Hi Paul, Your (1) is Theorem 4.2 in Marco Grandis, Finite sets and symmetric simplicial sets, Theory and Applications of Categories, Vol. 8, 2001, No. 8, pp 244-252. Richard On Tue, Aug 14, 2018, at 5:41 AM, Paul Blain Levy wrote:
Hi,
The following statements seem plausible.
1.
Let Fin be the category of natural numbers and functions, i.e. the full subcategory of Set on natural numbers, identifying n with {0, ... , n-1}.
For i+1 < n, let swap_i be the function n --> n that swaps i with i+1.
For i <= n, let discard_i be the function n --> n+1 that increments everything >= i.
For i < n, let copy_i be the function n+1 --> n that?? decrements everything > i.
Then the category Fin (not monoidal category, just category) is generated by these operations, subject to a list of equations that treat every possible pair of operations.
2.
The same for the category of natural numbers and injections, using just swap and discard.
3.
The same for the category of natural numbers and bijections, using just swap.
Statement 3 is a standard theorem presenting the symmetric group.?? Is there a reference for statements 1 and 2?
Paul
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