A filtered bialgebra $(A,m,u,\delta,\epsilon)$ with $A_j,j\in\mathbb Z$ filtration is defined as a bialgebra so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$. The assotiated graded space $\text {gr}A=\bigoplus_{j}A_{j}/A_{j-1}$ has a multiplication and a comultiplication induced from $A$, but the unit and conit might not exists. To fix this one can require $u(1)\subset A_0$ to make sure $\mathrm{gr} A$ has a unit, but how to make sure the counit $\epsilon$ is well-defined on $\text{gr}A$?(The only thing I can think of is to require $\epsilon$ factors thru $A_0$, but I'm not sure)
Any reference or comment is wellcome
