2019/01/06 by Ivan Di Liberti, Fosco Loregiàn, Di Liberti, Ivan +1 · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1901.01594
We prove an equivalence between cocomplete Yoneda structures and certain proarrow equipments on a 2-category \mathcal K. In order to do this, we recognize the presheaf construction of a cocomplete Yoneda structure as a relative, lax idempotent monad sending each admissible 1-cell f :A → B to an adjunction \boldsymbolP_!f\dashv\boldsymbolP^*f. Each cocomplete Yoneda structure on \mathcal K arises in this way from a relative lax idempotent monad "with enough adjoint 1-cells", whose domain generates the ideal of admissibles, and the Kleisli category of such a monad equips its domain with proarrows. We call these structures "yosegi". Quite often, the presheaf construction associated to a yosegi generates an ambidextrous Yoneda structure; in such a setting there exists a fully formal version of Isbell duality.