User:Lord Farin/Sandbox

This page exists for me to be able to test out features I am developing. Also, incomplete proofs may appear here.

Feel free to comment.

Over time, stuff may move to User:Lord_Farin/Sandbox/Archive.

= Definition of metacategory without objects =

Definition
Let $\mathbf C_1$ be a collection of objects called morphisms.

Let $\mathbf C_2$ be a collection of pairs of morphisms; write $\mathbf C_2 \left({g, f}\right)$ to express that $\left({g, f}\right)$ is a member of $\mathbf C_2$.

Let $\circ$ be an operation symbol which must assign to every pair of morphisms $\left({g, f}\right)$ in $\mathbf C_2$ a morphism $g \circ f$, called the composition of $g$ with $f$.

An arrows-only metacategory is a metamodel for the language of category theory subject to the following axioms:


 * $h \circ \left({g \circ f}\right)$ is defined whenever $\left({h \circ g}\right) \circ f$ is
 * Whenever both are defined, $h \circ \left({g \circ f}\right) = \left({h \circ g}\right) \circ f$
 * $h \circ \left({g \circ f}\right)$ is defined whenever $h \circ g$ and $g \circ f$ are
 * For all morphisms $f$ there exist morphisms $u$ and $u'$ such that $u \circ f$ and $f \circ u'$ are defined, and $u \circ f = f = f \circ u'$

Improvement of Sequence of Implications of Connectedness Properties
For brevity, let us introduce the following acronyms:


 * $\mathrm{AC}$: Arc-Connected
 * $\mathrm{UC}$: Ultraconnected
 * $\mathrm{PC}$: Path-Connected
 * $\mathrm{HC}$: Hyperconnected
 * $\mathrm C$:  Connected

Then the following sequence of implications holds:


 * $\begin{xy}

<-3em,3em>*+{\mathrm{AC}} = "AC", <3em,3em>*+{\mathrm{UC}} = "UC", <0em,0em>*+{\mathrm{PC}} = "PC", <6em,0em>*+{\mathrm{HC}} = "HC", <3em,-3em>*+{\mathrm C}  = "C",

"AC";"PC" **@2{-} ?><>(1.2)*@2{>}, "UC";"PC" **@2{-} ?><>(1.2)*@2{>}, "PC";"C" **@2{-} ?><>(1.2)*@2{>}, "HC";"C" **@2{-} ?><>(1.2)*@2{>}, \end{xy}$

What do you think? Now that I have created the moulds, it will be easy to adapt to the other 'Sequences of Implication'. --Lord_Farin (talk) 10:21, 31 August 2012 (UTC) It be noted that it will forever be impossible to endow diagrams (and indeed, any TeX rendered with MathJax) with internal links; sorry. --Lord_Farin (talk) 10:22, 31 August 2012 (UTC)


 * No response? :( --Lord_Farin (talk) 21:31, 7 September 2012 (UTC)


 * It's very nice! I don't know the first thing about what it meant, though :) --GFauxPas (talk) 21:53, 7 September 2012 (UTC)


 * Sorry, only just noticed it. Very nice - one caveat: you need to refer to the key to (a) work out what the codes mean, and (b) to get to the link explaining them. The somewhat clumsier page from which the original of this came does have the full map as one self-contained unit. Might be interesting to put the two presentations up on the same page as alternative renditions. --prime mover (talk) 22:21, 7 September 2012 (UTC)