# 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…

## 60 Citations

Enhanced 2-categories and limits for lax morphisms

- Mathematics
- 2012

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

- Mathematics, Computer ScienceArXiv
- 2019

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

- Mathematics, Computer ScienceAppl. Categorical Struct.
- 2019

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

- Mathematics, Computer ScienceElectronic 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

- Mathematics
- 2002

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

- Mathematics
- 2011

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

- Mathematics
- 2014

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

- Mathematics
- 2013

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

- Mathematics
- 2000

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

- Mathematics
- 2014

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…

## References

SHOWING 1-10 OF 65 REFERENCES

A unified treatment of transfinite constructions for free algebras

- Mathematics
- 1980

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

- Mathematics
- 1989

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

- Mathematics
- 1989

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

- Mathematics
- 1993

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

- Mathematics
- 1983

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

- Computer Science
- 1992

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

- Mathematics
- 1989

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

- Mathematics
- 1993

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

- Mathematics
- 1989

[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

- Mathematics
- 1995

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…