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edited the entry orthogonality a bit, for instance indicated that there are other meanings of orthogonality. This should really be a disambiguation page.
And what makes the category-theoretic notion of orthogonality not be merged with weak factorization system? And why is orthogonal factorization system the first example at orthogonality if in fact that imposes unique lifts, while in orthogonality only existence of lifts is required?
I think the entry-situation here deserves to be further harmonized.
You misread orthogonality, it does indeed say that there “exists a unique diagonal filler”.
Perhaps the category-theoretic notion should be at orthogonal morphisms or something.
You misread orthogonality, it does indeed say that there “exists a unique diagonal filler”.
Ah, right, sorry.
Hm, but the notation $\perp$ is also used for non-unique LLPs.
Perhaps the category-theoretic notion should be at orthogonal morphisms or something.
Yes, maybe. I won’t do it now, however.
I really like Cisinski’s notation for non-unique llp (resp. rlp) (l(X) (resp. r(X)) for a class of morphisms X). It’s simple, clean, clear, and doesn’t require left exponents.
In fact, I think that this notation should be extended to orthogonality as $l_\perp(X)$ (resp. $r_\perp(X)$).
Hm, but the notation $\perp$ is also used for non-unique LLPs.
It shouldn’t be. That notation for unique lifting is older, and also is the only one that matches the terminology ($\perp$ = orthogonal = unique lifting).
The notation I like for non-unique lifting properties is \boxslash
, which is very easy to remember because it is nothing other than a picture of a lifting in a square! Unfortunately iTex doesn’t support it yet. (I am not a fan of appropriating lowercase roman letters to have a specific global meaning – I want to be able to use letters like “l” and “r” as local variables.)
Do you mean ⧄? If so, you can get it in iTeX by using the entity syntax: ⧄
. Not sure what the context would be, but you could say $a ⧄ b$ if you wanted. (If it’s the other way, then that’s ⧅
: ⧅. I’m hoping that the unicode version of detexify will help a lot with finding symbols like this.)
If you think it would be a useful addition to iTeX, it’s really easy to extend the symbol set so just drop a line to Jacques. Make sure you tell him what sort of object it should be (standard symbol, operator, relation etc). If there’s similar characters, it’s nice to provide the “whole set” rather than just one or two.
Totally serious, how about $llp(X)$ (resp. $rlp(X)$)?
Oh, and by the way, Urs. The $\perp$ notation you’re thinking of is in Lurie, but for nonunique lifting, he uses it as a subscript.
Meanwhile, the reason why I hate the $\perp$ notation is that you have to write stupid things like ${}^\perp (M^\perp)$ or ${}_\perp (M_\perp)$ with parentheses, which look kinda silly.
Thanks, Andrew. I’m kinda busy right now but maybe I’ll mention it to Jacques.
llp(X) and rlp(X) wouldn’t be too bad. I don’t think $^\perp(M^\perp)$ looks silly though; I think it describes what’s going on pretty well.
Also, the nice thing about $\perp$ and ⧄ is that they let you use the same notation for the relation between two morphism and the operation on sets of morphisms. E.g. we have $f \perp g$ if and only if $g \in \{f\}^\perp$ if and only if $f \in {}^{\perp}\{g\}$.
Okay.
I think many of our entries use LLP and RLP. Such as small object argument.
(They do because I typed that, but after Mike made me not use $\perp$ ;-)
@Mike: I don’t know. I just really dislike how ${}^\perp (X^\perp)$ looks. It’s an ugly string of symbols.
If I had to rationalize why I don’t like it, I think it looks silly to have an operator acting on the left as a superscript/subscript with a parenthesized argument.
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