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1. • CommentRowNumber1.
   • CommentAuthorLinde Wester
   • CommentTimeDec 18th 2015
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   Author: Linde Wester
   Format: TextCould anyone explain the correspondence between the definition of a club in terms of a monad, and the explicit definition of a club given in https://ncatlab.org/nlab/show/club I am lost in the very beginning, the section 'Action by substitution product', where the substitution product is described in terms of a pullback, which makes use of a 'cartesian monad T on Cat whose algebras are symmetric strict monoidal categories, and T1=P, where 1 is the terminal category, and P is the category of finite sets'. It would probably help to know how T is defined explicitly.

   Could anyone explain the correspondence between the definition of a club in terms of a monad, and the explicit definition of a club given in https://ncatlab.org/nlab/show/club
   
   I am lost in the very beginning, the section 'Action by substitution product', where the substitution product is described in terms of a pullback, which makes use of a 'cartesian monad T on Cat whose algebras are symmetric strict monoidal categories, and T1=P, where 1 is the terminal category, and P is the category of finite sets'. It would probably help to know how T is defined explicitly.
2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 29th 2015
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   Author: Todd_Trimble
   Format: MarkdownItexI'm very sorry for not responding earlier. Here the monad $T$ takes a category $C$ to the free symmetric monoidal category generated by $C$. The objects of $T C$ are finite tuples of objects of $C$, say $(c_1, \ldots, c_m)$; a morphism from $(c_1, \ldots, c_m)$ to $(d_1, \ldots, d_n)$ consists of a bijection $\pi: \{1, \ldots, m\} \to \{1, \ldots, n\}$ (so $m = n$) together with morphisms $f_i: c_i \to d_{\pi(i)}$ for $i = 1, \ldots, m$. Hopefully it is clear how morphisms compose. The construction $T C$ is akin to a wreath product from group theory. When applied to the terminal category $1$, this construction $T 1$ is the category (or rather groupoid) of finite permutations $\mathbf{P}$. Objects of $\mathbf{P}$ may be identified with natural numbers $n$ (we identify $n$ with the unique $n$-tuple of objects of $1$), and morphisms just amount to bijections $\{1, \ldots, m\} \to \{1, \ldots, n\}$, i.e., permutations of $\{1, \ldots, n\}$. This category is equivalent to the category of finite sets and bijections between them. Let me know if other points need clarification.

   I’m very sorry for not responding earlier.

   Here the monad takes a category to the free symmetric monoidal category generated by . The objects of are finite tuples of objects of , say ; a morphism from to consists of a bijection (so ) together with morphisms for . Hopefully it is clear how morphisms compose. The construction is akin to a wreath product from group theory.

   When applied to the terminal category , this construction is the category (or rather groupoid) of finite permutations . Objects of may be identified with natural numbers (we identify with the unique -tuple of objects of ), and morphisms just amount to bijections , i.e., permutations of . This category is equivalent to the category of finite sets and bijections between them.

   Let me know if other points need clarification.