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Journal Article

Regular patterns, substitudes, Feynman categories and operads


Batanin,  Michael
Max Planck Institute for Mathematics, Max Planck Society;

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Batanin, M., Kock, J., & Weber, M. (2018). Regular patterns, substitudes, Feynman categories and operads. Theory and Applications of Categories, 33: 7, pp. 148-192.

Cite as: https://hdl.handle.net/21.11116/0000-0003-D404-9
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.