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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.

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  • $\begingroup$ There is quasiseparatedness of condensed $\mathbb R$-modules in condensed mathematics. $\endgroup$
    – Z. M
    Commented Oct 9, 2021 at 11:11

1 Answer 1

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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 $X$.

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.

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  • $\begingroup$ This is really ad-hoc to topological vector spaces, in which things are eased as $T_0=T_1=T_2$, and taking advantage of the non-Hausdorff 1-dimensional space. In the category of topological groups we keep the first feature but don't have the second one. For topological spaces, the first feature fails of course. $\endgroup$
    – YCor
    Commented Oct 7, 2021 at 18:19
  • $\begingroup$ For topological spaces, $T_0$ and $T_1$ seem not hard to characterize: one can first characterize the space on 2 elements with indiscrete topology, and the space on 2 elements with neither indiscrete nor discrete topology, and characterize in terms of homomorphisms from these spaces. For $T_2$, I don't see right away. $\endgroup$
    – YCor
    Commented Oct 7, 2021 at 20:21
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    $\begingroup$ perhaps one can characterise enough of the needed small finite spaces to apply the machinery at ncatlab.org/nlab/show/… $\endgroup$
    – David Roberts ♦
    Commented Oct 8, 2021 at 5:05
  • $\begingroup$ Thank you, Yves. Are these just ad hoc definitions or is there a reference for this? Although your answer is a solution for the first question it is somewhat artificial at leat from the analytical point of view. In the hope for other solutions, I wait a while before acceptingyours. $\endgroup$
    – Jochen Wengenroth
    Commented Oct 8, 2021 at 9:27
  • $\begingroup$ As you characterized the field with the trivial topology in categorical terms, one gets also a formal characterization of $\mathbb K$ with the natural topology: It is neither terminal nor pre-final and every morphism $f:\mathbb K\to X$ is either zero or monic. But again, this is not very satisfying. $\endgroup$
    – Jochen Wengenroth
    Commented Oct 8, 2021 at 9:34

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