A conjecture about atomic funcoids
A new conjecture: Conjecture $latex \langle f \rangle \mathcal{X} = \bigsqcup_{F \in \mathrm{atoms}\, f} \langle F \rangle \mathcal{X}$ for every funcoid $latex f$ and $latex \mathcal{X} \in \mathfrak{F} (\mathrm{Src}\, f)$. This conjecture seems important for the notion of exponential object in the category of continuous maps between endofuncoids, which I am investigating now.
The category Fcd has small co-products
Two days ago I have proved that the category Fcd of continuous maps between endofuncoids has small products. Today I have also proved that this category has small co-products. The draft article is now available online. I’m yet to check whether product functors preserve co-products and whether my category has exponential objects and so is […]
The category of continuous maps between endofuncoids has small products
I have proved that Theorem The category of continuous maps between endofuncoids has small products. See my draft article for a proof.
I’ve fixed the error in my proof
I have quickly corrected the error in my proof of an important theorem. Now it is even more beautiful.
Error in my proof
That proof which I claimed in this blog post is with an error: I have messed product of objects and product of morphisms. Now I desperately attempt to repair the proof.
Direct product in the category of continuous maps between endofuncoids
I released a rough draft of my article Direct product in the category of continuous maps between endofuncoids. This (among other) solves the problem I proposed in this blog post. Previously I have said that my research got stuck. Now I see how to continue it! I am again blessed.
Conjecture solved
I’ve published in my book’s preprint a theorem (currently numbered 6.100) which solves a former conjecture. Theorem $latex g \circ \left( \bigsqcup R \right) = \bigsqcup \left\{ g \circ f \,|\, g \in R \right\} = \bigsqcup \langle g \circ \rangle R$ if $latex g$ is a complete funcoid. My shame, I have earlier overlooked […]
Open mappings between endo-funcoids
Let $latex \mu$ and $latex \nu$ are endofuncoids and $latex f$ is a funcoid from $latex \mathrm{Ob}\,\mu$ to $latex \mathrm{Ob}\,\nu$. Then we can generalize Bourbaki’s notion of open mapping between topological spaces (that is a mapping for which images of open sets are open) by the following formula (where $latex x$ is a variable which […]
The (candidate) construction of direct product in the category of continuous maps between endo-funcoids
Consider the category of (proximally) continuous maps (entirely defined monovalued functions) between endo-funcoids. Remind from my book that morphisms $latex f: A\rightarrow B$ of this category are defined by the formula $latex f\circ A\sqsubseteq B\circ f$ (here and below by abuse of notation I equate functions with corresponding principal funcoids). Let $latex F_0, F_1$ are […]
My fool topology study
I am (re)reading Bourbaki “General Topology” (in Russian language). Despite I am a general topologist, I have never had a systematic general topology study. I think now I should fill this hole. Maybe after reading Bourbaki I will return to Johnstone “Stone Spaces” which I weakly if at all understand.