Complexity class corresponding to sorting
The sorting problem is actually complete for $\mathsf{TC}^0$ (under $\mathsf{AC}^0$-reduction). A standard source for this is Section 1.4.3 of Vollmer's book. 

Note that $\mathsf{TC}^0$ is the class of decision problems, but we usually think of sorting as a function problem, i.e., we want to output the numbers, say, in nondecreasing order. However, we can also define sorting as a decision problem as follows:


  Given a sequence of numbers $a_1,\ldots,a_n$ and two numbers $k,p\in [n]$, we want to decide if $a_k$ is at position $p$ in the sequence we get by sorting $a_1,\ldots,a_n$ in nondecreasing order. Note that to avoid ambiguity, when $a_i=a_j$, we want $a_i$ to precede $a_j$ if $i&lt;j$.