Call a category $C$ *rigid* if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example sets, commutative monoids, groups, abelian groups, commutative rings, but also the category of topological spaces. The category of monoids (or rings) is not rigid because $M \mapsto M^{\mathrm{op}}$ is an equivalence which is not isomorphic to the identity. See [M] for a survey and the general strategy for proving rigidity. The case of commutative rings was discussed recently on MO [here](https://mathoverflow.net/questions/55931/invariance-of-zx-under-a-self-equivalence-of-the-category-of-commutative-ring/56183#56183). The philosophy is that a category is rigid if every object can be defined in a categorical way, which is a quite interesting property.


**Question**: Is the category of schemes rigid?


Here is what I've done so far: The initial scheme is $\emptyset$ and the terminal scheme is $\text{Spec}(\mathbb{Z})$. Spectra of fields are characterized by the property that they are non-initial and and every morphism from a non-initial object to them is an epimorphism, see Kevin's answer [here](https://mathoverflow.net/questions/56564/what-are-the-epimorphisms-in-the-category-of-schemes). The underlying set $|X|$ of a scheme is the set of equivalence classes of morphisms $Y \to X$, where $Y$ is the spectrum of a field. So this recovers $|X|$ from $X$ in a categorical manner. If $x \in |X|$, then $\text{Spec}(\kappa(x))$ is the terminal spectrum of a field which maps to $X$ and has (set) image $x$.


However, I'm not able to recover the topology from $X$. I don't know how to characterize open or closed immersions. They are exactly the étale resp. proper monomorphisms, see [this](https://mathoverflow.net/questions/56591/what-are-the-monomorphisms-in-the-category-of-schemes) MO question, but it seems to be hard to characterize étale and proper categorically. After all, if are able to characterize affine schemes, then we will be done, since the category of affine schemes is rigid and every scheme is the canonical colimit of the affine schemes mapping into it.


In order to characterize affine schemes, it is enough to characterize the ring object $\mathbb{A}^1\_\mathbb{Z}$ in the category of schemes, since we can then define the ring of global sections of a scheme categorically and then say that affine schemes $Y$ are characterized by the property that for all schemes $X$ the map $Hom(X,Y) \to Hom(\mathcal{O}(Y),\mathcal{O}(X))$ is bijective.


Other approaches: 1. First show that the category of fields is rigid. I've already shown that the notions of prime field, $\mathbb{F}\_p$, $\mathbb{Q}$, finite, characteristic, normal, separable, algebraic, galois, transcendent, transcendence degree are categorical, but this is not enough to distinguish, for example, $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$. If $F$ is a self-equivalence of the category of fields, then $F$ maps $K(X)$ to $F(K)(X)$, so taking automorphisms there is a natural isomorphism $\text{PGL}(2,K) \cong \text{PGL}(2,F(K))$, but I wonder if this already implies that $K \cong F(K)$ naturally. 2. Characterize local schemes as a special full reflective subcategory containing the spectra of fields. 3. Try to categorify cohomology theory and use Serre's criterion for affineness.


EDIT (May '11): I've restarted this project in the last days. If $k$ is a field with only trivial endomorphisms, then I can show that every self-equivalence of $\text{Sch}/k$ preserves $\text{Spec}(k[\epsilon]/\epsilon^2)$, but also $\text{Spec}(k[[t]])$. But I still have no idea how to approach $\text{Spec}(k[t])$ categorically. Even basic notions such as "closed point" or "quasicompact" remain unclear.


EDIT (Feb '12): Let's work with $\mathrm{Sch}/k$ for some algebraically closed field $k$. Then $F$ maps $\mathbb{A}^1\_k$ to a ring object in $\mathrm{Sch}/k$. If we already knew that it is of finite type over $k$ and irreducible, then a Theorem by Greenberg (Cor. 4.4 in *Algebraic Rings*, Trans. AMS, Vol. 111, No. 3, pp. 472 - 481) will imply that the underlying scheme is just $\mathbb{A}^n\_k$ for some $n$. Now using my question about [factorization](https://mathoverflow.net/questions/65438/factorization-of-schemes) we should be able to conclude $n=1$. Of course, many details are missing here; for example it is not clear at all why $F$ should preserve schemes of finite type.


Any ideas concerning the categorical characterization of other properties / objects are appreciated. Feel free to add every piece as a single answer even if it does not answer the whole question.


[M] E. Makai jun, *Automorphisms and Full Embeddings of Categories in Algebra and Topology*, [online](http://www.heldermann.de/R&E/RAE18/ctw13.pdf)



Rigidity of the category of schemes
As requested by the OP in the comments of the (correct and complete) accepted answer of user131755: it's possible to say more.



