Polynomial functors and trees

Details Polynomial functors and trees Polynomial functors and trees

2008-07-17

arxiv.org/abs/0807.2874v3

Mathematics

Category Theory

50 pages, LaTeX; uses Paul Taylor's diagrams and Peter Kabal's texdraw; does not compile with pdlaftex. v2: expository improve

Scientific paper

We explore the relationship between polynomial functors and (rooted) trees. In the first part we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category $\Omega$ of Moerdijk and Weiss; its main properties are described in terms of some factorisation systems. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a certain projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.

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