If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with degree-preserving morphisms. More precisely, let $V$ be a purely odd vector space (i.e. $V_0 = 0$ and $V_1 = V$) and form the exterior algebra $\Lambda(V)$ with its natural $\mathbb{Z}/2$ grading. This is a superalgebra, i.e. an algebra in $\mathrm{SVect}$. Then $\Lambda(V) \underline{\otimes} \Lambda(V)$ is an algebra in $\mathrm{SVect}$, where $\underline{\otimes}$ is the graded tensor product of graded algebras.

Now consider the map $\Delta : V \to \Lambda(V) \underline{\otimes} \Lambda(V)$ given by $$\Delta(v) = v \otimes 1 + 1 \otimes v.$$ With the sign conventions coming from the graded tensor product, you get $\Delta(v)^2 = 0$, and so according to the universal property of the exterior algebra, $\Delta$ extends to an algebra homomorphism $\Delta : \Lambda(V) \to \Lambda(V) \underline{\otimes} \Lambda(V)$. Coassociativity is clear. You can get the counit and antipode similarly using the universal property.

Another good example is the shuffle Hopf algebra, which is discussed in this question. Let $V$ be a vector space and $T(V)$ its tensor algebra. The shuffle Hopf algebra is a Hopf structure on $T(V)$ which uses neither the standard algebra nor coalgebra structures on the tensor algebra.

The comultiplication is given by deconcatenation: $$ \Delta(v_1 \dots v_n) = \sum_{j=1}^{n+1} v_1 \dots v_{j-1} \otimes v_j \dots v_n, $$ while the multiplication is given by the shuffle product: $$ (v_1 \dots v_k) \cdot (v_{k+1} \dots v_n) = \sum_{\sigma \in S_{k,n-k}} v_{\sigma^{-1}(1)} \dots v_{\sigma^{-1}(n)},$$ where $S_{k,n-k}$ is the set of $(k,n-k)$ shuffle permutations, i.e. $$\sigma(1) < \dots < \sigma(k)$$ and $$\sigma(k+1) < \dots < \sigma(n).$$ I haven't really worked much with the shuffle algebra myself, but the answers to the question linked above have some discussion of what it is good for.

If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with degree-preserving morphisms. More precisely, let $V$ be a purely odd vector space (i.e. $V_0 = 0$ and $V_1 = V$) and form the exterior algebra $\Lambda(V)$ with its natural $\mathbb{Z}/2$ grading. This is a superalgebra, i.e. an algebra in $\mathrm{SVect}$. Then $\Lambda(V) \underline{\otimes} \Lambda(V)$ is an algebra in $\mathrm{SVect}$, where $\underline{\otimes}$ is the graded tensor product of graded algebras.

Now consider the map $\Delta : V \to \Lambda(V) \underline{\otimes} \Lambda(V)$ given by $$\Delta(v) = v \otimes 1 + 1 \otimes v.$$ With the sign conventions coming from the graded tensor product, you get $\Delta(v)^2 = 0$, and so according to the universal property of the exterior algebra, $\Delta$ extends to an algebra homomorphism $\Delta : \Lambda(V) \to \Lambda(V) \underline{\otimes} \Lambda(V)$. Coassociativity is clear. You can get the counit and antipode similarly using the universal property.

Another good example is the shuffle Hopf algebra, which is discussed in this question. Let $V$ be a vector space and $T(V)$ its tensor algebra. The shuffle Hopf algebra is a Hopf structure on $T(V)$ which uses neither the standard algebra nor coalgebra structures on the tensor algebra.

The comultiplication is given by deconcatenation: $$ \Delta(v_1 \dots v_n) = \sum_{j=1}^{n+1} v_1 \dots v_{j-1} \otimes v_j \dots v_n, $$ while the multiplication is given by the shuffle product: $$ (v_1 \dots v_k) \cdot (v_{k+1} \dots v_n) = \sum_{\sigma \in S_{k,n-k}} v_{\sigma^{-1}(1)} \dots v_{\sigma^{-1}(n)},$$ where $S_{k,n-k}$ is the set of $(k,n-k)$ shuffle permutations, i.e. $$\sigma(1) < \dots < \sigma(k)$$ and $$\sigma(k+1) < \dots < \sigma(n).$$ I haven't really worked much with the shuffle algebra myself, but the answers to the question linked above have some discussion of what it is good for.

