Weak bialgebras, weak Hopf algebras, and counital maps

Fix a field $k$, an associative unital $k$-algebra $H$ (as in Algebras over a commutative ring, central structure maps, and algebra homomorphisms), and a coassociative counital coalgebra structure $(\mathrm{\Delta },\epsilon )$ as in Coalgebra and coalgebra map over a field. Write $\mathrm{\Delta }(h)=\sum h_{(1)}\otimes h_{(2)}$ and suppress the finite Sweedler sums. A weak bialgebra is these structures with multiplicative coproduct $\mathrm{\Delta }(ab)=\mathrm{\Delta }(a)\mathrm{\Delta }(b)$, not necessarily a unital algebra map, and with the following weak unit and weak counit identities: $(\mathrm{\Delta }\otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }(1)=(\mathrm{\Delta }(1)\otimes 1)(1\otimes \mathrm{\Delta }(1))=(1\otimes \mathrm{\Delta }(1))(\mathrm{\Delta }(1)\otimes 1),$ $\epsilon (abc)=\sum \epsilon (a{b}_{(1)})\epsilon (b_{(2)}c)=\sum \epsilon (a{b}_{(2)})\epsilon (b_{(1)}c)(a,b,c\in H)\mathrm{.}$ In general $\mathrm{\Delta }(1)\mathrm{\ne }1\otimes 1$ and $\epsilon$ is not multiplicative.

For $\mathrm{\Delta }(1)=\sum 1_{(1)}\otimes 1_{(2)}$, define the target and source counital maps ${\epsilon }_t(h)=\sum \epsilon (1_{(1)}h)1_{(2)},{\epsilon }_s(h)=\sum 1_{(1)}\epsilon (h{1}_{(2)})\mathrm{.}$ Write $H_t={\epsilon }_t(H)$ and $H_s={\epsilon }_s(H)$. The source-and-target-bases lemma below proves these ranges are commuting unital subalgebras. We also use the variant ${\epsilon }_s^{\mathrm{\prime }}(h)=\sum 1_{(1)}\epsilon (1_{(2)}h)\mathrm{.}$ The same lemma proves that its restriction to $H_t$ is the inverse anti-isomorphism to ${\epsilon }_t{\mathrm{\mid }}_{H_s}$.

A weak Hopf algebra is a weak bialgebra with a linear map $S:H\to H$ satisfying $\sum h_{(1)}S(h_{(2)})={\epsilon }_t(h),\sum S(h_{(1)})h_{(2)}={\epsilon }_s(h),\sum S(h_{(1)})h_{(2)}S(h_{(3)})=S(h)\mathrm{.}$ No bijectivity of $S$ is imposed in this definition. The module tensor product below uses the source-target maps and does not require an antipode.