Is the absolute Galois group $\mathrm{Gal}(\overline{\mathbf{Q}}|\mathbf{Q})$ the same as the group $\mathrm{Aut}_{\mathbf{Q}}(\overline{\mathbf{Q}})$ the automorphism group in the category of $\mathbf{Q}$-algebras?
The first group is profinite and the second one seems to be an ordinary group.
Edit: The question generates a very interesting comments and discussion. At some point I don't understand why it was closed and why so many down votes. Anyway, Thank you for your clarifications and useful comments!