I have written a short article with my response on Todd Trimble’s commentary on my book.

In this response I present these of Todd Trimble’s results which are new for me.

Note that I skipped specifically category-theoretic results (such as that the category of endofuncoids is topological). I am going to amend my article with categorical results later.

Todd Trimble has notified me that he has written a “commentary” (notes) on my theory of funcoids presented in my monograph.

His commentary is available at this nLab wiki page.

I’ve started to read his notes. First I needed to lookup into Wikipedia to know what Chu space is. He uses category theory however as it now seems to me not very advanced. So I hope to understand his writing in soon time.

Conjecture For every funcoid $f$ and filter $\mathcal{X}\in\mathfrak{F}(\mathrm{Src}\,f)$, $\mathcal{Y}\in\mathfrak{F}(\mathrm{Dst}\,f)$:

1. $\mathcal{X} \mathrel{[(\mathsf{FCD}) f]} \mathcal{Y} \Leftrightarrow \forall F \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} f : \mathcal{X} \mathrel{[F]} \mathcal{Y}$;
2. $\langle (\mathsf{FCD}) f \rangle \mathcal{X} = \bigsqcap_{F \in \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)} f} \langle F \rangle \mathcal{X}$.

Conjecture $(\mathsf{FCD}) f = \bigsqcap^{\mathsf{FCD}} (\Gamma (A ; B) \cap \mathrm{GR}\, f)$ for every reloid $f \in \mathsf{RLD} (A ; B)$.

Conjecture $(\mathsf{RLD})_{\mathrm{in}} f = \bigsqcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\mathrm{Src}\, f ; \mathrm{Dst}\, f)}\, f$ for every funcoid $f$.

(I use notation from this note and this draft article.)

Today I’ve proved that this bijection preserves composition.

See this note (updated) for the proofs.

I have just proven the following two new theorems:

Theorem Composition of complete reloids is complete.

Theorem $(\mathsf{RLD})_{\mathrm{out}} g \circ (\mathsf{RLD})_{\mathrm{out}} f = (\mathsf{RLD})_{\mathrm{out}} (g \circ f)$ if $f$ and $g$ are both complete funcoids (or both co-complete).

See this note for the proofs.