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1

Intuitively I understand the first definition but not the second. In fact the standard notation for a polynomial with integer coefficients and variables taken from the set X suggests first taking products, then integer linear combinations so that is exactly what a free abelian group on a free abelian monoid means. The second definition is less intuitive. ...

0

I think your attempt to construct an inverse does not work. Already in your definition I encounter some problems. You write: Take $\eta \in \mathrm{End}(F_1\otimes F_2)$. Define $\eta_1|_{F(X)}=\eta|_{F(X)\otimes F(1)}$ and $\eta_2|_{F(Y)}=\eta|_{F(1)\otimes F(Y)}$ for objects $X, Y$ in $C$. In your question you do not assume that $C$ is a monoidal ...

2

You need to set up the correct hom–tensor adjunction. Recall that, for rings $R, S, T$, an $(R, S)$-bimodule $M$, $(S, T)$-bimodule $N$, and $(R, T)$-bimodule $P$, we have: $$\mathrm{Hom}_{(R, T)} (M \otimes_S N, P) \cong \mathrm{Hom}_{(R, S)} (M, \mathrm{Hom}_{(\mathbb{Z}, T)} (N, P))$$ $$\mathrm{Hom}_{(R, T)} (M \otimes_S N, P) \cong \mathrm{Hom}_{(S, T)} ... 0 Just some thoughts. Let's consider a very simple fusion algebra, just some matrices. We can define the square matrix: E_{ij} = e_i \otimes e_j.$$ \left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{array} \right) = \left( \begin{array}{c} 1 & 0 & 0 \end{array} \right) \otimes \left( \begin{array}{c} 0 \\ ...

0

The proof was in Balmer's older 'Supports and filtration...' (Prop. 2.4). It is quite simple: by the unit-counit relation, every rigid x is a direct summand of x \otimes Dx \otimes x. So if x \otimes x belongs to an ideal then so does x. Etc.

2

You can recover $\phi$ from $\nu'=(\mathrm{id}\otimes\phi)\circ \nu$ by applying $\epsilon$ and using the relation between $\epsilon$ and $\nu$. This forces $\phi$ to be unique. In fact, you can just write down what $\phi$ has to be: $$(\epsilon\otimes\mathrm{id})\circ(\mathrm{id}\otimes\nu')$$ (Or rather $\phi$ is the map $X^*\rightarrow X^{\wedge}$ ...

4

I think you are pointing to two different phenomena, and I don't know which one you're more interested in. Can you clarify? In the second example you're just unwrapping the definition of a functor $BM \to \text{Set}$. You can always describe what a functor $C \to D$ looks like in a manner "internal to $D$" in a sort of trivial way: of course it's the same ...

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