Monad interleaving: a construction of the operad for Leinster's weak $ω$-categories

Details Monad interleaving: a construction of the operad for Leinster's weak $ω$-categories Monad interleaving: a construction of the operad for Leinster's weak $ω$-categories

2003-09-20

arxiv.org/abs/math/0309336v2

Mathematics

Category Theory

24 pages. Fuller explanations, updated bibliography. To appear in JPAA

Scientific paper

We show how to "interleave" the monad for operads and the monad for contractions on the category \coll of collections, to construct the monad for the operads-with-contraction of Leinster. We first decompose the adjunction for operads and the adjunction for contractions into a chain of adjunctions each of which acts on only one dimension of the underlying globular sets at a time. We then exhibit mutual stability conditions that enable us to alternate the dimension-by-dimension free functors. Hence we give an explicit construction of a left adjoint for the forgetful functor $\owc \lra \coll$, from the category of operads-with-contraction to the category of collections. By applying this to the initial (empty) collection, we obtain explicitly an initial operad-with-contraction, whose algebras are by definition the weak $\omega$-categories of Leinster.

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