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Weak units, universal cells, and coherence via universality for bicategories

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Amar Hadzihasanovic

Poly-bicategories generalise planar polycategories in the same way as
bicategories generalise monoidal categories. In a poly-bicategory, the
existence of enough 2-cells satisfying certain universal properties
(representability) induces coherent algebraic structure on the 2-graph
of single-input, single-output 2-cells. A special case of this theory
was used by Hermida to produce a proof of strictification for
bicategories. No full strictification is possible for
higher-dimensional categories, seemingly due to problems with 2-cells
that have degenerate boundaries; it was conjectured by C. Simpson that
semi-strictification excluding units may be possible.

We study poly-bicategories where 2-cells with degenerate boundaries are
barred, and show that we can recover the structure of a bicategory
through a different construction of weak units. We prove that the
existence of these units is equivalent to the existence of 1-cells
satisfying lower-dimensional universal properties, and study the
relation between preservation of units and universal cells.

Then, we introduce merge-bicategories, a variant of poly-bicategories
with more composition operations, which admits a natural monoidal
closed structure, giving access to higher morphisms. We derive
equivalences between morphisms, transformations, and modifications of
representable merge-bicategories and the corresponding notions for
bicategories. Finally, we prove a semi-strictification theorem for
representable merge-bicategories with a choice of composites and units.

Keywords:
bicategories, polycategories, multicategories, merge-bicategories,
polygraphs, coherence, strictification, weak units

2010 MSC:
18D05

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 29, pp 883-960.

Published 2019-09-25.

http://www.tac.mta.ca/tac/volumes/34/29/34-29.pdf

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