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# Polynomial functors and trees

By Joachim Kock
*Int. Math. Res. Not.* **2011** (2011), 609-673.
ArXiv:0807.2874

#### Abstract

We explore the relationship between polynomial functors and trees.
In the first part we characterise trees as certain polynomial
functors and obtain a completely formal but at the same time
conceptual and explicit construction of two categories of rooted
trees, whose main properties we describe in terms of some
factorisation systems. The second category is the category Ω
of Moerdijk and Weiss.
Although the constructions are motivated and
explained in terms of polynomial functors, they all amount to
elementary manipulations with finite sets. Included in part 1 is also an
explicit construction of the free monad on a polynomial
endofunctor, given in terms of trees. 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 projectivity
condition, which serves also to
characterise polynomial endofunctors and monads among (coloured)
collections and operads.
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Last updated: 2011-02-03 by
Joachim Kock.