# ItaCaFest

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

The seminar will be live on Zoom at https://cesnet.zoom.us/j/92938745753

The password of the meeting is **ItaCa**.

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 doodling 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

In the table below you will notice that every talk comes with some additional material: an abstract, the slides and a video .

### First meeting ⎯ June 17 2020 ⎯ 14:00 CEST

#### a3poster, a4poster, flyer

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

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

14:45 | Question time | |||

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

15:45 | Question time | |||

16:00 | Free chat |

#### Andrea Gagna

##### (Charles University Prague) *Oplax 3-functors*.

We motivate the introduction of a notion of normalized oplax 3-functors from a homotopical point of view. We explain the algebra of trees needed for the definition and show that they induce a canonical simplicial morphism. Finally, we characterize the simplicial morphisms between nerves coming from normalized oplax 3-functors.

#### Nicola Gambino

##### (University of Leeds) *Variations on distributive laws*.

The notion of a distributive law between monads goes back to fundamental work of Jon Beck from the late ‘60s. Just as a monad describes a kind of algebraic structure, a distributive law between two monads describes how the algebraic structure for one monad distributes over the algebraic structure for the other, as in the notion of a ring (where products distribute over sums).

I will give a survey of variations of distributive laws, according to three orthogonal directions: replacing monads with relative monads (in the sense of Altenkirch et al), replacing categories with objects of a 2-category (à la Street), and increasing categorical dimension. One application is to substitution monoidal structures and operads. This is based on joint work with Fiore, Hyland and Winskel and recent joint work with Lobbia.

### Second meeting ⎯ July 16 2020 ⎯ 14:00 CEST

#### a3poster, a4poster, flyer

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

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

14:45 | Question time | |||

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

15:45 | Question time | |||

16:00 | Free chat |

#### Milly Maietti

##### (Università di Padova) * Predicative generalizations of topos-like structures *.

Predicative generalizations of the notion of elementary topos had been already investigated in the literature starting from the work by I. Moerdijk and E. Palmgren and later by B. van der Berg. In this talk we propose generalizations of the concept of arithmetic quasi-topos and hence of elementary topos which enjoy an internal language which is predicative a’ la Russell and are such that when applying suitable reducibility axioms we get the original (impredicative) notions back. The main difference with previous notions of predicative toposes is that in our notions we just require a non-iterative power-object construction. Genuine examples of quasi-toposes and predicative toposes may be built by employing the notion of elementary quotient completion introduced in joint work with P. Rosolini and applied to predicative versions of triposes in replacement to the usual Hyland-Johnstone-Pitts’s tripos-to-topos construction.

#### Alessio Santamaria

##### (Università di Pisa) * Towards a Calculus of Substitution for Dinatural Transformations *.

In 1972, with the papers *Many-Variable Functorial Calculus, I* and *An Abstract Approach of Coherence*, Kelly started a long-term project on achieving an abstract theory of coherence. To do so, he argues that a “tidy calculus of substitution” of functors in many variables and appropriately general natural transformations is in order; such a calculus ought to generalise the usual Godement calculus of functors in one variable and ordinary natural transformations. Inspired by the work he did with Eilenberg on extranatural transformations in 1966, he developed this calculus for (many-variable) covariant functors and natural transformations. Upon trying the mixed-variance case, he ran into problems linked to the fact that extranaturals do not compose. In this talk, I will show how we realised that the full mixed-variance case wanted by Kelly involves a simple generalisation of dinatural transformations; a sufficient and essentially necessary condition for two consecutive dinatural transformations to compose will be mentioned, and I will present a new definition of horizontal composition of dinaturals. Armed with these results, I will show how to achieve he first steps made by Kelly (in the mixed-variance case, this time) towards a full substitution calculus. There are still some conceptual difficulties about the remaining steps which are yet to be overcome.