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Details of talk

TitleUniversal properties of polynomials via doctrinal Yoneda structures
PresenterCharles Walker (Macquarie University)
Author(s)Mr Charles Walker
SessionCategory Theory, Algebraic Topology, K-Theory
Time14:30:00 2017-12-12
Abstract


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Given a category $\mathcal{E}$ with pullbacks, one may form the bicategory of
spans in $\mathcal{E}$, denoted $\mathbf{Span}(\mathcal{E})$.
The universal properties satisfied by this construction, as established by
Dawson, Par{\'e}, Pronk and Hermida, simply describe what data one needs in
order to construct a homomorphism of bicategories from this bicategory
$\mathbf{Span}(\mathcal{E})$ to another bicategory $\mathcal{C}$.

In this talk, it is our primary goal to describe an analogue of these results
for the bicategory of polynomials, denoted $\mathbf{Poly}(\mathcal{E})$, both
with the simpler cartesian 2-cells, and the more complex general 2-cells between
polynomials.

However, we do not prove these results directly; indeed, it is the secondary
goal of this talk to show how one may avoid most of the coherence conditions
which stem from the complicated nature of polynomial composition. 
This is to be done through the use of Yoneda structures.
Our approach will thus lead to a proof of the universal properties of the
polynomial construction, whilst avoiding many of the coherence conditions that
would have appeared in a direct proof.