#
Operads as polynomial 2-monads

##
Mark Weber

In this article we give a construction of a polynomial 2-monad from an
operad and describe the algebras of the 2-monads which then arise. This
construction is different from the standard construction of a monad
from an operad in that the algebras of our associated 2-monad are the
categorified algebras of the original operad. Moreover it enables us to
characterise operads as categorical polynomial monads in a canonical
way. This point of view reveals categorical polynomial monads as a
unifying environment for operads, Cat-operads and clubs. We recover the
standard construction of a monad from an operad in a 2-categorical way
from our associated 2-monad as a coidentifier of 2-monads, and
understand the algebras of both as weak morphisms of operads into a
Cat-operad of categories. Algebras of operads within general symmetric
monoidal categories arise from our new associated 2-monad in a
canonical way. When the operad is sigma-free, we establish a Quillen
equivalence, with respect to the model structures on algebras of
2-monads found by Lack, between the strict algebras of our associated
2-monad, and those of the standard one.

Keywords:
operads; polynomial functors

2010 MSC:
18D20; 18D50; 55P48

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 49, pp 1659-1712.

Published 2015-12-02.

http://www.tac.mta.ca/tac/volumes/30/49/30-49.pdf

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