Definition talk:Metacategory

From ProofWiki
Jump to navigation Jump to search

This is really a list of axioms not a definition. I don't know how to phrase it clearly in an axiomatic way. Linus44 13:54, 19 February 2011 (CST)

Good question. It looks coherent enough as it is. If you wish to axiomatise (eugh, horrible word) it formally, take a look at the Axiom Index and see what I did for that.--prime mover 16:51, 19 February 2011 (CST)

Multiple definitions

I would like to add at least two other definitions of metacategory. One uses no objects (replacing them by their identity morphisms) and the other is used to make things formal and practically define a language in which we can prove formal duality for categorical statements. How to proceed? Pointers to instances of such multiple definitions are also fine. I think it is however best to keep the current, accessible and clear definition as the main one (the others are really reformulations not best suited for a first introduction). --Lord_Farin (talk) 18:57, 12 September 2012 (UTC)

If the definitions are equivalent, pick one and use that as the definition, then prove the other by means of an iff proof and link to it in an "also" section (we haven't decided how to do this consistently yet: we need a standard section which is to say "Some authorities / sources use this as the definition for (etc.) and from it prove (whatever equivalent statement)" but I have not come up with anything definitive for this.)
If the definitions are not equivalent but genuinely different specifications, then add them both but with different labels, e.g. "Metacategory (Heineken's Version)" and "Metacategory (Theakston's Derivation)" etc. We have some instances of this in Topology where I think I did a pretty good job of what is a bit of a mess, especially around Separation Axioms.
The trouble is with new areas of mathematics like this is that a definitive approach has not yet happened. Eventually a canonical approach will evolve, based (usually) on what ultimately works best - but till now we have multiple threads to keep aligned. Nightmare. --prime mover (talk) 19:39, 12 September 2012 (UTC)
Approaches are equivalent. Check my sandbox, and please comment. --Lord_Farin (talk) 21:48, 12 September 2012 (UTC)
Way outside my comfort zone I'm afraid - but as I say, if the definitions are equivalent then use the first of the two approaches above. That would be what I would do. --prime mover (talk) 22:09, 12 September 2012 (UTC)
I did not entirely follow the suggested course of action, but I think that my approach is justified by the fact that it founds category theory as a first-order language (i.e., some appropriate instance of PredCalc). Now I finally feel like I can put up the formal and conceptual duality theorems for CatTh, which make proving stuff *a lot* easier. --Lord_Farin (talk) 12:52, 21 September 2012 (UTC)


Braces have been added around the identity morphisms. The original and the new thing look identical to me. What's your set-up, abcxyz? --Lord_Farin (talk) 22:22, 8 January 2013 (UTC)

Okay, yes, just one of them looks different to me. $\operatorname{id}_X \circ g$ vs. ${\operatorname{id}_X} \circ g$. Do they look identical to you? --abcxyz (talk) 22:25, 8 January 2013 (UTC)
Hm, no. Had missed that one. I like $\operatorname{id}_X \mathop \circ g$ better as a solution, though. It gives more information on the intent of the source writer and even (ever so slightly) seems to reduce the now somewhat oversized gap between the first two entities. --Lord_Farin (talk) 22:29, 8 January 2013 (UTC)
The "mathop" technique is what has evolved over the course of development in the last few months. It's only used when it's needed, but works well when it's used. --prime mover (talk) 22:39, 8 January 2013 (UTC)
Sorry, I can't tell the difference between the spacings of $f \circ {\operatorname{id}_X}$ and ${\operatorname{id}_X} \circ g$ (even when I zoomed in a lot). I'm not sure that the added space isn't merely an illusion.
By the way, \mathop actually seems to decrease space (in this case), e.g. $f \circ g$ (f \circ g) vs. $f \mathop \circ g$ (f \mathop \circ g). --abcxyz (talk) 22:54, 8 January 2013 (UTC)
There's definitely a difference on my screen; our setups might differ, causing this difference. As for the behaviour of \mathop, it seems to designate its operand as a mathematical operator. Since MathJax is not a perfect TeX clone, such small "quirks" (it may well be intentional, I don't know) can always occur. --Lord_Farin (talk) 23:00, 8 January 2013 (UTC)