Regular patterns, substitudes, Feynman categories and operads

Michael Batanin; Joachim Kock; Mark Weber

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arXiv · 1510.08934

Regular patterns, substitudes, Feynman categories and operads

Michael Batanin·Joachim Kock·Mark Weber

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Abstract

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.

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Michael Batanin, Joachim Kock, Mark Weber. 2018-03-05. Regular patterns, substitudes, Feynman categories and operads. https://arxiv.org/abs/1510.08934

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