Reference on inductive (direct) limit of ordered vector spaces and vector lattices

I looked in all textbooks on vector lattices (Riesz spaces) as well as ordered vector spaces, but couldn't find any mentions of neither inductive nor projective limit for these structures. Googling also didn't help. Since there is some ambiguity in terminology let me state the definition.

Let $$\Gamma$$ be a directed set and let $$\left(E_{\gamma}\right)_{\gamma\in\Gamma}$$ be a family of partially ordered vector spaces, such that for every $$\alpha\le\beta$$ there is a designated positive linear operator $$T_{\alpha\beta}:E_{\alpha}\to E_{\beta}$$, such that $$T_{\gamma\gamma}=Id_{E_{\gamma}}$$ and $$T_{\beta\gamma}T_{\alpha\beta}=T_{\alpha\gamma}$$, for every $$\alpha,\beta,\gamma$$. The inductive limit then is an ordered space $$E$$ and a collection of positive linear operators $$T_{\gamma}:E_{\gamma}\to E$$ such that $$T_{\beta}T_{\alpha\beta}=T_{\alpha}$$, and whenever $$F,\left(S_{\gamma}\right)_{\gamma\in\Gamma}$$ has this property, there is a unique $$S:E\to F$$ such that $$S_{\gamma}=ST_{\gamma}$$.

If we drop "ordered" and "positive", we will get the definition of the inductive limit for vector spaces. In fact, the inductive limit of ordered vector spaces can be constructed as the inductive limit of the underlying vector spaces endowed with the weakest linear order the makes all $$T_{\gamma}$$'s positive. It is also not hard to show that if all $$E_{\gamma}$$'s are vector lattices, and $$T_{\alpha\beta}$$'s are lattice homomorphisms, then $$E$$ is itself a vector lattice.

I need some permanence properties of this construction, which I suspect are known, so I am asking for a reference where it is studied systematically.

• If nobody knows a reference you might ask explicitely for the permanence properties. The construction of the inductive limit (category theorists nowadays seem to prefer the name colimit) involves the forgetful functor into the category of vector spaces. You might look for a right adjoint of that functor. This is sometimes useful to get permanence properties for free. Commented Feb 8, 2022 at 16:21
• @JochenWengenroth I seem to be able to prove the properties myself, I just don't want to dedicate a third of a short note that I am writing to developing this theory from scratch. Could you please give an example in a different category (locally convex spaces perhaps?) where the permanence properties are obtained for free? In my rudimentary understanding of category theory, I find it hard to see how the adjoint functor can "detect" the subcategories of the target category
– erz
Commented Feb 9, 2022 at 1:36
• This does not exactly fulfil your request but nevertheless, I propose the following example: The forgetful functor $U:$ LCS $\to$ Vect has a right adjoint (endow a vector space with the trivial topology generated by the seminorm $0$). Therefore, $U$ preserves colimits. This implies and (perhaps even more importantly) explains that colimits in LCS are the colimits in Vect endowed with a suitable locally convex topology. If you consider instead the category of Hausdorff LCS, the forget functor does not have a right adjoint and colimits in that category are algebraically different. Commented Feb 9, 2022 at 7:55
• @JochenWengenroth but this does not prove existence of colimits, only that they are the way they are, right?
– erz
Commented Feb 10, 2022 at 0:22
• Yes, that's right. It is however important to have a candidate for colimits, this helps a lot either to verify that the cadidate is (or rather, can be made) a colimt or that a colimit does not exist. Commented Feb 10, 2022 at 12:09