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• CommentRowNumber1.
• CommentAuthorHarry Gindi
• CommentTimeJun 24th 2010

Over at orthogonal subcategory problem, it’s not clear to me whether or not the “objects orthogonal to $\Sigma$” should be morphisms orthogonal to $\Sigma$, or if it should mean objects of $X$ of $C$ such that $X\to *$ is orthogonal to $\Sigma$ (where $*$ denotes the terminal object). (Hell, it could even mean objects that are the source of a map orthogonal to $\Sigma$). I was in the process of changing stuff to fit the first interpretation, but I rolled it back and decided to ask here.

If it should in fact be the second (or third) definition, I would definitely suggest changing the notation $\Sigma^\perp$, which is extremely misleading, since that is the standard notation for the first notion.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJun 24th 2010

It means objects $X$ such that $X \to 1$ is orthogonal to $\Sigma$. Sorry if that was confusing to you, but I was under the impression that this is fairly standard, even if it hasn’t been standardized within the nLab. (It’s also deducible from the proof given on that page.) For example, one may define a $j$-sheaf w.r.t. a topology $j$ to be a presheaf which is orthogonal to the class of $j$-dense inclusions of presheaves.

For those who know what I meant by “object orthogonal to $\Sigma$”, I don’t think the usage $\Sigma^\perp$ would be “extremely misleading”, particularly since the intended usage is spelled out right on that page (and that’s true even if the notation $\Sigma^\perp$ is elsewhere used to denote a class of morphisms). I think also that the notation is standardly used for both notions (not at the same time of course!). Am I wrong about that?

I think one can simply make a note of “object orthogonal to $\Sigma$” at the article orthogonality, and note that this usually means $f \perp (X \to 1)$ for all morphisms $f$ in $\Sigma$ (as opposed to $(0 \to X) \perp f$).

• CommentRowNumber3.
• CommentAuthorHarry Gindi
• CommentTimeJun 24th 2010
• (edited Jun 24th 2010)

Sorry! I was going to actually edit the page orthogonality based on the answer to this question! =)

I also noticed that the definition is essentially given in the proof after I posted this topic.

As far as I’ve seen, in the 4-5 books I’ve seen the notion of orthogonality mentioned, $\Sigma^\perp$ always means “the class of morphisms orthogonal to $\Sigma$”. I’ll take your word for it, since you know a lot more about the subject than I do.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJun 24th 2010

Yeah, unfortunately the same notation is used “standardly” to mean two different things in different contexts.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 24th 2010

I edited orthogonal subcategory problem:

added subsections and hyperlinks. Also linked to it from orthogonality, where it might have been a subsection otherwise.