Скачать с ютуб Juan Orendain --- Canonical squares in fully faithful and absolutely dense equipments. в хорошем качестве

Juan Orendain --- Canonical squares in fully faithful and absolutely dense equipments.

A talk on May 8, 2024 on Zoom. Abstract: Equipments are categorical structures of dimension 2 having two separate types of 1-arrows vertical and horizontal and supporting restriction and extension of horizontal arrows along vertical ones. Equipments were defined by Wood in [W] as 2-functors satisfying certain conditions, but can also be understood as double categories satisfying a fibrancy condition as in [Sh]. In the zoo of 2-dimensional categorical structures, equipments nicely fit in between 2-categories and double categories, and are generally considered as the 2-dimensional categorical structures where synthetic category theory is done, and in some cases, where monoidal bicategories are more naturally defined. In a previous talk in the seminar, I discussed the problem of lifting a 2-category into a double category along a given category of vertical arrows, and how this problem allows us to define a notion of length on double categories. The length of a double category is a number that roughly measures the amount of work one needs to do to reconstruct the double category from a bicategory along its set of vertical arrows. In this talk I will review the length of double categories, and I will discuss two recent developments in the theory: In the paper [OM] a method for constructing different double categories from a given bicategory is presented. I will explain how this construction works. One of the main ingredients of the construction are so-called canonical squares. In the preprint [O] it is proven that in certain classes of equipments fully faithful and absolutely dense every square that can be canonical is indeed canonical. I will explain how from this, it can be concluded that fully faithful and absolutely dense equipments are of length 1, and so they can be 'easily' reconstructed from their horizontal bicategories. References: [O] Length of fully faithful framed bicategories. arXiv:2402.16296. [OM] J. Orendain, R. Maldonado-Herrera, Internalizations of decorated bicategories via π-indexings. To appear in Applied Categorical Structures. arXiv:2310.18673. [W] R. K. Wood, Abstract Proarrows I, Cahiers de topologie et géométrie différentielle 23 3 (1982) 279-290. [Sh] M. Shulman, Framed bicategories and monoidal fibrations. Theory and Applications of Categories, Vol. 20, No. 18, 2008, pp. 650–738.