degree $1$. Denote by $I$ the $({\mathbb{Z}}/2{\mathbb{Z}})$-graded vector space defined by 0 in degree 0 and ${\Bbbk}$ in degree 1. Up to isomorphism, ${\mathbb{1}}$ and $I$ are the only invertible objects of $\mathrm{SVect}_{\Bbbk}$. Their associated scalar are $\sigma_{\mathbb{1}}=1$ and $\sigma_I=-1$.

Hopf algebras {#sect-Hopf-alg}
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An *algebra* in ${\mathcal{S}}$ is an object $A$ of ${\mathcal{S}}$ endowed with morphisms $\mu{\colon}A \otimes A \to A$ (the product) and $\eta{\colon}{\mathbb{1}}\to A$ (the unit) such that $$\mu(\mu \otimes {\mathrm{id}}_A)=\mu({\mathrm{id}}_A \otimes \mu) \quad \text{and} \quad \mu({\mathrm{id}}_A \otimes \eta)={\mathrm{id}}_A=\mu(\eta \otimes {\mathrm{id}}_A).$$ A *coalgebra* in ${\mathcal{S}}$ is an object $C$ of ${\mathcal{S}}$ endowed with morphisms $\Delta{\colon}C \to C \otimes C$ (the coproduct) and $\varepsilon{\colon}C \to {\mathbb{1}}$ (the counit) such that $$(\Delta \otimes {\mathrm{id}}_C)\Delta=({\mathrm{id}}_C \otimes \Delta)\Delta \quad \text{and} \quad ({\mathrm{id}}_C \otimes \varepsilon)\Delta={\mathrm{id}}_C=(\varepsilon \otimes {\mathrm{id}}_C)\Delta.$$ A *bialgebra* in ${\mathcal{S}}$ is an object $A$ of ${\mathcal{S}}$ endowed with an algebra structure and a coalgebra structure such that its coproduct $\Delta$ and counit $\varepsilon$ are algebra morphisms (or equivalently, such that its product $\mu$ and unit $\eta$ are coalgebra morphisms), i.e., $$\begin{aligned}
\Delta \mu&=(\mu \otimes \mu)({\mathrm{id}}_A \otimes \tau_{A,A} \otimes {\mathrm{id}}_A)(\Delta \otimes \Delta),  & \Delta \eta&=\eta \otimes \eta, \\  \varepsilon \mu &=\varepsilon \otimes \varepsilon, & \varepsilon \eta&={\mathrm{id}}_{\mathbb{1}}.\end{aligned}$$ An *antipode* for a bialgebra $A$ 