Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 2.2.6.13. The $2$-categories $\mathrm{Bimod}$ and $\operatorname{Corr}(\operatorname{\mathcal{C}})$ of Examples 2.2.2.3 and 2.2.2.1 both depend on certain auxiliary choices:

  • Let $A$, $B$, and $C$ be associative rings, and suppose we are given a pair of bimodules $M = {}_{A}^{}M_{B}$ and $N = {}_{B}^{}N_{C}$. Then we can regard $M$ and $N$ as $1$-morphisms in the $2$-category $\mathrm{Bimod}$, whose composition is defined to be the relative tensor product $M \otimes _{B} N$. This tensor product is well-defined up to (unique) isomorphism: it is universal among abelian groups $P$ which are equipped with a $B$-bilinear map $M \times N \rightarrow P$. However, it is possible to give many different constructions of an abelian group with this universal property, each of which gives a (slightly) different composition law for the $1$-morphisms in the $2$-category $\mathrm{Bimod}$.

  • Let $\operatorname{\mathcal{C}}$ be a category which admits fiber products, and suppose we are given a pair of correspondences

    \[ X \leftarrow M \rightarrow Y \quad \quad Y \leftarrow N \rightarrow Z \]

    in $\operatorname{\mathcal{C}}$. Then $M$ and $N$ can be regarded as $1$-morphisms in the $2$-category $\operatorname{Corr}(\operatorname{\mathcal{C}})$, whose composition is given by the fiber product $M \times _{Y} N$ (regarded as a correspondence from $X$ to $Z$). This fiber product is well-defined up to (unique) isomorphism as an object of $\operatorname{\mathcal{C}}$, but there is generally no way to choose a preferred representative of its isomorphism class. Consequently, different choices of fiber product lead to (slightly) different definitions for the composition of $1$-morphisms in the $2$-category $\operatorname{Corr}(\operatorname{\mathcal{C}})$.

By making a different choice of conventions in these examples, one can obtain $2$-categories $\mathrm{Bimod}'$ and $\operatorname{Corr}'(\operatorname{\mathcal{C}})$ having the same objects, $1$-morphisms, and $2$-morphisms as the $2$-categories $\mathrm{Bimod}$ and $\operatorname{Corr}(\operatorname{\mathcal{C}})$, but different composition laws for $1$-morphisms. In this case, the $2$-categories $\mathrm{Bimod}'$ and $\operatorname{Corr}'(\operatorname{\mathcal{C}})$ can be obtained from $\mathrm{Bimod}$ and $\operatorname{Corr}(\operatorname{\mathcal{C}})$ (respectively) by the twisting procedure of Construction 2.2.6.8. In particular, the resulting $2$-categories $\mathrm{Bimod}'$ and $\operatorname{Corr}'(\operatorname{\mathcal{C}})$ are isomorphic (though not necessarily strictly isomorphic) to the $2$-categories $\mathrm{Bimod}$ and $\operatorname{Corr}(\operatorname{\mathcal{C}})$, respectively.