This may be a silly question, but I have no intuition in this direction. Every category internal to a [Mal'cev category](http://ncatlab.org/nlab/show/Mal%27cev+category) is a groupoid (this is why categories internal to $Grp$ are groupoids). If this was true it would put restrictions on generalising algebraic stacks (which are stacks of groupoids by default) to algebraic stacks *of categories*.


So my question is:



> 
> Is the opposite of the category of commutative rings (= category of affine schemes) Mal'cev?
> 
> 
> 



Do affine schemes form a Mal'cev category?
(This is a repeat of an above comment.) The category of affine schemes is not Mal'cev. This can be disproven by producing an reflexive, non-symmetric relation on an affine scheme $X$ whose graph is a closed subscheme of $X\times X$.


Take $X=\mathbb{A}^1=Spec(\mathbb{C}[x])$. The relation $(x,x)$ and $(x,0)$ (as $x$ runs over all points) is reflexive and asymmetric. If we identify $X\times X$ with 
$$Spec(\mathbb{C}[x]\otimes \mathbb{C}[y])=Spec(\mathbb{C}[x,y]),$$
then the graph of the quotient is the union of the lines $x=y$ and $y=0$. This is a closed subscheme. In terms of rings, this looks like the quotient $\mathbb{C}[x,y]\rightarrow \mathbb{C}[x,y]/(xy-y^2)$.


There's also a simpler and less-satisfying example. Since the category of affine schemes contains the category of finite sets (with $[n]\rightarrow Spec(\mathbb{C}^n)$, one can choose any reflexive, non-symmetric relation on a finite set (since they are all algebraic).


