### Esquisse et contrôle

"... Nous reprenons ici un exemple de contrôle développé par Yves Ledru du L.S.R de Grenoble [Ledru 98]. Cet exemple a déjà fait l’objet d’un nombre d’articles important lors du congrès:“Approches Formelles dans l’Assistance au Développement de Logiciels ” qui a eu lieu à Grenoble en janvier 2000 [AFADL’ ..."

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Nous reprenons ici un exemple de contrôle développé par Yves Ledru du L.S.R de Grenoble [Ledru 98]. Cet exemple a déjà fait l’objet d’un nombre d’articles important lors du congrès:“Approches Formelles dans l’Assistance au Développement de Logiciels ” qui a eu lieu à Grenoble en janvier 2000 [AFADL’2000]. Ainsi, diverses techniques ont été illustrées sur ce même exemple.

### Mosaics for Specifications with Implicit State

, 2000

"... In this paper, we develop a new framework for algebraicspecifications with implicit state. Among other results, this bridges the gap between specifications with implicit state and with explicit state. ..."

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In this paper, we develop a new framework for algebraicspecifications with implicit state. Among other results, this bridges the gap between specifications with implicit state and with explicit state.

### Scetches and Specifications . . .

, 2000

"... SKETCHES AND SPECIFICATIONS is a common denomination for several papers which deal with applications of Ehresmann’s sketch theory to computer science. These papers can be considered as the first steps towards a unified theory for software engineering. However, their aim is not to advocate a unificat ..."

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SKETCHES AND SPECIFICATIONS is a common denomination for several papers which deal with applications of Ehresmann’s sketch theory to computer science. These papers can be considered as the first steps towards a unified theory for software engineering. However, their aim is not to advocate a unification of computer languages; they are designed to build a frame for the study of notions which arise from several areas in computer science. These papers are arranged in two complementary families: REFERENCE MANUAL and USER’S GUIDE. The reference manual provides general definitions and results, with comprehensive proofs. On the other hand, the user’s guide places emphasis on motivations and gives a detailed description of several examples. These two families, though complementary, can be read independently. No prerequisite is assumed; however, it can prove helpful to be familiar either with specification techniques in computer science or with category theory in mathematics. These papers are under development, they are, or will be, available at:

### Scetches and Specifications User'S Gude -- First . . .

, 2000

"... SKETCHES AND SPECIFICATIONS is a common denomination for several papers which deal with applications of Ehresmann’s sketch theory to computer science. These papers can be considered as the first steps towards a unified theory for software engineering. However, their aim is not to advocate a unificat ..."

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SKETCHES AND SPECIFICATIONS is a common denomination for several papers which deal with applications of Ehresmann’s sketch theory to computer science. These papers can be considered as the first steps towards a unified theory for software engineering. However, their aim is not to advocate a unification of computer languages; they are designed to build a frame for the study of notions which arise from several areas in computer science. These papers are arranged in two complementary families: REFERENCE MANUAL and USER’S GUIDE. The reference manual provides general definitions and results, with comprehensive proofs. On the other hand, the user’s guide places emphasis on motivations and gives a detailed description of several examples. These two families, though complementary, can be read independently. No prerequisite is assumed; however, it can prove helpful to be familiar either with specification techniques in computer science or with category theory in mathematics. These papers are under development, they are, or will be, available at:

### Higher Gauge Theory II: 2-Connections

"... Connections and curvings on gerbes are beginning to play a vital role in differential geometry and theoretical physics — first abelian gerbes, and more recently nonabelian gerbes and the twisted nonabelian gerbes introduced by Aschieri and Jurčo in their study of M-theory. These concepts can be el ..."

