I have two questions about the definition of a Hopf algebra in Hatcher's book on algebraic topology. He defines it as follows (see Section 3.C, page 283):


**Definition**: A *Hopf algebra* is a graded algebra $A = \oplus\_{n \geq 0} A^n$ over a commutative base ring $R$ satisfying the following two conditions:


1. There is an identity element $1 \in A^0$ such that the map $R \rightarrow A^0$, $r \mapsto r \cdot 1$, is an isomorphism; one says that $A$ is *connected*.
2. There is a *diagonal* or *coproduct* $\Delta\colon A \rightarrow A \otimes A$, a homomorphism of graded algebras satisfying $\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha + \sum\_{0<i<n} \alpha'\_i \otimes \alpha''\_{n-i}$ for $\alpha \in A^n$, $n >0$, and $\alpha'\_j,\alpha''\_j \in A^j$.


Of course, in other settings Hopf algebras don't need to be graded, but I guess graded ones are the ones he cares about. Here are my two questions:


1. In condition 2, one could instead ask for the a priori weaker condition that $\Delta(\alpha) = \alpha \otimes 1 + 1 \otimes \alpha + x$ with $x \in \oplus\_{0<i<n} A^i \otimes A^{n-i}$. It seems to me that in his examples (all coming from H-spaces) this is all that holds, at least naively. Is this a mistake, or is this really the right definition?
2. When I look at definitions of Hopf algebras elsewhere, there is also an *antipode* map. How can you derive this in Hatcher's setup?



Question about definition of Hopf algebra in Hatcher
You have not transcribed Hatcher's definition correctly. He has a sum $\sum\_i\alpha'\_i\otimes\alpha''\_i$ over some unspecified index set, not over $\{1,\dotsc,n-1\}$, and he requires that the terms $\alpha'\_i,\alpha''\_i$ are homogeneous of arbitrary positive degree (although the requirement that $\Delta$ preserves gradings forces $|\alpha'\_i|+|\alpha''\_i|=|\alpha|$. With these corrections, Hatcher's definition is equivalent to your "a priori weaker condition".


Also, in the connected graded context, it is not hard to check by induction on $n$ that there is a unique way to define the antipode $\chi\colon A^n\to A^n$ such that the defining property $\mu\circ(\chi\otimes 1)\circ\Delta=\eta\epsilon$ is satisfied.


