## Australian Kittens 2021: A Meeting of Early Career Researchers in Category Theory and Homotopy Theory

December 2-3, 2021

This two day virtual conference is a meeting and celebration of early career researchers in Australian category theory, homotopy theory and related areas. The meeting will consist of a series of 20 minute contributed talks from Australian graduate students and postdocs, virtual/hybrid social events, career advice and mentoring activities.
If you are interested in participating or giving a talk please register here. |

**Schedule: **

Time (UTC+11) |
Thursday |
Friday |

10:00-10:30 |
Adrian Miranda |
Edmund Heng |

10:30-11:00 |
Patrick Elliott |
Giacomo Tendas |

11:00-11:30 |
Break |
Olivia Borghi |

11:45-12:45 |
Activity (CV workshop) |
Activity (Research Statement workshop) |

2:00-2:30 |
Kurt Stoeckl |
Paul Lessard |

2:30-3:00 |
Ruben Zilibowitz |
Eli Hazel |

3:00-3:30 |
Ivo Vekemans |
Break |

4:00-5:00 |
Career Panel |
Trivia and Prizes |

**Thursday:**

We are pleased to welcome Dr. Michelle Strumila (Monash), Dr. Daniel Stevenson (University of Adelaide) and Prof. Arum Ram (University of Melbourne) for a discussion on academic careers.

**10:00-10:30: Adrian Miranda****(Macquarie University)****Title:****Enrichment over Bijective on Objects Functors with applications to explicit free cocompletions with applications to the Formal Theory of Monads****Abstract:**The full subcategory of Cat^2 on bijective on objects functors (hereafter BO) is complete, cocomplete and cartesian closed. Categories enriched over BO are 2-functors which are bijective on objects and arrows. A prime example of such is the 2-functor from Mnd*(C) to KL(C); the latter being the free cocompletion of a 2-category C under Kleisli objects for monads. When C = Cat, this BO category is isomorphic to the full sub-BO-category of BO itself on left adjoints. Alternatively, Mnd*(Cat) --> KL(Cat) can be seen as the BO-enriched free cocompletion of Id_Cat under "loose kleisli objects", which we will introduce. Time permitting, we will also see two other perspectives on BO, one via double categories and one via free cocompletion under codescent objects of cateads.**10:30-11:00 Patrick Elliott (University of Melbourne)**

Title: The Cyclic W-construction and the Cyclic Homotopy Coherent NerveTitle: The Cyclic W-construction and the Cyclic Homotopy Coherent Nerve

**Abstract:**Coloured Cyclic operads are an important and natural generalisation of coloured operads in which operations have no preferred input and output. Examples arise naturally in low dimensional topology and for algebraic structures with pairings. Hackney, Robertson, and Yau recently laid the groundwork for a theory of up-to-homotopy cyclic operads, building on the dendroidal approach to infinity operads of Cisinski, Moerdijk, and Weiss. In this talk we will present work in progress on the homotopy theory of higher cyclic operads in which the colour set is equipped with an involution, including a cyclic dendoridal nerve theorem and the construction of a homotopy coherent cyclic dendroidal nerve.**11:45-12:45: CV Workshop**

Abstract:What goes on your academic CV? This activity goes through the basics of academic CV's and will involve an interactive exercise in reading some dummy CV's and developing strategies for improving the example CV's.Abstract:

**2:00-2:30: Kurt Stoeckl (University of Melbourne)**

Title: Gröbner Bases for Operads Cannot be Generalised to Wheeled StructuresTitle: Gröbner Bases for Operads Cannot be Generalised to Wheeled Structures

**Abstract:**The theory of Gröbner basis for operadic structures provides an algorithmic method of proving that certain structures are Koszul. First introduced for operads by Dotsenko and Khoroshkin, the theory has recently been generalised to coloured operads by Kharitonov and Khoroshkin. In this talk we will discuss the existing theory, some further possible generalisations, and present a simple counter example showing the theory cannot be generalised to wheeled operads, properads and props.**2:30-3:00: Ruben Zilibowitz****(Macquarie University)**

Title:Monos and epis in free Cartesian closed categoriesTitle:

**Abstract:**For a set of atoms X one can define the free Cartesian closed category (CCC) F_X on this. The universal property of F_X says that for any assignment of X to ob(D), where D is any other CCC, there is a unique, up to iso, Cartesian closed (CC) functor F_X --> D extending this. There are non-trivial monomorphisms and epimorphisms in F_X, however, whose monicness and epicness is not necessarily preserved by the CC functor. In this talk, I will define one such epimorphism and sketch the proof that it is epic. We will also see that the faithful CC functor into Set does not preserve epicness for this morphism.**3:00-3:30:****Ivo Vekemans (ANU)**

Title:Mackey functors as G-commutative monoidsTitle:

**Abstract:**Given a finite group G, G-coefficient systems, G-Mackey functors and G-Tambara functors arise in the context of G-actions, encoding operations which behave like induction, restriction, conjugation, and tensor induction. For example in equivariant homotopy theory Mackey and Tambara functors replace abelian groups and commutative rings respectively. In this talk we see how Mackey functors are G-commutative monoids in the G-symmetric monoidal category of coefficient systems, defining these terms as we go. We will also see a similar statement relating Mackey functors and Tambara systems.**4:00-5:00 Career Panel**We are pleased to welcome Dr. Michelle Strumila (Monash), Dr. Daniel Stevenson (University of Adelaide) and Prof. Arum Ram (University of Melbourne) for a discussion on academic careers.

