During seminar the other day, when speaking about subobject classifiers, I asked if the subobject classifier for the category of multisets would have integer truth values, corresponding to the number of times and element is in the set. We attempted to show this, but quickly realized that we were not even sure of the "correct" category for multisets.

To clarify, when I say correct I want my category to

- Have objects identified by multisets
- Have maps between the multisets be on the level of elements in the multiset, and forget the order of those elements, e.g. there is only one map {111223}->{55}
- The subobject classifier will behave as I had hoped, with {1} having truth value 3 in {111}

My question is;

Can you construct a category satisfying these properties?

Thanks in advance!

EDIT: First, sorry about not checking nLab, I forget about that site far too often. Second, I should say that I have a little bit of motivation for my property two. So let me clarify what I meant in property two. Given a multiset, it can be thought of as a pair $S\times\mathbb{N}$ for a set $S$. Now, when considering morphisms between multisets I want the maps $f,g:\lbrace 1122\rbrace\rightarrow\lbrace34\rbrace$ such that $f$ sends

$\begin{eqnarray*} 1&\mapsto& 3,\\ 1&\mapsto& 4,\\ 2&\mapsto& 3,\\ 2&\mapsto& 4\\ \end{eqnarray*}$

and $g$ sends

$\begin{eqnarray*} 1&\mapsto& 4,\\ 1&\mapsto& 3,\\ 2&\mapsto& 4,\\ 2&\mapsto& 3\\ \end{eqnarray*}$

to be the same morphism. But if $h$ sends

$\begin{eqnarray*} 1&\mapsto& 4,\\ 1&\mapsto& 4,\\ 2&\mapsto& 4,\\ 2&\mapsto& 3\\ \end{eqnarray*}$

then $h$ is not the same as $g$ or $f$. Further I would like it such that $\lbrace 112\rbrace$ is not a subobject of $\lbrace 12\rbrace$ but it is a subobject of $\lbrace 11122\rbrace$.

Hopefully this will clear it up.