Abstract
We introduce a theory of coherence for symmetric monoidal categories inthe spirit of Segal and show that it is equivalent, in an appropriate sense,to MacLane’s original notion. More precisely, we prove that“special Γ categories”, the analogue ofspecial Γ spaces, and coherently symmetric monoidalcategories are one and the same. This is analogous to the situation intopology where special Γ spaces are precisely homotopicalcommutative monoids. In light of the obervation that the category of smallcategories Cat bears a functorial Quillen model structure with respect tothe class of categorical equivalences: in fact, is a homotopy theory in thesense of Heller, we may reinterpret the theorem as stating that coherentlysymmetric monoidal categories are precisely the homotopical commutativemonoids within this new homotopy theory.
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Arroyo, F.H. Coherent Homotopical Algebras: “Special Gamma-Categories”. Applied Categorical Structures 5, 339–362 (1997). https://doi.org/10.1023/A:1008615904492
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DOI: https://doi.org/10.1023/A:1008615904492