Free extensivity via distributivity

  • Fernando Lucatelli Nunes

    Utrecht University, Utrecht, The Netherlands; University of Coimbra, Coimbra, Portugal
  • Rui Prezado

    University of Coimbra, Coimbra, Portugal
  • Matthijs Vákár

    Utrecht University, Utrecht, The Netherlands
Free extensivity via distributivity cover

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Abstract

We consider the canonical pseudodistributive law between various free limit completion pseudomonads and the free coproduct completion pseudomonad. When the class of limits includes pullbacks, we show that this consideration leads to notions of extensive categories. More precisely, we show that extensive categories with pullbacks and infinitary lextensive categories are the pseudoalgebras for the pseudomonads resulting from two of these pseudo­distributive laws. Moreover, we introduce the notion of doubly-infinitary lextensive category, and we establish that the freely generated ones are cartesian closed. From this result, we further deduce that, in freely generated infinitary lextensive categories, the objects with a finite number of connected components are exponentiable. We conclude our work with remarks on examples, descent theoretical aspects of this work, results concerning non-canonical isomorphisms, and relationship with other work.

Cite this article

Fernando Lucatelli Nunes, Rui Prezado, Matthijs Vákár, Free extensivity via distributivity. Port. Math. (2024), published online first

DOI 10.4171/PM/2129