> 
> **Theorem** [Mochizuki 2004, vDdB 2019]. *Let $S$ and $S'$ be schemes. Then the natural functor
> $$\operatorname{Isom}(S,S') \to \mathbf{Isom}(\mathbf{Sch}\_{S'},\mathbf{Sch}\_S)$$
> is an equivalence of categories, where $\operatorname{Isom}(S,S')$ is a discrete category and $\mathbf{Isom}$ denotes the category whose objects are equivalences and whose morphisms are natural isomorphisms.*
> 
> 
> 


The version where $\mathbf{Sch}$ denotes the category of locally Noetherian schemes with finite type morphisms is due to Mochizuki [Mochizuki 2004], and the general statement appears in a preprint of myself [vDdB 2019].


In particular, taking $S = S' = \operatorname{Spec} \mathbf Z$ answers the question, since $\operatorname{Aut}(\operatorname{Spec} \mathbf Z) = 1$.




---


**Some ideas of the proof.**


Here is a broad overview of the proof; more details can be found in [vDdB 2019]. As will become clear, most of the ideas were already present in some form, but there were some key tricks missing.


**1. Underlying set.**


The underlying set of $X \in \mathbf{Sch}\_S$ is reconstructed as the set of isomorphism classes of simple subobjects.


**2. Topology.**


Although we don't know if regular monomorphisms in $\mathbf{Sch}$ are the same as (locally closed) immersions (see also [this question](https://mathoverflow.net/q/66279/82179)), we do know:


* Every open immersion is a regular monomorphism;
* Every closed immersion is a regular monomorphism;
* Every regular monomorphism is an immersion.


Thus, a morphism $f \colon X \to Y$ is an immersion if and only if it can be written as a composition of two regular monomorphisms.


Next, one shows:



> 
> **Proposition.** *Let $(X,x)$ be a pointed scheme. Then $(X,x) \cong (\operatorname{Spec} R, \mathfrak m)$ for a valuation ring $R$ with maximal ideal $\mathfrak m$ if and only if all of the following hold:*
> 
> 
> 1. *$X$ is reduced and connected;*
> 2. *the category of immersions $Z \hookrightarrow X$ containing $x$ is a linear order;*
> 3. *there exists a subset $V \subseteq |X|$ that is the support of infinitely many pairwise non-isomorphic immersions $Z \hookrightarrow X$ containing $x$.*
> 
> 
> 


Together with the characterisation of immersions, this leads to categorical criteria for closed immersions and open immersions in $\mathbf{Sch}\_S$.


**3. Quasi-coherent sheaves.**


A variant of the Beck cogroup argument (see also user131755's post) realises nilpotent thickenings $\mathbf{Spec}\_X(\mathcal O\_X \oplus \mathscr F) \to X$ as cogroups in $X/\mathbf{Sch}\_X$. This gives (loosely speaking) a pseudofunctor
\begin{align\*}
\mathbf{Sch}\_S &\to \mathbf{Cat}^{\operatorname{op}}\\
X &\mapsto \mathbf{Qcoh}(\mathcal O\_X),
\end{align\*}
reconstructed from $\mathbf{Sch}\_S$ using only categorical data.


**4. The structure sheaf.**


Now we run an enhanced version of [this argument of the OP](https://mathoverflow.net/a/65509/82179) (that took place in the ring setting). We would like to say that the '(pre)sheaf End' $\mathscr End(\mathbf{Qcoh}(\mathcal O\_{-}))$ on $\mathbf{Sch}\_S$ is isomorphic to the structure (pre)sheaf $\mathcal O$ on the big Zariski site $\mathbf{Sch}\_S$.


This is possible, but the difficulty is to say what exactly this presheaf End (or really prestack End) should be (also since it all takes place on the big Zariski site $\mathbf{Sch}\_S$, not just the small Zariski site $S$).


**5. Proof of main theorem.**


By 2 and 4 above, we have reconstructed from $\mathbf{Sch}\_S$ the topology on $|S|$ together with its structure sheaf $\mathcal O\_S$. This gives (roughly speaking) some sort of lax functor of $2$-categories
\begin{align\*}
\{\text{categories equivalent to } \mathbf{Sch}\_S \text{ for some } S\} &\to \mathbf{Sch}\\
\mathbf{Sch}\_S &\mapsto S.
\end{align\*}
But in fact the reconstruction of the scheme $X \in \mathbf{Sch}\_S$ (with its structure morphism $X \to S$) from categorical data in $\mathbf{Sch}\_S$ is functorial in $X$. With some work, this shows that this lax functor is a lax inverse of $S \mapsto \mathbf{Sch}\_S$. $\square$


(Because I don't really speak $n$-category, I phrase the last part a little differently in my paper.)




---


**References.**


[vDdB 2019] Remy van Dobben de Bruyn, *Automorphisms of categories of schemes*, 2019. Submitted. arXiv:[1906.00921](https://arxiv.org/abs/1906.00921).


[Mochizuki 2004] Shinichi Mochizuki, [*Categorical representation of locally Noetherian log schemes*](http://dx.doi.org/10.1016/j.aim.2003.10.005). Adv. Math. **188**.1, p. 222-246 (2004). [ZBL1073.14002](https://zbmath.org/?q=an:1073.14002).


