Let $\mathbb{K}$ be a field (for example $\mathbb{K} = \mathbb{Q}$). An algebraic group $G$ can be thought of as a functor from $\mathbb{K}$-algebras to groups. Is $G$ exact, i.e. does it preserve exact sequences? Does it preserve inclusions? Does it preserve surjections?
References are appreciated.
There's no such thing as an exact sequence of $K$-algebras. The functor associated to an algebraic group generally does not preserve surjections. For example, there is a surjection $\mathbb{Q}[x] \to \mathbb{Q}[x]/(x^2+1) \cong \mathbb{Q}[i]$, but applying $GL_1(-)$ produces a map
$$\mathbb{Q}^{\times} \cong GL_1(\mathbb{Q}[x]) \to GL_1(\mathbb{Q}[i]) \cong \mathbb{Q}[i]^{\times}$$
which is not a surjection because e.g. its image does not contain $i$. On the other hand, the functor associated to an affine algebraic group, being representable, preserves all limits, which you can use to show that it preserves injections. So it is "left exact" in this case, although using these terms outside of the abelian context risks some misunderstandings.