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Connections and curvings on gerbes are beginning to play a vital role in differential geometry and theoretical physics — first abelian gerbes, and more recently nonabelian gerbes and the twisted nonabelian gerbes introduced by Aschieri and Jurčo in their study of M-theory. These concepts can be elegantly understood using the concept of ‘2bundle’ recently introduced by Bartels. A 2-bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a ‘2-connection’ on a principal 2-bundle. We describe principal 2-bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for twisted nonabelian gerbes with connection and curving subject to a certain constraint — namely, the vanishing of the ‘fake curvature’, as defined by Breen and Messing. This constraint also turns out to guarantee the existence of ‘2-holonomies’: that is, parallel transport over both curves and surfaces, fitting together to define a 2-functor from the ‘path 2-groupoid’ of the base space to the structure 2-group. We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base

### Corrections to Toposes, Triples and Theories

, 1994

"... The corrections are listed by page number. The name in parentheses after the page number shows who told us of the error. GENERAL COMMENT Our text is intended primarily as an exposition of the mathematics, not a historical treatment of it. In particular, if we state a theorem without attribution we d ..."

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The corrections are listed by page number. The name in parentheses after the page number shows who told us of the error. GENERAL COMMENT Our text is intended primarily as an exposition of the mathematics, not a historical treatment of it. In particular, if we state a theorem without attribution we do not in any way intend to claim that it is original with this book. We note specifically that most of the material in Chapters 4 and 8 is an extensive reformulation of ideas and theorems due to C. Ehresmann, J. Bénabou, C. Lair and their students, to Y. Diers, and to A. Grothendieck and his students. We learned most of this material second hand or recreated it, and so generally do not know who did it first. We will happily correct mistaken attributions when they come to our attention. p. 9 (Peter Johnstone). Exercise (SGRPOID) is incorrect as it stands; a semilattice without identity satisfies (i) through (iii) but is not a category. Condition (iii) must be strengthened to read: Say an element e has the identity property if e ◦f =f whenever e ◦f is defined and g ◦e =g whenever g ◦e is defined. Then we require that for any elementf, there is an elementewith the identity property for whiche ◦f is defined and an element e ′ with the identity property for which f ◦e ′ is defined. p. 26 (D. Čubrić). Property (ii) of Exercise (SUBF) should read “If f: A − → B, then F(f) restricted to

### Preprint typeset in JHEP style- HYPER VERSION Higher Gauge Theory II: 2-Connections

"... Abstract: Connections and curvings on gerbes are beginning to play a vital role in differential geometry and theoretical physics — first abelian gerbes, and more recently nonabelian gerbes and the twisted nonabelian gerbes introduced by Aschieri and Jurčo in their study of M-theory. These concepts c ..."

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Abstract: Connections and curvings on gerbes are beginning to play a vital role in differential geometry and theoretical physics — first abelian gerbes, and more recently nonabelian gerbes and the twisted nonabelian gerbes introduced by Aschieri and Jurčo in their study of M-theory. These concepts can be elegantly understood using the concept of ‘2bundle’ recently introduced by Bartels. A 2-bundle is a generalization of a bundle in which the fibers are categories rather than sets. Here we introduce the concept of a ‘2-connection’ on a principal 2-bundle. We describe principal 2-bundles with connection in terms of local data, and show that under certain conditions this reduces to the cocycle data for twisted nonabelian gerbes with connection and curving subject to a certain constraint — namely, the vanishing of the ‘fake curvature’, as defined by Breen and Messing. This constraint also turns out to guarantee the existence of ‘2-holonomies’: that is, parallel transport over both curves and surfaces, fitting together to define a 2-functor from the ‘path 2-groupoid’ of the base space to the structure 2-group. We give a general theory of 2-holonomies and show how they are related to ordinary parallel transport on the path space of the base

### Remarks on 2-Groups

, 2008

"... A 2-group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m:G× G → G has been replaced by a functor. A number of precise definitions of this notion have already been explored, but a full treatment of their relationship ..."

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(Show Context)
A 2-group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m:G× G → G has been replaced by a functor. A number of precise definitions of this notion have already been explored, but a full treatment of their relationships is difficult to extract from the literature. Here we describe the relation between two of the most important versions of this notion, which we call ‘weak ’ and ‘coherent ’ 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix:1 → x⊗¯x, ex: ¯x⊗x → 1 forming an adjunction. We define 2-categories of weak and coherent 2-groups and construct an ‘improvement ’ 2-functor which turns weak 2-groups into coherent ones; using this one can show that these 2-categories are biequivalent. We also internalize the concept of a coherent 2-group. This gives a way of defining topological 2-groups, Lie 2-groups, and the like. 1