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The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras. (See the edit below for a reference.) Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete category is complete, the category of bialgebras and the category of Hopf algebras are complete.

Indeed, one nice way of thinking of coalgebras (which falls out from its being locally finitely presentable) is that it's equivalent to the category of left exact functors from the category of finite-dimensional algebras to $Set$. Limits of such left exact functors may be computed pointwise.

We can say more: any category that is algebraic over a locally presentable category is also locally presentable. It follows that the category of bialgebras and the category of Hopf algebras are also cocomplete.

However, I claim neither the category of bialgebras nor the category of Hopf algebras are algebraic over set, i.e., the underlying set functors are not monadic. In fact, the underlying set functors don't even have left adjoints. If they did, then they would preserve the terminal object, but in each of these cases the ground field $k$ is the terminal bialgebra/Hopf algebra, and since the underlying set of $k$ is not terminal, the claim is proven.

Edit: I am not sure whether Hopf algebras are coalgebraic over the category of algebras; this is perhaps a difficult problem. The status of this and related universal properties (including the case of bialgebras, which is easier) are well-presented in a paper by Porst.

The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras. (See the edit below for a reference.) Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete category is complete, the category of bialgebras and the category of Hopf algebras are complete.

Indeed, one nice way of thinking of coalgebras (which falls out from its being locally finitely presentable) is that it's equivalent to the category of left exact functors from the category of finite-dimensional algebras to $Set$. Limits of such left exact functors may be computed pointwise.

We can say more: any category that is algebraic over a locally presentable category is also locally presentable. It follows that the category of bialgebras and the category of Hopf algebras are also cocomplete.

However, I claim neither the category of bialgebras nor the category of Hopf algebras are algebraic over set, i.e., the underlying set functors are not monadic. In fact, the underlying set functors don't even have left adjoints. If they did, then they would preserve the terminal object, but in each of these cases the ground field $k$ is the terminal bialgebra/Hopf algebra, and since the underlying set of $k$ is not terminal, the claim is proven.

Edit: I am not sure whether Hopf algebras are coalgebraic over the category of algebras; this is perhaps a difficult problem. The status of this and related universal properties (including the case of bialgebras, which is easier) are well-presented in a paper by Porst. (Link fixed)

Edit: The category of weak bialgebras is also complete and cocomplete; this is proved by similar methods from the theory of accessible categories. Namely, the category of weak bialgebras (like that of bialgebras and of Hopf algebras) can be constructed as an equifier between two natural transformations in the 2-category of accessible categories, just as in the bialgebra case. I'll refer you to another paper of Porst for details (section 2).

The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras, Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete category is complete, the category of bialgebras and the category of Hopf algebras are complete.

Indeed, one nice way of thinking of coalgebras (which falls out from its being locally finitely presentable) is that it's equivalent to the category of left exact functors from the category of finite-dimensional algebras to $Set$. Limits of such left exact functors may be computed pointwise.

However, I claim neither the category of bialgebras nor the category of Hopf algebras are algebraic over set, i.e., the underlying set functors are not monadic. In fact, the underlying set functors don't even have left adjoints. If they did, then they would preserve the terminal object, but in each of these cases the ground field $k$ is the terminal bialgebra/Hopf algebra, and since the underlying set of $k$ is not terminal, the claim is proven.

Edit: Of course we can also view bialgebras and Hopf algebras as coalgebraic over the category of algebras, and since the category of algebras is cocomplete, so is the category of bialgebras, and likewise the category of Hopf algebras.

The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras. (See the edit below for a reference.) Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete category is complete, the category of bialgebras and the category of Hopf algebras are complete.

Indeed, one nice way of thinking of coalgebras (which falls out from its being locally finitely presentable) is that it's equivalent to the category of left exact functors from the category of finite-dimensional algebras to $Set$. Limits of such left exact functors may be computed pointwise.

We can say more: any category that is algebraic over a locally presentable category is also locally presentable. It follows that the category of bialgebras and the category of Hopf algebras are also cocomplete.

However, I claim neither the category of bialgebras nor the category of Hopf algebras are algebraic over set, i.e., the underlying set functors are not monadic. In fact, the underlying set functors don't even have left adjoints. If they did, then they would preserve the terminal object, but in each of these cases the ground field $k$ is the terminal bialgebra/Hopf algebra, and since the underlying set of $k$ is not terminal, the claim is proven.

Edit: I am not sure whether Hopf algebras are coalgebraic over the category of algebras; this is perhaps a difficult problem. The status of this and related universal properties (including the case of bialgebras, which is easier) are well-presented in a paper by Porst.

The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras, Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete category is complete, the category of bialgebras and the category of Hopf algebras are complete.

Indeed, one nice way of thinking of coalgebras (which falls out from its being locally finitely presentable) is that it's equivalent to the category of left exact functors from the category of finite-dimensional algebras to $Set$. Limits of such left exact functors may be computed pointwise.

However, I claim neither the category of bialgebras nor the category of Hopf algebras are algebraic over set, i.e., the underlying set functors are not monadic. In fact, the underlying set functors don't even have left adjoints. If they did, then they would preserve the terminal object, but in each of these cases the ground field $k$ is the terminal bialgebra/Hopf algebra, and since the underlying set of $k$ is not terminal, the claim is proven.

The category of bialgebras and the category of Hopf algebras are algebraic over the category of coalgebras, Since the category of coalgebras is locally presentable (as remarked in the paper Ralph referred to), it is complete, and since any category that is algebraic over a complete category is complete, the category of bialgebras and the category of Hopf algebras are complete.

Indeed, one nice way of thinking of coalgebras (which falls out from its being locally finitely presentable) is that it's equivalent to the category of left exact functors from the category of finite-dimensional algebras to $Set$. Limits of such left exact functors may be computed pointwise.

However, I claim neither the category of bialgebras nor the category of Hopf algebras are algebraic over set, i.e., the underlying set functors are not monadic. In fact, the underlying set functors don't even have left adjoints. If they did, then they would preserve the terminal object, but in each of these cases the ground field $k$ is the terminal bialgebra/Hopf algebra, and since the underlying set of $k$ is not terminal, the claim is proven.

Edit: Of course we can also view bialgebras and Hopf algebras as coalgebraic over the category of algebras, and since the category of algebras is cocomplete, so is the category of bialgebras, and likewise the category of Hopf algebras.