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touched the formatting in congruence, fixed a typo on the cartesian square, added a basic example
A kind soul pointed out to me by email that there was a typo in the third clause of the Definition (an exchange $q_1 \leftrightarrow q_2$ either in the diagram or in the equation following it). I have fixed it.
In this vein, I have added a brief remark relating to internal categories, which could be much expanded on.
I also fixed the formatting (lack of whitespace may get the bullet lists mixed up).
Quick Q from a beginner: What is $\times_X$? I’m unfamiliar with this notation and can’t seem to find any ref to it in my (admittedly introductory) books on Category Theory. I assume it somehow embodies the notion of head-to-tail needed for transitivity (ex. for SET, within a given object $X$ we would want to confine ourselves to pairs $a\sim b$ and $b\sim c$ with the same $b$), but would like to see its precise definition in category theory.
Cheers, Ken
This is meant to be the fiber product
of $R \hookrightarrow X \times X \overset{pr_2}{\to} X$
with $R \hookrightarrow X \times X \overset{pr_1}{\to}$,
i.e. the subobject of pairs of composable pairs in relation.
I have added this explanation to the entry here.
But it may (mor may not) be easier to first understand the notion of an internal groupoid and for that first the notion of an internal category (which is maybe currently the only page on these matters with enough detail to be intelligible). What the page on congruences here tries to very tersely describe is a special case of such internal categories.
How does the material on this page relate to congruences in Euclidean geometry, which are usually the first notion of congruence encountered in mathematics?
Yeah, I don’t want to defend the existence of this entry under this title. It better ought to be titled “internal equivalence relation”.
But at least I have now started an Idea-section (here) with the following two sentences:
In Euclidean geometry, by congruence one means the equivalence relation on the collection of subsets of a Euclidean space which regards two of these as equivalent if one is carried into the other by an isometry (of the ambient Euclidean space).
For better or worse, the following entry is only tangentially related to this standard meaning of congruence. What is discussed in the following is the general issue of formulating the notions of an equivalence on (the generalized elements of) any object in any finitely complete category.
@Urs Thanks for clarifying the $\times_X$ notation. I was trying to understand the categorical notion which captures normal subgroup -> quotient group in GRP, ideal -> quotient ring in RING, and partition -> quotient set in SET (as well as hopefully gaining insight into what the analogues would be for other categories). I assume that in those cases, the relevant subobject of $X$ would somehow correspond to a normal subgroup of $X$ in GRP, an ideal of $X$ in RING , and the set $X$ itself in SET?
FYI, the only def of “congruence” I could find elsewhere (ex. Barr’s book and a couple of others) was that of a partition on the set of arrows (for a locally small category). I imagine this is just your def confined to CAT and where the subobject of $X$ is its Arrow category?
I’ll take a look at the internal groupoid and internal category pages and see if those shed some light!
Just for reference, I’m a different Guest than asked the Euclidean Q.
Cheers, Ken
Yes, for $G \,\in\, Groups$ and $N \subset G$ a normal subgroup, take $R \hookrightarrow G \times G$ to be the subgroup of the direct product on pairs of elements which differ by right multiplication with an element in $N$.
Regarding “Barr’s book”: I am not going to guess and hunt references; in fact I am going to call it quits for tonight (I am on GST…). But if tomorrow morning I find in this thread a hyperlink to the precise page in the exact book that you want me to inspect, I promise to click on it and take a look. :-)
Oh, I was just trying to wrap my head around it — no need to trouble yourself unless you’re so inclined. In case you’re curious about the Barr’s ref, it looks like it’s online at https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf p.89 (definition 3.5.1).
Your definition strikes me as an internal definition as opposed to the Barr’s one. On the other hand, Barr’s one seems to map to yours in the case of CAT with the subobject of $X$ being its Arrow category (I think).
Cheers, Ken
what is “$i$” in this remark? It doesn’t seem to be defined anywhere.
Since $i$ is a monomorphism, the maps $r$, $s$, and $t$ are necessarily unique if they exist.
Is it suppose to be referring ro this morphism?
a subobject $R\stackrel{(p_1,p_2)}\hookrightarrow X \times X$ of the Cartesian products of $X$ with itself
re #11:
Thanks for the url.
Yes, the notion of congruence in Barr & Wells 1995, Def. 3.5.1 on p. 89 is not really related:
[Incidentally, click the button labeled “Source” on the top right of this comment to check out how to code these hyperlinks here — it’s easy.]
They also use “congruence” to mean “some equivalence relation respecting some ambient algebraic structure”. But I don’t see that there would be a choice of $C$ and $X$ in the entry under discussion that would reduce its definition strictly to theirs. (But I haven’t thought about it all too much.)
Please notice that the entry we are speaking about is by no means my entry, if that’s what you meant by saying “your entry.” I didn’t write it (you can see the original authors in the page history) and I don’t approve of how it is written. For one, it is lacking references, as we see now.
(Note: I’m the Guest from above. I signed up for an account and now appear under my new username)
re #14: Ah, I see. I had assumed the Barr def was a common special case of the one on nlab, but with CAT, $X$ a category, and $R$ just $Arr(X)$. I’ll have to think about whether that even makes sense. Thanks for the technical tips about using the site!
As for “your”, I meant it in the general sense of nlab rather than any particular author. I figured nlab was some type of wiki so there could be lots of people editing various parts (whether a small group of practitioners or the public at large).
Is there an easy way to tell whether the nlab def appearing on this page is well-established or is just somebody’s well-intentioned but homegrown terminology? The only reason I care is that I’ve been trying to understand the category theory generalization of the type of twisting which occurs in a group-quotient (i.e. the semi-direct product or group-extension twisting in a short-exact-sequence) and how it manifests in other quotient types of constructions (ring/ideal, set/equiv-relation, etc).
Your (in the @Urs sense) suggestions about internal categories and internal groupoids were very helpful, btw. I basically want to figure out what the proper notion of “internal equivalence class” or “internal quotient” is commonly called so I can dig for a more thorough treatment of the subject (ex. in a textbook) and see what it looks like in categories other than the three I mentioned.
Cheers, Ken
here are some classical references which use “congruence” more or less as in the entry:
Jiri Adamek, Horst Herrlich, George Strecker, p. 195 of: Abstract and Concrete Categories, John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories 17 (2006) 1-507 [tac:tr17, book webpage]
Francis Borceux, Exp. 2.10.3.b in: Handbook of Categorical Algebra Vol. 1 Basic Category Theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [doi:10.1017/CBO9780511525858]
Francis Borceux, Exps. 2.5.6 in: Handbook of Categorical Algebra Vol. 2 Categories and Structures, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]
Thanks so much! Those refs are incredibly helpful. I really appreciate it.
Cheers, Ken
Glad it helped.
By the way, I found these by going through the list of textbooks here and doing a text search for “congruence” in each (or in many of them, at least).
Incidentally, doing so, I noticed that:
quite a few authors use the term “congruence” in that other sense,
no author bothers to cite any references, for either notion,
some speak as if the internal notion needs no introduction (including Borceux, to my disappointment),
none point out that there is a potential terminology clash — within the same field, no less.
If I am not mistaken, the Elephant even uses both definitions, without comment nor warning (nor reference).
added pointer to:
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