If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with degree-preserving morphisms. More precisely, let $V$ be a purely odd vector space (i.e. $V_0 = 0$ and $V_1 = V$) and form the exterior algebra $\Lambda(V)$ with its natural $\mathbb{Z}/2$ grading. This is a superalgebra, i.e. an algebra in $\mathrm{SVect}$. Then $\Lambda(V) \underline{\otimes} \Lambda(V)$ is an algebra in $\mathrm{SVect}$, where $\underline{\otimes}$ is the graded tensor product of graded algebras.

Now consider the map $\Delta : V \to \Lambda(V) \otimes \Lambda(V)$ given by $$\Delta(v) = v \otimes 1 + 1 \otimes v.$$ With the sign conventions coming from the graded tensor product, you get $\Delta(v)^2 = 0$, and so according to the universal property of the exterior algebra, $\Delta$ extends to an algebra homomorphism $\Delta : \Lambda(V) \to \Lambda(V) \otimes \Lambda(V)$. Coassociativity is clear. You can get the counit and antipode similarly using the universal property.

Another good example is the shuffle Hopf algebra, which is discussed in this question. Let $V$ be a vector space and $T(V)$ its tensor algebra. The shuffle Hopf algebra is a Hopf structure on $T(V)$ which uses neither the standard algebra nor coalgebra structures on the tensor algebra.

The comultiplication is given by deconcatenation: $$ \Delta(v_1 \dots v_n) = \sum_{j=1}^{n+1} v_1 \dots v_{j-1} \otimes v_j \dots v_n, $$ while the multiplication is given by the shuffle product: $$ (v_1 \dots v_k) \cdot (v_{k+1} \dots v_n) = \sum_{\sigma \in S_{k,n-k}} v_{\sigma^{-1}(1)} \dots v_{\sigma^{-1}(n)},$$ where $S_{k,n-k}$ is the set of $(k,n-k)$ shuffle permutations, i.e. $$\sigma(1) < \dots < \sigma(k)$$ and $$\sigma(k+1) < \dots < \sigma(n).$$ I haven't really worked much with the shuffle algebra myself, but the answers to the question linked above have some discussion of what it is good for.

If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with degree-preserving morphisms. More precisely, let $V$ be a purely odd vector space (i.e. $V_0 = 0$ and $V_1 = V$) and form the exterior algebra $\Lambda(V)$ with its natural $\mathbb{Z}/2$ grading. This is a superalgebra, i.e. an algebra in $\mathrm{SVect}$. Then $\Lambda(V) \underline{\otimes} \Lambda(V)$ is an algebra in $\mathrm{SVect}$, where $\underline{\otimes}$ is the graded tensor product of graded algebras.

Now consider the map $\Delta : V \to \Lambda(V) \underline{\otimes} \Lambda(V)$ given by $$\Delta(v) = v \otimes 1 + 1 \otimes v.$$ With the sign conventions coming from the graded tensor product, you get $\Delta(v)^2 = 0$, and so according to the universal property of the exterior algebra, $\Delta$ extends to an algebra homomorphism $\Delta : \Lambda(V) \to \Lambda(V) \underline{\otimes} \Lambda(V)$. Coassociativity is clear. You can get the counit and antipode similarly using the universal property.

Another good example is the shuffle Hopf algebra, which is discussed in this question. Let $V$ be a vector space and $T(V)$ its tensor algebra. The shuffle Hopf algebra is a Hopf structure on $T(V)$ which uses neither the standard algebra nor coalgebra structures on the tensor algebra.

The comultiplication is given by deconcatenation: $$ \Delta(v_1 \dots v_n) = \sum_{j=1}^{n+1} v_1 \dots v_{j-1} \otimes v_j \dots v_n, $$ while the multiplication is given by the shuffle product: $$ (v_1 \dots v_k) \cdot (v_{k+1} \dots v_n) = \sum_{\sigma \in S_{k,n-k}} v_{\sigma^{-1}(1)} \dots v_{\sigma^{-1}(n)},$$ where $S_{k,n-k}$ is the set of $(k,n-k)$ shuffle permutations, i.e. $$\sigma(1) < \dots < \sigma(k)$$ and $$\sigma(k+1) < \dots < \sigma(n).$$ I haven't really worked much with the shuffle algebra myself, but the answers to the question linked above have some discussion of what it is good for.