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Operads and phylogenetic trees

##
John C. Baez and Nina Otter

We construct an operad Phyl whose operations are the edge-labelled
trees used in phylogenetics. This operad is the coproduct of Com, the
operad for commutative semigroups, and $[0,\infty)$, the operad with unary
operations corresponding to nonnegative real numbers, where composition is
addition. We show that there is a homeomorphism between the space of
n-ary operations of Phyl and $\T_n\times [0,\infty)^{n+1}$, where
$\T_n$ is the space of metric n-trees introduced by Billera, Holmes and
Vogtmann. Furthermore, we show that the Markov models used to reconstruct
phylogenetic trees from genome data give coalgebras of Phyl. These
always extend to coalgebras of the larger operad Com + $[0,\infty]$,
since Markov processes on finite sets converge to an equilibrium as time
approaches infinity. We show that for any operad O, its coproduct with
$[0,\infty]$ contains the operad W(O) constructed by Boardman and
Vogt. To prove these results, we explicitly describe the coproduct of
operads in terms of labelled trees.

Keywords:
operads, trees, phylogenetic trees, Markov processes

2010 MSC:
18D50

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 40, pp 1397-1453.

Published 2017-10-04.

http://www.tac.mta.ca/tac/volumes/32/40/32-40.pdf

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