**Friday:**

**10:00-10:30: Edmund Heng (ANU)**

Title:

Title:

**Categorification and algebras in fusion categories**

****

**Abstract:**Categorical representation theory studies actions of mathematical objects (groups, algebras, etc) on categories. One way to obtain these actions are through the categorification of known classical representations. In this talk we focus on categorifying certain representations of generalised braid groups called the Burau representations, and how one can construct algebra (monoid) objects in fusion categories to achieve this.

**10:30-11:00: Giacomo Tendas**

**Enriched accessibility and flatness**

Title:

**(Macquarie University)**Title:

**Abstract:**The importance of accessible categories in ordinary categry theory has been widely recognised; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as Ab, SSet, Cat, and Met. The problem in this context is that the equivalence between (a) and (b) is no longer true in general.

In this talk, after a short introduction to the theory of accessible categories, we'll give sufficient conditions on the base of enrichment V so that (a) iff (b) holds; this will include, among others, the bases Cat, SSet, Pos and 2-Cat. Time permitting, we'll also see what happens when the base of enrichment has an algebraic flavour; i.e. when V is Ab, R-Mod, GAb, and DGAb.

**10:30-11:00: Olivia Borghi**

Title:Operadic Unitarization

**(University of Melbourne)**Title:

**Abstract:**In his paper, Quasi-Hopf Algebras, Vladimir Drinfe'ld describes a process by which we can supply certain braided hopf algebras with a coboundary structure. He calls this "unitarization." The categories of modules over braided hopf algebras are braided monoidal categories. The categories of modules over a coboundary hopf algebra are coboundary monoidal. These are monoidal categories whose commutativity isomorphisms are governed by the cactus groups. An operadic analog of the unitarization process allows us to supply certain braided monoidal categories with a coboundary monoidal structure.

**11:45-12:45: Research Statement Workshop**

Abstract:This activity introduces the various types of research statements one needs to write (eg: job applications, grant applications, etc). We will give tips for writing research statements with a specific focus on the challenges in describing category theory and homotopy theory to a more general mathematical audience.

Abstract:

**2:00-2:30: Paul Lessard (Macquarie University)**

Title: Weighted Limits in the 2-Category of Categories with Arities and a 2-Categorical Treatment of Spectrification

Title: Weighted Limits in the 2-Category of Categories with Arities and a 2-Categorical Treatment of Spectrification

**Abstract:**We provide a nearly formal treatment of the 2-category of categories with arities of Berger-Mellies-Weber with important consequences for the computation of weighted limits in that 2-category. As an example application of that theory we'll demonstrate that two “obvious” notions of

**Z**-category are equivalent. Along the way we'll observe that the spectrification functors of classical algebraic topology admit a natural 2-categorical generalization.

**2:30-3:00: Eli Hazel (Macquarie University)**

**Title:**Towards an equivalence of categories for relation algebras

**Abstract:**Relation algebras are a variety of Boolean algebras with operators, introduced by Tarski in 1941. The axioms of relation algebras were intended to capture the calculus of (binary) relations, in the same way that Boole's algebras were intended to capture the so-called

*calculus of classes*. Boole's program was highly successful, culminating in the late 1930s with Stone's Representation Theorem for Boolean Algebras; i.e., that every Boolean algebra can be represented as a subalgebra of a powerset, with elements given by the clopen sets of some Boolean space (nowadays usually referred to as Stone spaces). It was hoped that an analogous representation theorem would hold for relation algebras, however, the existence of non-representable relation algebras was proved by Lyndon in 1950.

In this talk, we will discuss our current work on relation algebras, in which we hope a modern categorical perspective will be fruitful. Our focus at present is an equivalence of categories for complete atomic relation algebras, the category of groupoids being a subcategory thereof.

This talk is based on joint work with Dirk Hofmann (University of Aveiro).

**4:00-5:00: Trivia and Prizes**

Scientific Committee: Bryce Clarke (Macquarie University) Nicola Di Vittorio (Macquarie University) Chandan Singh (University of Melbourne) Paula Verdugo (Macquarie University) |
Prize Committee:Olivia Borghi (University of Melbourne) Ivo Vekemans (ANU) Primary Contact: marcy.robertson <at>unimelb.edu.au |