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!

definedby taking an inverse limit of groups of finite Galois extensions. $\endgroup$ – Todd Trimble♦ Mar 15 '16 at 15:51