I want to characterize Hausdorffness of a locally convex space only using categorical terms of the additive category LCS of locally convex spaces and continuous linear maps, i.e., terms like mono- or epimorphisms, categorical limits or colimits, or images and kernels are allowed but the toplological definition *Distinct points have disjoint neighbourhoods* is forbidden.


Using the field $\mathbb K$ (either real or complex) as a special object, two characterizations of Hausdorffness are


* Every morphism $f:\mathbb K\to X$ is strict (i.e., its canonical factorization $\dot f:$ coimage$(f) \to$ im$(f)$ is an isomorphism)
* There is a monomorphism $X\to \mathbb K^I$ for some set $I$ (where $\mathbb K^I$ is a categorical product, this characterization uses Hahn-Banach.)


These two characterizations would fit the bill if $\mathbb K$ is characterized in categorical terms.


The questions are thus:


* Is there a characterization of Hausdorffness in terms of LCS without using the field $\mathbb K$?
* Is there a characterization of $\mathbb K$ in LCS?


A similar question could of course be asked for the categories of all topological spaces or (to have enough morphisms) all completely regular spaces. Mayby a reference in this direction would help for the questions in LCS.



Is Hausdorffness a categorical property in the category of locally convex spaces?
For a category $\mathcal{C}$, let $\mathcal{C}'$ denote the full subcategory of $\mathcal{C}$ whose objects are the non-terminal objects of $\mathcal{C}$.


In a category, say that an object $Y$ is final if for every object $X$ there exists an epimorphism $X\to Y$.


In turn, say that an object of $\mathcal{C}$ is pre-final if it is a final object of $\mathcal{C}'$.


Then say that an object $Y$ of $\mathcal{C}$ is pseudo-Hausdorff if $\mathrm{Hom}(X,Y)$ is reduced to a singleton for every pre-final $X$.




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Then in the category $\mathcal{C}$ of locally convex spaces (and also topological vector spaces over an arbitrary Hausdorff field), the terminal objects are those spaces reduced to $\{0\}$. In both $\mathcal{C}$ and $\mathcal{C}'$, epimorphisms are just surjective maps (this uses the existence of non-Hausdorff objects). In turn, the pre-final objects are those 1-dimensional non-Hausdorff spaces. And the pseudo-Hausdorff objects are then the Hausdorff spaces.


