• Corpus ID: 14512629

ON PROPERTY-LIKE STRUCTURES

@article{Kelly1997ONPS,
  title={ON PROPERTY-LIKE STRUCTURES},
  author={G. M. Kelly and Stephen Lack},
  journal={Theory and Applications of Categories},
  year={1997},
  volume={3},
  pages={213-250}
}
A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of "essentially unique" and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads… 
directory.umm.ac.id
60 Citations
Highly Influential Citations
6
Background Citations
28
Methods Citations
8

60 Citations

Citation Type

Citation Type

Enhanced 2-categories and limits for lax morphisms
Abstract We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or…
Probability, valuations, hyperspace: Three monads on Top and the support as a morphism
TLDR
It is obtained that taking the support of a $\tau$-smooth probability measure is also given by a morphism of monads, which implies that every H- algebra (topological complete semilattice) is also a V-algebra.
Distributive Laws via Admissibility
  • Charles Walker
  • Mathematics, Computer Science
    Appl. Categorical Struct.
  • 2019
TLDR
It is shown that this problem of lifting a KZ doctrine P to the 2-category of pseudo T-algebras for some pseudomonad T is equivalent to giving a pseudo-distributive law, and that such distributive laws may be simply described algebraically and are essentially unique.
*-Autonomous Envelopes and Conservativity
  • Michael Shulman
  • Mathematics, Computer Science
    Electronic Proceedings in Theoretical Computer Science
  • 2021
We prove 2-categorical conservativity for any {0,⊤}-free fragment of MALL over its corresponding intuitionistic version: that is, that the universal map from a closed symmetric monoidal category to…
The formal theory of monads II
Abstract We give an explicit description of the free completion EM ( K ) of a 2-category K under the Eilenberg–Moore construction, and show that this has the same underlying category as the…
Pseudo-commutativity of KZ 2-monads
Abstract In this paper we prove that KZ 2-monads (also known as lax-idempotent 2-monads) are pseudo-commutative. The main examples of KZ 2-monads for us will be 2-monads whose algebras are V…
Fibrations of predicates and bicategories of relations
We reconcile the two different category-theoretic semantics of regular theories in predicate logic. A 2-category of `regular fibrations' is constructed, as well as a 2-category of `regular proarrow…
TIGHTLY BOUNDED COMPLETIONS
By a 'completion' on a 2-category K we mean here an idempotent pseu- domonad on K. We are particularly interested in pseudomonads that arise from KZ- doctrines. Motivated by a question of Lawvere, we…
ON THE MONADICITY OF CATEGORIES WITH CHOSEN COLIMITS
There is a 2-category J -Colim of small categories equipped with a choice of colimit for each diagram whose domain J lies in a given small class J of small categories, functors strictly preserving…
OPERADS AS POLYNOMIAL 2-MONADS
In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is dierent from the standard construction…
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 65 REFERENCES
A unified treatment of transfinite constructions for free algebras
Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” T k A → A on A of one or more endofunctors of A, subjected to equational axioms.…
Flexible limits for 2-categories
Many important 2-categories — such as Lex, Fib/B, elementary toposes and logical morphisms, the dual of Grothendieck toposes and geometric morphisms, locally-presentable categories and left adjoints,…
Two-dimensional monad theory
Abstract We consider a 2-monad T with rank on a complete and cocomplete 2-category, and write T-Alg for the 2-category given the T-algebras, the morphisms preserving the structure to within coherent…
Introduction to extensive and distributive categories
Abstract In recent years, there has been considerable discussion as to the appropriate definition of distributive categories. Three definitions which have had some support are: (1) A category with…
A presentation of topoi as algebraic relative to categories or graphs
A functor is conservatiae if it reflects isomorphisms, and is fznitary if it preserves filtered colimits; a monad is finitary if its functor-part is so. Whenever U: I(; -+.iz/ is conservative, it…
Applications of Categories in Computer Science: On clubs and data-type constructors
TLDR
The author worked out a general notion of “club”, as a monad with certain properties, not necessarily on Cat now, but on any category with finite limits, which was further developed in [13], and applied later to other coherence problems in [16] and elsewhere.
Elementary observations on 2-categorical limits
With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important…
Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads
Abstract A right adjoint functor is said to be of descent type if the counit of the adjunction is pointwise a coequalizer. Building on the results of Tholen's doctoral thesis, we give necessary and…
Accessible categories : the foundations of categorical model theory
[F-S] D. Fremlin and S. Shelah, Pointwise compact and stable sets of measurable functions, manuscript, 1990. [G-G-M-S] N. Ghoussoub, G. Godefroy, B. Maurey, W. Schachermayer, Some topological and…
Monads for which Structures are Adjoint to Units
Abstract We analyse the 2-dimensional categorical algebra underlying the process of completing categories, or posets. The algebra explains why and how completeness of a category is describable in…
...
1
2
3
4
5
...