symmetric monoidal (∞,1)-category of spectra
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A wreath is a natural generalization of distributive law. Like the latter, it produces a sort of composite monad, the wreath product.
Formally, let $EM(\mathcal{K})$ be the free completion of a 2-category under $EM$ objects. A wreath is an object in $EM(EM(\mathcal{K}))$, and since $EM$ is a monad, it admits a multplication which is indeed the wreath product $\wr : EM(EM(\mathcal{K})) \to EM(\mathcal{K})$.
Let $(A,(t,\eta,\mu),(s, \lambda, \sigma, \nu))$ be a wreath in $\mathcal{K}$. Its wreath product is the monad on $A$ in $\mathcal{K}$ so defined:
Every distributive law is a wreath, and the wreath product of a distributive law qua wreath indeed coincides with the composite monad one would get out of the distributive law.
J. Pure Appl. Algebra 175 (2002), no. 1-3, 243–265.
Created on July 18, 2024 at 08:08:18. See the history of this page for a list of all contributions to it.