# ItaCa Fest

ItaCa Fest is an online webinar aimed to gather the community of ItaCa.

The seminar will be live on Zoom at this link.

The main event will last a couple of hours, but everyone is invited to stay longer for the (scientific) chat. The sound of chalk on a blackboard is inimitable, but we will be using Whiteboard to allow people to chat and doodle at the same time. We host the video content of this meeting on our YT channel.

If you would like to be a speaker, or you want to contact the organizers of the seminar, you can reach us via email at **webmasters.itaca@gmail.com**. Make sure to include the word **FEST20** in the subject.

## Schedule

### ItaCa Fest XMAS EDITION ⎯ December 16, 2020 ⎯ 10:00 CET

(Note the change of time!)

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

10:00 | TENDAS |
Macquarie University | Equivalent characterizations of accessible V-categories |
▤ ▶ |

10:45 | Question time | |||

11:00 | PAOLI |
University of Leicester | Weakly globular double categories and weak units |
▤ ▶ |

11:45 | Question time | |||

12:00 | Lunch break | |||

14:00 | LOREGIAN |
Tallinn University of Technology | Coends of higher arity |
▤ ▶ |

14:45 | Question time | |||

15:00 | ROSOLINI |
Università di Genova | The Comonad of Identities |
▤ ▶ |

15:45 | Question time | |||

16:00 | Free chat |

### Giacomo Tendas

#### Equivalent characterizations of accessible V-categories

Ordinary accessible categories have been studied by various authors and, since their introduction, have been characterized in several ways. When moving to the enriched context, though, many aspects of the theory are still very little developed. Moreover, two different notions of accessibility for V-categories have been introduced: one is a direct generalization of the ordinary definition, and involves only conical filtered colimits; the other (introduced by Borceux, Quinteiro, and Rosický) involves colimits weighted by flat V-functors, a larger class of “filtered” colimits which includes the conical ones. In the first part of the talk we will compare these two notions and exhibit several examples of base of enrichment for which these coincide or differ. In the second part we will provide a characterization of enriched accessible categories (for both notions) using “virtual” orthogonality and reflectivity conditions. The idea is to generalize the known characterization of locally presentable categories (as orthogonality classes and accessibly embedded reflective subcategories of presheaves) to our non-cocomplete case. This is part of a joint work with Steve Lack.

### Simona Paoli

#### Weakly globular double categories and weak units

Weakly globular double categories are a model of weak 2-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani 2-categories. Fair 2-categories, introduced by J. Kock, model weak 2-categories with strictly associative compositions and weak unit laws. In this talk I will illustrate how to establish a direct comparison between weakly globular double categories and fair 2-categories and prove they are equivalent after localisation with respect to the 2-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.

### Fosco Loregian

#### Coends of higher arity

We specialise a recently introduced notion of generalised dinaturality to the case where the domain (resp., codomain) is constant, obtaining notions of ends (resp., coends) of higher arity, dubbed herein (p,q)-ends (resp., (p,q)-coends). While higher arity co/ends are particular instances of “totally symmetrised” (ordinary) co/ends, they serve an important technical role in the study of a number of new categorical phenomena, which may be broadly classified as two new variants of category theory. The first of these, weighted category theory, consists of the study of weighted variants of the classical notions and construction found in ordinary category theory, besides that of a limit. This leads to a host of varied and rich notions, such as weighted Kan extensions, weighted adjunctions, and weighted ends. The second, diagonal category theory, proceeds in a different (albeit related) direction, in which one replaces universality with respect to natural transformations with universality with respect to dinatural transformations, mimicking the passage from limits to ends. In doing so, one again encounters a number of new interesting notions, among which one similarly finds diagonal Kan extensions, diagonal adjunctions, and diagonal ends. This is joint work with Théo de Oliveira Santos.

### Pino Rosolini

#### The Comonad of Identities

Lawvere’s hyperdoctrines mark the beginning of applications of category theory to logic. In particular, existential elementary doctrines proved essential to give models of non-classical logics. The clear connection between (typed) logical theories and certain Pos-valued functors is exemplified by the embedding of the category of elementary doctrines into that of primary doctrines, which has a right adjoint given by a completion which freely adds quotients for equivalence relations. We extend that result in two ways: first we show that, in fact, the embedding is 2-functorial and 2-comonadic. Next, we show that the same applies to the more general case of elementary fibrations: the embedding of elementary fibrations into fibrations with finite products is 2-comonadic. This is joint work with Jacopo Emmenegger and Fabio Pasquali.

## Past Events

### November 25, 2020 ⎯ 14:00 CET

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:00 | TROTTA |
Università di Verona | The existential completion, choice principles and applications |
▤ ▶ |

14:45 | Question time | |||

15:00 | NARDIN |
Universität Regensburg | Equivariant multiplicative structures |
▤ ▶ |

15:45 | Question time | |||

16:00 | Free chat |

### October 21, 2020 ⎯ 14:00 CEST

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:00 | ROVELLI |
ANU | Towards an explicit comparison between globular and simplicial models of (∞,2)-categories |
▤ ▶ |

14:45 | Question time | |||

15:00 | VIRILI |
Università di Udine | Factorization systems on derivators |
▤ ▶ |

15:45 | Question time | |||

16:00 | Free chat |

### September 23, 2020 ⎯ 14:00 CEST

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:00 | GHIORZI |
Appalachian State University | Complete internal categories |
▤ ▶ |

14:45 | Question time | |||

15:00 | PERRONE |
MIT Department of Mathematics | Colimits as algebraic operations |
▤ ▶ |

15:45 | Question time | |||

16:00 | Free chat |

### June 17, 2020 ⎯ 14:00 CEST

#### a3poster, a4poster, flyer

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:00 | GAGNA |
Univerzita Karlova | Oplax 3-functors |
▤ |

14:45 | Question time | |||

15:00 | GAMBINO |
University of Leeds | Variations on distributive laws |
▤ |

15:45 | Question time | |||

16:00 | Free chat |

### July 16, 2020 ⎯ 14:00 CEST

#### a3poster, a4poster, flyer

Time | Speaker | Affiliation | Talk | Material |
---|---|---|---|---|

14:00 | MAIETTI |
Università di Padova | Predicative generalizations of topos-like structures |
▤ ▶ |

14:45 | Question time | |||

15:00 | SANTAMARIA |
Università di Pisa | Towards a Calculus of Substitution for Dinatural Transformations |
▤ ▶ |

15:45 | Question time | |||

16:00 | Free chat |