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Regular patterns, substitudes, Feynman categories and operads

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Michael Batanin, Joachim Kock, and Mark Weber

We show that the regular patterns of Getzler (2009) form a 2-category
biequivalent to the 2-category of substitudes of Day and Street (2003),
and that the Feynman categories of Kaufmann and Ward (2013) form a
2-category biequivalent to the 2-category of coloured operads (with
invertible 2-cells). These biequivalences induce equivalences between
the corresponding categories of algebras. There are three main
ingredients in establishing these biequivalences. The first is a
strictification theorem (exploiting Power's General Coherence Result)
which allows to reduce to the case where the structure maps are
identity-on-objects functors and strict monoidal. Second, we subsume the
Getzler and Kaufmann-Ward hereditary axioms into the notion of Guitart
exactness, a general condition ensuring compatibility between certain left
Kan extensions and a given monad, in this case the
free-symmetric-monoidal-category monad. Finally we set up a biadjunction
between substitudes and what we call pinned symmetric monoidal categories,
from which the results follow as a consequence of the fact that the
hereditary map is precisely the counit of this biadjunction.

Keywords:
operads, symmetric monoidal categories

2010 MSC:
18D10, 18D50

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 7, pp 148-192.

Published 2018-02-19.

http://www.tac.mta.ca/tac/volumes/33/7/33-07.pdf

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