**Background:**

(1) If **C** and **D** are categories and there is a forgetful functor U:**C**→**D**, then a * C-ification* functor F:

**D**→

**C**is an adjoint to U. For example, the (left) adjoint to the forgetful functor from groups to monoids is "groupification" of a monoid, given by formally adjoining inverses. The (left) adjoint to the forgetful functor from presheaves to sheaves is the usual "sheafification" functor.

Note that whenever you have a (left adjoint) **C**-ification functor F (whenever you have an adjunction, for that matter), you get a universal property. For any object X∈**D**, there is a canonical morphism (called the *unit of adjunction*) ε_{X}:X→U(F(X)) with the property that any morphism f:X→U(Y) factors as f=U(g)\circ ε_{X} for a unique morphism g:F(X)→Y in **C**.

(2) A scheme X is *separated* if the diagonal morphism X→XxX is a closed immersion. It is enough to check that the image of the diagonal is closed. Being separated is the algebro-geometric analogue of being hausdorff, which nothing in algebraic geometry ever is.

My question is whether there exists a "separification" functor adjoint to the forgetful functor U from the category of separated schemes to the category of schemes. Note that the forgetful functor U does not respect colimits (you can glue together separated schemes to get a non-separated scheme), so it has no hope of having a right adjoint. But U *does* respect limits (it's enough to show that an arbitrary product of separated schemes is separated and that fiber products of separated schemes are separated), so it might have a left adjoint.

To put it another way, given a scheme X, is there a canonically defined separated scheme X^{s} and a morphism X→X^{s} so that any morphism from X to a separated scheme factors uniquely through X→X^{s}?

Related questions I'd like to know the answer to:

- Is there a "relative separification" functor. That is, does an arbitrary morphism of schemes f:X→Y admit a canonical factorization through a separated morphism f
^{s}:X'→Y. This would be analogous to Stein factorization, which I regard as "relative affinification". An arbitrary (quasi-compact and quasi-separated) morphism f:X→Y canonically factors through the affine morphism Spec_{Y}(f_{*}O_{X})→Y - Is there a separification functor for algebraic spaces? Is it possible that the separification of a scheme is naturally an algebraic space?
- Is there a separification functor for algebraic stacks? (An algebraic stack is separated if the diagonal is proper.)

affineschemes $T$ the induced map $Hom(T,X) \to lim_i Hom(T,X_i)$ is bijective, then this is already true for all schemes $T$. But this is easy because both sides are sheaves in $T$ with respect to the Zariski Topology. $\endgroup$ – Martin Brandenburg Jun 14 '11 at 15:29