# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorJohn Baez
• CommentTimeJul 24th 2013
• (edited Jul 24th 2013)

… the limit

$L c := \lim_{c\to R d} d$

over the comma category $c/R$ (whose objects are pairs $(d,f:c\to R d)$ and whose morphisms are arrows $d\to d'$ in $D$ making the obvious triangle commute in $C$) of the projection functor

$L c = \lim_{\leftarrow} (c/R \to D ) \,.$

I don’t really understand this (and while I could figure it out, it’s probably not good to make readers do so). At first it sounds like someone is saying “the limit $L c$ over the comma category of the projection functor $L c$”, which would be circular. But it must be that both formulas are intended as synonymous definitions of $L c$. At that point one is left wondering why one has a backwards arrow under it and the other does not. I guess old-fashioned people prefer writing limits with backwards arrows under them, so someone is trying to cater to all tastes? I think it’s better in this website to use $lim$ and $colim$ for limit and colimit.

I could probably guess how to fix this, but I won’t since I might screw something up.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 24th 2013

I went ahead and made some changes per your comment. See if that looks better. (I think I’d try a different explanation if I were writing this – or writing this today in case I was the one who wrote that then! – but never mind.)

• CommentRowNumber3.
• CommentAuthorJohn Baez
• CommentTimeAug 2nd 2013

Thanks.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeApr 3rd 2018
• (edited Apr 3rd 2018)

Clarified some language in the statements that characterize adjoints between locally presentable categories, in response to a comment made by user Hurkyl in another thread (here).

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeApr 3rd 2018

Clarified the language in another relevant spot (where a counterexample was given).

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJun 27th 2019

Change notation in the statement of the theorem to match its proof (the functor is $R:C\to D$ instead of $G:D\to C$).

• CommentRowNumber7.
• CommentAuthorDmitri Pavlov
• CommentTimeNov 16th 2019

1. Explained the non-standard notation for the limit.

Bartosz Milewski

2. Further explanation of syntax

3. Changed notation for presheaves

Bartosz Milewski

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 26th 2020

The above edits concern the section Examples – In presheaf categories.

Bartosz wanted to make notationally explicit the Yoneda embedding in the various formulas shown there. I have now touched the section myself, added the remaining instances of the Yoneda embedding; and also made some further cosmetic changes to the typesetting, such as height-aligned parenthesis etc.

4. When reading the presheaf example, I was curious if one could make an argument that representables form a solution set, and this justifies the restriction of the coend to representables only.
• CommentRowNumber13.
• CommentAuthorThomas Holder
• CommentTimeJul 1st 2020
• (edited Jul 1st 2020)

• Duško Pavlović, On completeness and cocompleteness in and around small categories , APAL 74 (1995) pp.121-152.
• CommentRowNumber14.
• CommentAuthorTodd_Trimble
• CommentTimeMar 13th 2021

Strengthened the first of the two statements for adjoint functors in the locally presentable case.