REGULAR PATTERNS, SUBSTITUDES, FEYNMAN CATEGORIES AND OPERADS
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REGULAR PATTERNS, SUBSTITUDES, FEYNMAN CATEGORIES AND OPERADS. / Batanin, Michael; Kock, Joachim; Weber, Mark.
I: Theory and Applications of Categories, Bind 33, 2018, s. 148-192.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - REGULAR PATTERNS, SUBSTITUDES, FEYNMAN CATEGORIES AND OPERADS
AU - Batanin, Michael
AU - Kock, Joachim
AU - Weber, Mark
PY - 2018
Y1 - 2018
N2 - 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.
AB - 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.
KW - operads
KW - symmetric monoidal categories
KW - YONEDA STRUCTURES
KW - ALGEBRA
M3 - Journal article
VL - 33
SP - 148
EP - 192
JO - Theory and Applications of Categories
JF - Theory and Applications of Categories
SN - 1201-561X
ER -
ID: 331498515