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First I have written and self-published a e-book with my research related with general topology, I welcome to read my book, it is a draft but near release quality now:
I have also started a VERY rough partial draft which I refer as “volume 2”. This volume contains things related with category theory.
I welcome you to look into volume 2 and judge whether my CT concepts are novel (or is novel only applying them to funcoids and reloids?)
For example, I define product of every family of endomorphisms of a category whose Hom-sets are complete lattices. (Need to check this my statement for errors however.) Is it a new idea? The product in this case is also an endomorphism (but for certain ordered dagger categories I define product of every family of morphisms, not only endomorphisms.)
Please copy ideas from my texts to nLab wiki.
I did look, and unfortunately there are concepts with unfamiliar names on the first page that are used with no definition, and unexplained notation. This makes me give up immediately.
@DavidRoberts: I have added definitions of metamonovalued, metainjective, metacomplete, and co-metacomplete morphisms to volume 2 draft.
Now you can read it till usage of continuous morphisms. Consult volume 1 for continuous morphisms.
Well, OK, I will give the definitions of continuous morphisms ($\mathrm{C}$) here ($\mu$ and $\nu$ are arbitrary endomorphisms of a category whose Hom-sets are ordered):
$f\in\mathrm{C}(\mu;\nu)\Leftrightarrow f\in\Hom(\Ob\mu;\Ob\nu)\text{ and } f\circ\mu\sqsubseteq\nu\circ f.$If the precategory is a partially ordered dagger precategory then continuity also can be defined in two other ways:
$f\in\mathrm{C}^'(\mu;\nu)\Leftrightarrow f\in\Hom(\Ob\mu;\Ob\nu)\text{ and } \mu\sqsubseteq f^{\dagger}\circ\nu\circ f;$ $f\in\mathrm{C}''(\mu;\nu)\Leftrightarrow f\in\Hom(\Ob\mu;\Ob\nu)\text{ and } f\circ\mu\circ f^{\dagger}\sqsubseteq\nu.$I’ve added also definitions of monovalued morphisms and entirely defined morphisms.
It seems now nothing prevents you to read it.
I updated my draft for clarity more.
It is not only applying CT to funcoids, but also the reverse: Funcoids are a useful tool for pure CT.
For example, product of morphisms of certain categories with ordered Hom-sets is defined by me as a pointfree funcoid.
See volume 1 of my book (almost ready draft).
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