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Homotopy theory for algebras over polynomial monads

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M. A. Batanin and C. Berger

We study the existence and left properness of transferred model structures
for ``monoid-like'' objects in monoidal model categories. These include
genuine monoids, but also all kinds of operads as for instance symmetric,
cyclic, modular, higher operads, properads and PROP's. All these
structures can be realised as algebras over polynomial monads.
We give a general condition for a polynomial monad which ensures the
existence and (relative) left properness of a transferred model structure
for its algebras. This condition is of a combinatorial nature and singles
out a special class of polynomial monads which we call tame polynomial.
Many important monads are shown to be tame polynomial.

Keywords:
Quillen model category, polynomial monad, coloured operad, graph

2010 MSC:
18D20, 18D50, 55P48

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 6, pp 148-253.

Published 2017-02-06.

http://www.tac.mta.ca/tac/volumes/32/6/32-06.pdf

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