Changes to my article about products in certain categories
There are two changes in Products in dagger categories with complete ordered Mor-sets draft article: 1. I’ve removed the section on relation of subatomic product with categorical product saying that for funcoids they are the same. No, they are not the same. My claim that they are the same was false. 2. Added section “Special […]
A wiki about applying generalized limit to singularities theory
I have created a wiki about development of theory of singularities using generalized limits. Please read my book (where among other I define generalized limits) and then participate in this research wiki. Study singularities in this novel approach and share Nobel Prize with me!
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} […]
Conjecture solved positively
I’ve proved this conjecture. See my book (which also contains a pointfree version of this theorem).
Pointfree funcoids as a generalization of frames/locales
I’ve put online my rough partial draft of the theory of bijective correspondence between frames/locales and certain pointfree funcoids. Pointfree funcoids are a massive generalization of locales and frames: They not only don’t require the lattice of filters to be boolean but these can be even not lattices of filters at all but just arbitrary […]
My study of pointfree topology
I have read The point of pointless topology today and am going to study the book Johnstone “Stone Spaces” which I purchased maybe a year or two ago. The purpose of this study is to integrate others’ pointless topology with my theory of pointfree funcoids. From my earlier comment on this blog: It seems that […]
Definition of order of pointfree funcoids changed in my book
In this blog post I announced that I am going to change the definition of order of pointfree funcoids in my book. Now in the last preprint the changes are done.
Changed the definition of order of pointfree funcoids
In my preprint I defined pre-order of pointfree funcoids by the formula $latex f\sqsubseteq g \Leftrightarrow [f]\subseteq[g]$. Sadly this does not define a poset, but only a pre-order. Recently I’ve found an other (non-equivalent) definition of an order on pointfree funcoids, this time this is a partial order not just a pre-order: $latex f \sqsubseteq […]
Pointfree funcoid induced by a locale or frame?
I have shown in my research monograph that topological (even pre-topological) spaces are essentially (via an isomorphism) a special case of endo-funcoids. It was natural to suppose that locales or frames induce pointfree funcoids, in a similar way. But I just spent a few minutes on defining the pointfree funcoid corresponding to a locale or […]
My further study plans
I remind that I am not a professional mathematician. Nevertheless I have written research monograph “Algebraic General Topology. Volume 1”. Yesterday I have asked on MathOverflow how to characterize a poset of all filters on a set. From the answer: the posets isomorphic to lattices of filters on a set are precisely the atomic compact […]