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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeAug 13th 2015
   • (edited Aug 13th 2015)
   • PermaLink
   Author: porton
   Format: MarkdownItexFirst 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: * [The book (first volume)](http://www.mathematics21.org/binaries/volume-1.pdf) * [Book homepage](http://www.mathematics21.org/algebraic-general-topology.html) * [LaTeX files under Git](https://bitbucket.org/portonv/algebraic-general-topology) 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](http://www.mathematics21.org/binaries/volume-2.pdf) 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.

   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:

   • The book (first volume)
   • Book homepage
   • LaTeX files under Git

   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.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeAug 14th 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexI 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorporton
   • CommentTimeAug 14th 2015
   • (edited Aug 14th 2015)
   • PermaLink
   Author: porton
   Format: MarkdownItex@DavidRoberts: I have added definitions of metamonovalued, metainjective, metacomplete, and co-metacomplete morphisms to [volume 2](http://www.mathematics21.org/binaries/volume-2.pdf) draft. Now you can read it till usage of continuous morphisms. Consult [volume 1](http://www.mathematics21.org/binaries/volume-1.pdf) 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: $$f\in\mathrm{C}^'(\mu;\nu)\Leftrightarrow f\in\Hom(\Ob\mu;\Ob\nu)\text{ and } \mu\sqsubseteq f^{\dagger}\circ\nu\circ f;$$ $$f\in\mathrm{C}''(\mu;\nu)\Leftrightarrow f\in\Hom(\Ob\mu;\Ob\nu)\text{ and } f\circ\mu\circ f^{\dagger}\sqsubseteq\nu.$$

   @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:

   $$f\in\mathrm{C}^'(\mu;\nu)\Leftrightarrow f\in\Hom(\Ob\mu;\Ob\nu)\text{ and } \mu\sqsubseteq f^{\dagger}\circ\nu\circ f;$$ $$f\in\mathrm{C}''(\mu;\nu)\Leftrightarrow f\in\Hom(\Ob\mu;\Ob\nu)\text{ and } f\circ\mu\circ f^{\dagger}\sqsubseteq\nu.$$
4. • CommentRowNumber4.
   • CommentAuthorporton
   • CommentTimeAug 14th 2015
   • PermaLink
   Author: porton
   Format: MarkdownItexI've added also definitions of monovalued morphisms and entirely defined morphisms. It seems now nothing prevents you to read it.

   I’ve added also definitions of monovalued morphisms and entirely defined morphisms.

   It seems now nothing prevents you to read it.

5. • CommentRowNumber5.
   • CommentAuthorporton
   • CommentTimeAug 14th 2015
   • PermaLink
   Author: porton
   Format: MarkdownItexI updated my draft for clarity more.

   I updated my draft for clarity more.

6. • CommentRowNumber6.
   • CommentAuthorporton
   • CommentTimeAug 14th 2015
   • PermaLink
   Author: porton
   Format: MarkdownItexIt 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](http://www.mathematics21.org/binaries/volume-1.pdf) (almost ready draft).

   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).