A new easy theorem in my draft
A new easy theorem in my draft. Theorem $latex \mathrm{DOM} (g \circ f) \supseteq \mathrm{DOM} f$, $latex \mathrm{IM} (g \circ f) \supseteq \mathrm{IM} g$, $latex \mathrm{Dom} (g \circ f) \supseteq \mathrm{Dom} f$, $latex \mathrm{Im} (g \circ f) \supseteq \mathrm{Im} g$ for every composable morphisms $latex f$, $latex g$ of a category with restricted identities. Proof […]
Responses to some accusations against style of my exposition
This is a very short addition to my book, in response to a person who criticized my style. He may be partly right, but: The proofs are generally hard to follow and unpleasant to read as they are just a bunch of equations thrown at you, without motivation or underlying reasoning, etc. I don’t think […]
New results about categories with restricted identities
I have rewritten the section “More results on restricted identities” of this draft. Now it contains some new (easy but important) formulas.
New concept in category theory: “unfixed morphisms”
I have developed my little addition to category theory, definition and research of properties of unfixed morphisms. Unfixed morphisms is a tool for turning a category (with certain extra structure) into a semigroup, that is abstracting away objects. Currently this research is available in this draft. I am going to rewrite my online book using […]
Removed some sections from my draft
I removed from my draft sections about “categories under Rel”. The removal happened because I developed a more general and more beautiful theory. The old version is preserved in Git history.
Restricted identity axioms changed
I announced that I have introduces axioms for “restricted identities”, a structure on a category which allows to turn the category into a semigroup (abstracting away objects). But I noticed that these axioms do not fit into concrete examples which I am going to research. So I have rewritten the text about restricted identities with […]
“Unfixed filters” rewritten
“Unfixed filter” section of my book was rewritten for more general lattices instead of old version with a certain lattice of sets.
Categories with restricted identities
In this draft (to be moved into the online book in the future, but the draft is nearing finishing this topic, not including functors between categories with restricted identities) I described axioms and properties of categories with restricted identities. Basically, a category with restricted identities is a category $latex \mathcal{C}$ together with morphisms $latex \mathrm{id}^{\mathcal{C}(A,B)}_X$ […]
“Unfixed filters” research now in the book volume-1
I essentially finished my research of unfixed filters. I moved all research of unfixed filters to volume-1.pdf. Particularly now it contains subsections “The lattice of unfixed filters” and “Principal unfixed filters and filtrator of unfixed filters”. Now I am going to research unfixed reloids and unfixed funcoids (yet to be defined).
Filters on a lattice are a lattice
I’ve proved that filters on a lattice are a lattice. See my book.