Offtopic: Formalized Gospel theology
This is partly an offtopic post in my math blog. It seems likely that I discovered a category in which such objects as the Father and the Son from the Gospel appear. I am not sure I really discovered God, but this seems likely. Consider a category (there seems to be multiple ways to add […]
Two new chapters in my math draft
I’ve added chapters “Cartesian closedness” and “Singularities” (from the site http://tiddlyspace.com which will be closed soon) to volume 2 draft. Both chapters are very rough draft and present not rigorous proofs but rough ideas.
Directed topological spaces and funcoids
I have researched relations between directed topological spaces and pair of funcoids. Here the first funcoid represents topology and the second one represents direction. Results are mainly negative: Not every directed topological space can be represented as a pair of funcoids. Different pairs of a topological space and its subfuncoid may generate the same directed […]
A new (but easy to prove) theorem in my research book: Theorem Let $latex \mu$ and $latex \nu$ be endomorphisms of some partially ordered dagger precategory and $latex f\in\mathrm{Hom}(\mathrm{Ob}\mu;\mathrm{Ob}\nu)$ be a monovalued, entirely defined morphism. Then $latex f\in\mathrm{C}(\mu;\nu)\Leftrightarrow f\in\mathrm{C}(\mu^{\dagger};\nu^{\dagger}).$
A new math abstraction, categories of sides
I introduce a new math abstraction, categories of sides, in order to generalize two theorems into one. Category of sides $latex \Upsilon$ is an ordered category whose objects are (small) bounded lattices and whose morphisms are maps between lattices such that every Hom-set is a bounded lattice and (for all relevant variables): $latex (a \sqcup […]
A new negative result in pointfree topology
I have proved the following negative result: Theorem $latex \mathsf{pFCD} (\mathfrak{A};\mathfrak{A})$ is not boolean if $latex \mathfrak{A}$ is a non-atomic boolean lattice. The theorem is presented in this file. $latex \mathsf{pFCD}(\mathfrak{A};\mathfrak{B})$ denotes the set of pointfree funcoids from a poset $latex \mathfrak{A}$ to a poset $latex \mathfrak{B}$ (see my free ebook). The theorem and its […]
A conjecture about product order and logic
The considerations below were with an error, see the comment. Product order $latex {\prod \mathfrak{A}}&fg=000000$ of posets $latex {\mathfrak{A}_i}&fg=000000$ (for $latex {i \in n}&fg=000000$ where $latex {n}&fg=000000$ is some index subset) is defined by the formula $latex {a \leq b \Leftrightarrow \forall i \in n : a_i \leq b_i}&fg=000000$. (By the way, it is a […]
Another star-category of funcoids
I’ve introduced another version of cross-composition of funcoids. This forms a category with star-morphisms. It is conjectured that this category is quasi-invertible, because I have failed to prove it. This should be included in the next version of my book.
The set of funcoids is a co-frame (without axiom of choice)
A mathematician named Todd Trimble has helped me to prove that the set of funcoids between two given sets (and more generally certain pointfree funcoids) is always a co-frame. (I knew this for funcoids but my proof required axiom of choice, while Todd’s does not require axiom of choice.) He initially published his proof here […]
Category theoretical generalization of reloids and funcoids
While walking home from McDonalds I conceived the following idea how we can generalize reloids and funcoids. Let $latex C$ be a category with finite products, the set of objects of which is a complete lattice (for the case of funcoids as described below it is enough to be just join-semilattice). One can argue which […]