Results 1 
6 of
6
Monads for which structures are adjoint to units
 Aarhus Preprint Series 1972/73 No.35
"... We present here the equational twodimensional categorical algebra which describes the process of freely completing a category under some class of limits or colimits. It is crystallized out of the authors 1967 dissertation [6] (revised form [7]). I presented a purely equational aspect of that alread ..."
Abstract

Cited by 56 (2 self)
 Add to MetaCart
(Show Context)
We present here the equational twodimensional categorical algebra which describes the process of freely completing a category under some class of limits or colimits. It is crystallized out of the authors 1967 dissertation [6] (revised form [7]). I presented a purely equational aspect of that already
The Constructive Lift Monad
 Informix Software, Inc
, 1995
"... ut by applying T to some poset (namely the original poset less the bottom). Both these properties fail to hold constructively, if the lift monad is interpreted as "adding a bottom"; see Remark below. If, on the other hand, we interpret the lift monad as the one which freely provides supre ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
ut by applying T to some poset (namely the original poset less the bottom). Both these properties fail to hold constructively, if the lift monad is interpreted as "adding a bottom"; see Remark below. If, on the other hand, we interpret the lift monad as the one which freely provides supremum for each subset with at most one element (which is what we shall do), then the first property holds; and we give a necessary and sufficient condition that the second does. Finally, we shall investigate the lift monad in the context of (constructive) locale theory. I would like to thank Bart Jacobs for guiding me to the litterature on Zsystems; to Gonzalo Reyes for calling my attention to Barr's work on totally connected spaces; to Steve Vickers for some pertinent correspondence. I would like to thank the Netherlands Science Organization (NWO) for supporting my visit to Utrecht, where a part of the present research was carried out, and for various travel support from
Flatness, preorders and general metric spaces
, 2008
"... This paper studies a general notion of flatness in the enriched context: Pflatness where the parameter P stands for a class of presheaves. One obtains a completion of a category A by considering the category FlatP(A) ofPflat presheaves over A. This completion is related to the free cocompletion un ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
This paper studies a general notion of flatness in the enriched context: Pflatness where the parameter P stands for a class of presheaves. One obtains a completion of a category A by considering the category FlatP(A) ofPflat presheaves over A. This completion is related to the free cocompletion under a class of colimits defined by Kelly. We define a notion of Qaccessible categories for a family Q of indexes. Our FlatP(A) for small A’s are exactly the Qaccessible categories where Q is the class of Pflat indexes. For a category A, for P =P0 the class of all presheaves, FlatP0(A) is the Cauchycompletion of A. Two classes P1 andP2 of interest for general metric spaces are considered. TheP1 andP2 flatness are investigated and the associated completions are characterized for general metric spaces (enrichments over IR+) ¯ and preorders (enrichments over Bool). We get this way two nonsymmetric completions for metric spaces and retrieve the ideal completion for preorders. 1
Completions of nonsymmetric metric spaces via enriched categories
 Georgian Math. J
"... Abstract. It is known from [13] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0,∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. It is known from [13] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0,∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0,∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the socalled dcpo completion.
ωInductive Completion of Monoidal Categories and Infinite Petri Net Computations
"... Abstract. There exists a KZdoctrine on the 2category of the locally small categories whose algebras are exactly the categories which admits all the colimits indexed by ωchains. The paper presents a wide survey of this topic. In addition, we show that this chain cocompletion KZdoctrine lifts smoo ..."
Abstract
 Add to MetaCart
Abstract. There exists a KZdoctrine on the 2category of the locally small categories whose algebras are exactly the categories which admits all the colimits indexed by ωchains. The paper presents a wide survey of this topic. In addition, we show that this chain cocompletion KZdoctrine lifts smoothly to KZdoctrines on (many variations of) the 2categories of monoidal and symmetric monoidal categories, thus yielding a universal construction of colimits of ωchains in those categories. Since the processes of Petri nets may be axiomatized in terms of symmetric monoidal categories this result provides a universal construction of the algebra of infinite processes of a Petri net.