le calcul differentiel des derivees d'ordre superieur. Dans cet article nous generalisons et developpons cette theorie pour le cas des varietes banachiques complexes. Etant donnes une algebre de Weil W et un ouvert B d'un espace de Banach, l'analyticite et la dimension infinie nous imposent des modifications dans la definition de B[W], le prolongement d'espece W de B, pour que ce dernier ait les proprietes souhaitees (Definition 2.8). Pour une fonction holomorphe f, nous demontrons une formule explicite en termes des derivees d'ordre superieur pour la fonction f[W] induite entre les prolongements d'espece W. Dans une seconde partie, nous considerons un modele analytique de la GDS muni d'un plongement j de la categorie des ouverts d'espaces de Banach, et nous montrons que le calcu...
In this paper we develop a general representation theory for mv-algebras. We furnish the appropri... more In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any mv-algebra is isomorphic to the mv-algebra of all global sections of a sheaf of mv-chains on a compact topological space. This result is intimately related to McNaughton's theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton's. On spite of the language utilized in this abstract, we wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory. preface We wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory. Our basic reference on mv-algebras is the book 'Algebraic Foundations of Many-valued Reasoning' [1], and we refer to this book and not to the original sources for the known specific results we utilize. Of sheaf theory we need only some basic general facts, the reader may consult the book 'Sheaves in Geometry and Logic' [9]. The basic reference for category theory is of course the classical 'Categories for the working mathematician' [8]. Contents 1. Background, terminology and notation 4 2. The prime spectrum of mv-algebras 8 3. Duality between mv-algebras and sheaves of mv-chains 9 4. The maximal and the [0, 1]-valued morphisms spectra 12 5. Hyperachimedian algebras 16 6. Finitely presented mv-algebras 20 7. The pushout-pullback lemma 24 8. The compacity lemma. 25 9. The theorem of McNaughton 27 10. Appendix. Sheaf theory of posets 29 References 34
In [1], Grothendieck develops the theory of pro-objects over a category C. The fundamental proper... more In [1], Grothendieck develops the theory of pro-objects over a category C. The fundamental property of the category Pro(C) is that there is an embedding C c −→ Pro(C), the category Pro(C) is closed under small cofiltered limits, and these limits are free in the sense that for any category E closed under small cofiltered limits, pre-composition with c determines an equivalence of categories Cat(Pro(C), E)+ ≃ Cat(C, E), (where the "+" indicates the full subcategory of the functors preserving cofiltered limits). In this paper we develop a 2-dimensional theory of pro-objects. Given a 2-category C, we define the 2-category 2-Pro(C) whose objects we call 2-pro-objects. We prove that 2-Pro(C) has all the expected basic properties adequately relativized to the 2-categorical setting, including the universal property corresponding to the one described above. We have at hand the results of Cat-enriched category theory, but our theory goes beyond the Cat-enriched case since we consider the non strict notion of pseudo-limit, which is usually that of practical interest.
A la mémoire de Charles Ehresmann Résumé. Nous définissons la notion de 2-catégorie 2-filtrante e... more A la mémoire de Charles Ehresmann Résumé. Nous définissons la notion de 2-catégorie 2-filtrante et donnons une construction explicite du bicolimite d'un 2-foncteurà valeurs dans les catégories. Une catégorie considérée une 2-catégorie triviale est 2-filtrante si et seulement si c'est une catégorie filtrante, et notre construction conduità une catégorieéquivalenteà la catégorie qui s'obtient par la construction usuelle de colimites filtrantes de catégories. Pour cette construction des axiomes plus faibles suffisent, et nous appelons la notion correspondante 2-catégorie pré 2-filtrante. L'ensemble complet des axiomes est nécessaire pour montrer que les bicolimites 2-filtrantes ont les propriétés correspondantes aux propriétés essentielles des colimites filtrantes.
Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C, W) that al... more Given a bicategory C and a family W of arrows of C, we give conditions on the pair (C, W) that allow us to construct the bicategorical localization with respect to W by dealing only with the 2-cells, that is without adding objects or arrows to C. We show that in this case, the 2-cells of the localization can be given by the homotopies with respect to W, a notion defined in this article which is closely related to Quillen's notion of homotopy for model categories but depends only on a single family of arrows. This localization result has a natural application to the construction of the homotopy bicategory of a model bicategory, which we develop elsewhere, as the pair (C f c , W) given by the weak equivalences between fibrant-cofibrant objects satisfies the conditions given in the present article.
We consider locales B as algebras in the tensor category sℓ of sup-lattices. We show the equivale... more We consider locales B as algebras in the tensor category sℓ of sup-lattices. We show the equivalence between the Joyal-Tierney descent theorem for open localic surjections sh(B) q −→ E in Galois theory [An extension of the Galois Theory of Grothendieck, AMS Memoirs 151] and a Tannakian recognition theorem over sℓ for the sℓ-functor Rel(E) Rel(q *) −→ Rel(sh(B)) (B-Mod) 0 into the sℓ-category of discrete B-modules. Thus, a new Tannaka recognition theorem is obtained, essentially different from those known so far. This equivalence follows from two independent results. We develop an explicit construction of the localic groupoid G : G
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