# zbMATH — the first resource for mathematics

Associativity as commutativity. (English) Zbl 1099.18006
Let $${\mathcal V}$$ be a monoidal category. For objects $$a$$ and $$c$$ of $${\mathcal V}$$, the associativity constraint can be viewed as a natural family of isomorphisms between $$(a\otimes -)\circ (-\otimes c)$$ and $$(-\otimes c)\circ (a\otimes -)$$ in the category of endofunctors of $${\mathcal V}$$. From this the authors were motivated to view associativity as a special kind of commutativity and to relate Mac Lane’s hexagon for commutativity to his pentagon for associativity. They discuss this using their insertion operator.

##### MSC:
 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 18A15 Foundations, relations to logic and deductive systems
##### Keywords:
monoidal category; associativity constraint; symmetry
Full Text:
##### References:
 [1] Categories for the working mathematician (1971) · Zbl 0232.18001 [2] Rice University Studies, Papers in Mathematics 49 pp 28– (1963) [3] Operads: Proceedings of Renaissance Conferences 202 (1997) [4] The Mathematical Intelligencer 20 pp 72– (1998) · Zbl 1052.00504 [5] DOI: 10.1016/0022-4049(72)90016-3 · Zbl 0244.18009 [6] Proof-Theoretical Coherence (2004) · Zbl 1153.03003 [7] DOI: 10.1016/S0021-8693(02)00510-0 · Zbl 1063.16044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.