Direct products in a category of funcoids
I’ve released a draft article about categorical products and coproducts of endo-funcoids, as well as products and coproducts of other kinds of endomorphisms. An open problem: Apply this to the theory of reloids. An other open problem: Whether the category described in the above mentioned article is cartesian closed.
New concept: metamonovalued morphisms
Let fix some dagger category every of Hom-sets of which is a complete lattice, and the dagger functor agrees with the lattice order. I define a morphism $latex f$ to be monovalued when $latex \circ f^{-1}\le \mathrm{id}_{\mathrm{Dst}\, f}$. I call a morphism $latex f$ metamonovalued when $latex (\bigwedge G) \circ f = \bigwedge_{g \in G} […]
Crucial error in my definition of product funcoids
I have noticed a crucial error in the article with my definition of product funcoids (I confused the direction of an implication). Thus it is yet not proved that my “product” is really a categorical product. The same applies to my definition of coproduct.
A section about relationships of different products in my book
I have added new section 16.13 “Relationships of cross-composition and subatomic products” with some important theorems (which I am going to use in the second volume of my book) to my preprint. I have also removed altogether the section “Displaced product” and the definition of displaced product, as it has turned out that displaced product […]
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 […]
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.
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 […]
One more conjecture about provability without axiom of choice
I addition to this conjecture I formulate one more similar conjecture: Conjecture $latex a\setminus^{\ast} b = a\#b$ for arbitrary filters $latex a$ and $latex b$ on a powerset cannot be proved in ZF (without axiom of choice). Notation (where $latex \mathfrak{F}$ is the set of filters on a powerset ordered reverse to set-theoretic inclusion): $latex […]