Homology, Homotopy and Applications

Comparing operadic theories of $n$-category

Eugenia Cheng

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We give a framework for comparing on the one hand theories of $n$-categories that are weakly enriched operadically, and on the other hand $n$-categories given as algebras for a contractible globular operad. Examples of the former are the definition by Trimble and variants (Cheng-Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We first provide a generalisation of Trimble’s original theory that allows for the use of other parametrising operads in a very general way, via the notion of categories weakly enriched in V where the weakness is parametrised by a $\mathcal{V}$-operad $P$. We define weak $n$-categories by iterated weak enrichment using a series of parametrising operads $P_i$. We then show how to construct from such a theory an $n$-dimensional globular operad for each $n \geqslant 0$ whose algebras are precisely the weak $n$-categories, and we show that the resulting globular operad is contractible precisely when the operads $P_i$ are contractible. We then show how the globular operad associated with Trimble’s topological definition is related to the globular operad used by Batanin to define fundamental $n$-groupoids of spaces.

Article information

Homology Homotopy Appl., Volume 13, Number 2 (2011), 217-249.

First available in Project Euclid: 30 April 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D05: Double categories, 2-categories, bicategories and generalizations 18D20: Enriched categories (over closed or monoidal categories) 18D50: Operads [See also 55P48]

$n$-category operad


Cheng, Eugenia. Comparing operadic theories of $n$-category. Homology Homotopy Appl. 13 (2011), no. 2, 217--249. https://projecteuclid.org/euclid.hha/1335806751

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