# Fifty seventh meeting of the Transpennine Topology Triangle

## Department of Mathematics and Computer Science

University of
Leicester

October 26th 2006

## Programme

Coffee and tea will be available from 11.00 am.
- 11.00 am (F9 Maths Building): Coffee
- 11.30 am (Engineering Lecture Theatre 1): David Gepner (Sheffield)

``Homotopy theory of orbispaces and elliptic cohomology.''
- 12.45 pm (eg Charles Wilson, 5th Floor): Lunch
- 2.30 pm (Engineering Lecture Theatre 1): Joachim Kock (UAB, Barcelona)

``Polynomial functors, trees, and opetopes''
- 3.45 pm (117 Michael Atiyah Building): Tea
- 4.30 pm (119 Michael Atiyah Building): Teimuraz Pirashvili (Leicester)

"Topological Hochschild cohomology and categorical rings"

Abstracts
Joachim Kock: Polynomial functors, trees, and opetopes

Polynomial functors are certain functors built from disjoint union,
product, and exponentiation. They are a categorification of the notion of
polynomial function: you can add and multiply them or substitute them into
each other; you can also differentiate them, and there is a Leibniz rule
and a chain rule. All these manipulations can be done just in terms of
coefficients and exponents. After this introductory material, I will
survey some ways in which polynomial functors are intimately linked with
trees and operads. Trees arise from free monads on polynomial
endofunctors. For such, the disjoint union of all coeffcients is a
collection of (decorated) trees, and they correspond to the set of
operations of a (coloured) operad, while the exponents correspond to their
arities. There is also a canonical way of associating a polynomial functor
to a tree, and a characterisation of the polynomial functors that arise in
this way; hence the very notion of tree is just a special case of
polynomial functor. To finish with some recent research (joint with Joyal,
Batanin, and Mascari), I will explain how polynomial functors provide an
elegant construction of 'higher trees', the so-called opetopes, which are
the basis for the Baez-Dolan approach to higher category theory.

Teimuraz Pirashvili: Topological Hochschild cohomology and categorical rings.

A categorical ring is a category equipped with two monoidal structure, addition and multiplication, such that some coherent conditions hold. We prove that this notion is relevant to get an algebraic model for two-stage ring spectra.

## How to get there

It is a short walk from the rail station. See maps and how to get to Leicester
University

The meeting is partially supported by a Scheme 3 grant from the London
Mathematical Society.

## Escape routes

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