The topics of this talk are motivated by two classic results: firstly, the fact that "Top^op is a quasi-variety" proven by Barr and Pedicchio in 1995; secondly, the identification of (constructively) completely distributive lattices as the nuclear objects in the autonomous category Sup of complete lattices and suprema-preserving maps (Higgs and Rowe 1989; Rosebrugh and Wood 1994). Regarding the former, arguably a more "perspicuous" argument is presented in Adámek and Pedicchio (1997) and further explored in Pedicchio and Wood (1999) to give a constructive proof showing that the dual of the category of pre-ordered sets and monotone maps is a quasi-variety. In this talk we recall the main ingredients and discuss these results in the context of quantale-enriched categories.

This talk is based on joint work with Carlos Fitas and Maria Manuel Clementino.

References:

Adámek, Jiří and Pedicchio, M. Cristina (1997). "A remark on topological spaces, grids, and topological systems". In: Cahiers de Topologie et Géométrie Différentielle Catégoriques 38.(3), pp. 217-226.

Barr, Michael and Pedicchio, M. Cristina (1995). "Top^op is a quasivariety". In: Cahiers de Topologie et Géométrie Différentielle Catégoriques 36.(1), pp. 3-10.

Higgs, Denis A. and Rowe, Keith A. (1989). "Nuclearity in the category of complete semilattices". In: Journal of Pure and Applied Algebra 57.(1), pp. 67-78.

Pedicchio, M. Cristina and Wood, Richard J. (1999). "Groupoidal completely distributive lattices". In: Journal of Pure and Applied Algebra 143.(1-3), pp. 339-350.

Rosebrugh, Robert and Wood, Richard J. (1994). "Constructive complete distributivity IV". In: Applied Categorical Structures 2.(2), pp. 119- 144.