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来自news.ycombinator.com/item?id=42913867
The set of natural transformations between two functors $F,G\colon\mathcal{C}\to\mathcal{D}$ can be expressed as the end
$$\mathrm{Nat}(F,G)\cong\int_{A}\mathrm{Hom}_{\mathcal{D}}(F(A),G(A)).$$Define set of natural cotransformations from $F$ to $G$ to be the coend
$$\mathrm{CoNat}(F,G)\cong\int^{A}\mathrm{Hom}_{\mathcal{D}}(F(A),G(A)).$$
Let:
- $F=\mathbf{B}_{\bullet}(\Sigma_{4})_{*/}$ be the under $\infty$-category of the nerve of the delooping of the symmetric group $\Sigma_{4}$ on 4 letters under the unique $0$-simplex $*$ of $\mathbf{B}_{\bullet}\Sigma_{4}$.
- $G=\mathbf{B}_{\bullet}(\Sigma_{7})_{*/}$ be the under $\infty$-category nerve of the delooping of the symmetric group $\Sigma_{7}$ on 7 letters under the unique $0$-simplex $*$ of $\mathbf{B}_{\bullet}\Sigma_{7}$.
How many natural cotransformations are there between $F$ and $G$? |
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