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Sep
19
comment Six object classes | products | co-products in search of a category
Would the fact that the construction of skeletons cannot be made functorial imply that in fact no way to define morphisms to get a category from the first case in my question? (See also the addendum I made to my question a few minutes ago.)
Sep
19
revised Six object classes | products | co-products in search of a category
added 653 characters in body
Sep
19
revised Six object classes | products | co-products in search of a category
added 653 characters in body
Sep
19
comment Six object classes | products | co-products in search of a category
Thanks for your answer! It arrived while I was writing some additional thoughts on the first case (though originally I incorrectly wrote $\mathbf{Set}$ instead of $\mathbf{FinSet}$). Those remarks already suggest that the non-negative integers ($\mathbb{N}_0$) can be viewed as the quotient of $\mathrm{ob}(\mathbf{FinSet})$ relative to $\mathrm{card}$. What I still don't have is a clear idea of how exactly how one extends this "quotienting" $\mathrm{ob}(\mathbf{FinSet}) \to \mathbb{N}_0$ to its morphisms (and thereby define the morphisms between elements of the quotient, $\mathbb{N}_0$).
Sep
19
revised Six object classes | products | co-products in search of a category
Fixed mistake by replacing $\mathbf{Set}$ with $\mathbf{FinSet}$.
Sep
19
revised Six object classes | products | co-products in search of a category
added 673 characters in body
Sep
19
asked Six object classes | products | co-products in search of a category
Sep
13
revised Looking for characterizations of the commutator subgroups of the “matrix groups”
deleted 26 characters in body
Sep
13
asked Looking for characterizations of the commutator subgroups of the “matrix groups”
Sep
12
accepted Can the choice of definition of morphisms for a slice category be justified categorically?
Sep
12
revised Can the choice of definition of morphisms for a slice category be justified categorically?
edited tags
Sep
12
asked Can the choice of definition of morphisms for a slice category be justified categorically?
Aug
15
comment On pentagonal tilings
Thanks, although the answers posted now prompt the question: what exactly is the difference between, e.g., patterns 4 (dark blue) and 8 (medium gray)?
Aug
15
accepted On pentagonal tilings
Aug
14
asked On pentagonal tilings
Aug
14
comment On counting and generating all $k$-permutations of a multiset
Thanks! In the meantime I realized that the generation problem can be "factored" into two generation problems, arranged as an outer loop and an inner loop; the outer loop is the generation of all $k$-combinations (i.e. $k$-submultisets) from the multiset; the inner one is the generation of all the permutations of the current $k$-submultiset. There are many non-recursive algorithms for the inner loop generation. I have not found a non-recursive algorithm for the outer loop one, but I guess it would not be difficult to convert one of these to a relatively simple iteration, as you did here.
Aug
14
accepted On counting and generating all $k$-permutations of a multiset
Aug
13
comment On counting and generating all $k$-permutations of a multiset
@hardmath: I need to think about it... Naively, I would have expected that such an algorithm would, in general, produce some configurations more than once. On the other hand, if the generation happens in lexicographic order, I suppose that it would be relatively inexpensive to weed these duplicates out. I need to work out the details. Thanks for your suggestion.
Aug
13
asked On counting and generating all $k$-permutations of a multiset
Aug
12
awarded  Popular Question