This question has been bugging me for a while and I can't seem to make sense of it on a clear conceptual level.

The theory of local rings is given by taking the theory of rings and adding the axioms \begin{eqnarray} (0=1) \vdash \bot \\ x + y = 1 \vdash \exists z : (xz = 1) \vee \exists z : (yz = 1) \end{eqnarray} This theory is geometric and thus has a classifying topos $S[\text{Local Rings}]$. With a bit of work one can show that this topos is actually given by the Zariski topos, i.e. given as the topos of Sheaves on the Zariski site $CRing_{fp}^{op}$ of finitely presented rings with coverings given by partitions of unity.

As an algebraic geometer with an acquaintance of the functor of points POV this is already a nice fact, as this basically means that scheme theory is a straight up consequence of the notion of a local ring (schemes can be identified with those sheaves that can be covered by representables).

But this is not where the story ends. Write $S[\text{Rings}]$ for the classifying topos of the theory of rings. A ring $R$ is equivalent to a geometric morphism $\text{Sets} \rightarrow S[\text{Rings}]$ (that's just how classifying toposes are defined after all). We also have a geometric morphism $S[\text{Local Rings}] \rightarrow S[\text{Rings}]$. It turns out that the topos theoretic pullback along those two geometric morphism is just the topos of Sheaves on $\text{Spec}(R)$. (This was mentioned in this mathoverflow post by Peter Arndt) (EDIT: As was pointed out by Simon Henry below, this should be taken with a grain of salt until it is clear in what sense this pullback is meant.)

$\require{AMScd}$ \begin{CD} Sh(Spec(R)) @>>> \text{Sets}\\ @V V V @VV V\\ S[\text{Local Rings}] @>>> S[\text{Rings}] \end{CD}

hence the construction of the spectrum of a ring follows directly from looking at the fibers of the geometric morphism $S[\text{Local Rings}] \rightarrow S[\text{Rings}]$. The slogan is "the spectrum of a ring is the universal way of making the ring into a local ring" (the corresponding local ring is given by the structure sheaf, which is a local ring internal to $Sh(Spec(R))$ and given by the geometric morphism $Sh(Spec(R)) \rightarrow S[\text{Local Rings}]$).

Moreover, locally ringed spaces as special cases of locally ringed topoi are considered by some writers as the right notion a "geometric" space - for example Jacob Lurie generalizes basic notions of geometry to higher geometry using locally ringed spaces. There's a natural notion of what it means to be locally isomorphic to a given locally ringed space and the concept of smooth manifold, complex analytic manifold and also scheme derive naturally from that. The category of locally ringed topoi is simply the overcategory (or slice category) of $\text{Topos}$ over $S[\text{Local Rings}]$ - a nice fact is that this automatically gives the right notion of a smooth map between smooth manifolds, holomorphic map between complex analytic manifolds, etc.

Some other remarks from the side of constructivist mathematics are that linear algebra in a constructive setting works best over local rings (as there are different notions of field in constructive mathematics).

**So to come to my question: What is going on here? Let me be specific: I can grasp the concept of a group as a syntactic description of what we mean by the symmetries of an object - what is the underlying essence of a local ring? What part of our human intuition do they describe naturally?**

A little remark: There is a partial answer to be found in the following: A local ring has a canonical *apartness relation* $x \# y$, where $x \# y$ iff $x-y$ is invertible. In fact a ring object in the category of sets equipped with apartness relations is the same thing as a local ring. I am however not yet satisfied with this.

laxpullback square. Note that it only commutes up to a non-invertible 2-morphism. $\endgroup$ – Marc Hoyois Nov 24 '16 at 17:2211more comments