Geodesic
Weak units, universal cells, and coherence via universality for bicategories
Theory and applications of categories, Tome 34 (2019), pp. 883-960.

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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.
Publié le :
Classification : 18D05
Keywords: bicategories, polycategories, multicategories, merge-bicategories, polygraphs, coherence, strictification, weak units 
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     author = {Amar Hadzihasanovic},
     title = {Weak units, universal cells, and coherence via universality for bicategories},
     journal = {Theory and applications of categories},
     pages = {883--960},
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     volume = {34},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a28/}
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Amar Hadzihasanovic. Weak units, universal cells, and coherence via universality for bicategories. Theory and applications of categories, Tome 34 (2019), pp. 883-960. http://geodesic.mathdoc.fr/item/TAC_2019_34_a28/