TITLE: Question about definition of Hopf algebra in Hatcher
QUESTION [13 upvotes]: 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:

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.
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:

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?

When I look at definitions of Hopf algebras elsewhere, there is also an antipode map.  How can you derive this in Hatcher's setup?

REPLY [8 votes]: 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.