## Abstract

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, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which 'structure is adjoint to unit', and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads.

Original language | English |
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Pages (from-to) | 213-250 |

Number of pages | 38 |

Journal | Theory and Applications of Categories |

Volume | 3 |

Publication status | Published - 1997 |

## Keywords

- 2-category
- Monad
- Property
- Structure