Geodesic
Regular patterns, substitudes, Feynman categories and operads
Theory and applications of categories, Tome 33 (2018), pp. 148-192

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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.

Publié le :
Classification : 18D10, 18D50
Keywords: operads, symmetric monoidal categories 
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     author = {Michael Batanin and Joachim Kock and Mark Weber},
     title = {Regular patterns, substitudes, {Feynman} categories and operads},
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Michael Batanin; Joachim Kock; Mark Weber. Regular patterns, substitudes, Feynman categories and operads. Theory and applications of categories, Tome 33 (2018), pp. 148-192. http://geodesic.mathdoc.fr/item/TAC_2018_33_a6/