Stylistic question. In category theory, if you have to write about sequences of the form $\dotsm\xrightarrow[]{f_2}o_2\xrightarrow[]{f_1}o_1\xrightarrow[]{f_0}o_0$, and you have to tell readers that there may be some equalities $o_i=o_{j}$ between objects, $i<j$, but that this is inessential and all that matters is that none of the $f_i$ is an identity arrow, what would you write?

EDIT: inexperienced readers, beware: there seems to be something like a consensus that there is no problem here, not even an expository problem, and you will not find much mathematics in all of this, and you might like to skip it. Thanks to the commenters.

  • EDIT: I propose the following solution: One could briefly write

"Note that this is a pasting scheme, and that $o_i=o_j$ is not an equality between objects, but rather an equality between 0-cells of a pasting scheme, and, as such, does not violate the principle of equivalence."

This, however, is not short, and it might confuse an audience who does not know about pasting schemes.

More briefly, one could say something informal about syntax and semantics, saying that $o_i=o_j$ is pure syntax.

It seems to me that only categorical properties formulated by way of mentioning equalitites between objects violate the principle of equivalence.

Do you agree with that?


  • In formulating the question, I have already given an example of what I would write.
  • The reason I am asking is that mentioning equalities between objects is held to be something to be avoided, with good reason. A related keyword is principle of equivalence.
  • Relatedly: it is well known that the very definition of categories (as it is presented nowadays) involves an equation between objects, namely when one stipulates that only for morphisms $f$ and $g$ with $\mathrm{dom}(f)=\mathrm{cod}(g)$ the composite $f\circ g$ is defined; however, this is not really an equation of objects, rather an equality between things of sort "codomain-sort of the cod- and dom- functions", in the sense of categorical logic. (I seem to mean it as I say it: both cod and dom are function-symbols, as such, they have types, which are lists of sorts, the last component of which is what I call "codomain-sort of the cod-and dom-functions".) So one way to defend the usual definition of categories against charges of violating the principle of equivalence is to say "I did not equate objects, I just equated things of "codomain-sort of the cod- and dom-functions", and stipulating that equating such is permissible.

  • I emphasize that part of the expostion-problem stated in the question is that I insist that readers have to be told, briefly, in a low-fuss fashion, that some, and even many, equalities between objects in the chain of morphisms are permitted, so some way of saying it has to be found. I see no better way than what I wrote in the question. To me this seems a reason for violating the principle of equicalence. Of course I am aware that one way of getting around explictly mentioning an equality between objects would be to disguise the objects, dressing them up in the dom- and cod- functions. But there is one precise reason why to me this seems not a solution: one then has to make the arbitrary choice whether to refer to $o_i$ by way of the proxy $\mathrm{cod}(f_i)$, or the proxy $\mathrm{dom}(f_{i-1})$. (Incidentally, I would appreciate a better technical term than "proxy" here, which is belong to another science, and using "representative" instead here seems not good, since this, in turn, usually means something else.) One might be tempted to let oneself be guided by the indexation in making this choice, but this is contingent notational criterion. I see no natural way to avoid explicitly speaking of equality of objects when telling the audience that some of the $o_i$ may be equal.

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    $\begingroup$ I cannot believe you wrote all that...why not just say «there may be some equalities oi=oj between objects, i<j, but that this is inessential and all that matters is that none of the fi is an identity arrow»?! $\endgroup$ – Mariano Suárez-Álvarez Aug 6 '17 at 8:36
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    $\begingroup$ Add the clause “where none of the $f_i$ is an identity morphism”; if your audience is not experienced in the subject, tell them (by voice) that it is not implied the objects are (pairwise) distinct. If you feel your audience is very inexperienced, add a couple of examples, one of which should be a counterexample. $\endgroup$ – egreg Aug 6 '17 at 8:52
  • $\begingroup$ @MarianoSuárez-Álvarez: thanks for your question and suggestion. Well, I wrote all that since (0) I am unsure how to get it across, (1) there is always something one does not know and I find expositional discussions important (within reason), and, most importantly for me, (2) the terminological apparatus I know from (di-)graph theory sort-of-fails me here: in the exposition problem I have, it is essential that there may even be some repetitions among the $f_i$, so I cannot just say "This is a trail in the standard graph-theoretic sense, since this would rule out any repetions of the arrows." $\endgroup$ – Peter Heinig Aug 6 '17 at 9:24
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    $\begingroup$ If you have that pesky condition, simply state it. What possible drawback can there be in saying what you mean?! You are overcomplicating this. $\endgroup$ – Mariano Suárez-Álvarez Aug 6 '17 at 10:26
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    $\begingroup$ As an aside, it's kind of weird you are emphasizing the principle of equivalence, but the very thing you are trying to discuss violates it -- you forbid identity arrows but allow isomorphism arrows. $\endgroup$ – user14972 Aug 7 '17 at 2:20

The constraints of the principle of equivalence don't apply to notation. That is, it is a constraint on what formulas we can write within the "object language" of category theory, not on meta-logical statements about formulas in that object language. This is especially the case as you are not actually adding any constraints! You are just pointing out that distinct variables in the generalized algebraic theory of categories are in no way required to be instantiated to different objects.

Personally, for a mathematical audience I probably wouldn't say anything in most cases. Mathematicians are used to not assuming that different variables imply different values. If I did want to emphasize that they could refer to the same objects, I'd probably say something like "... the $o_i$ not necessarily distinct".

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    $\begingroup$ many thanks for that. So this is an advice in the direction of keeping quiet about pasting schemes (at least if these are not central to the exposition), and rather talk about the old hat "syntax" versus "semantics", right? Your suggestion "formulas [...] withing the object language" means "statements about objects of models of the syntax of category theory", right? I am a bit confused about this usage "formulas in the object language of category theory". Also, by "meta-logical statements about formulas in that object language" you mean "formulas over the signature of category theory"? $\endgroup$ – Peter Heinig Aug 6 '17 at 9:33
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    $\begingroup$ I'm not suggesting talking about anything in the first paragraph at all. It's meant as an explanation why no one is going to jump on you due to the principle of equivalence (though no one was going to do that anyway since evil definitions are pretty common). My advice is basically the same as egreg's. You need to say something equivalent to "none of the $f_i$ are identities". (I'd probably say "a sequence of non-identity arrows..." but that's a personal choice.) The most powerful way of conveying that the $o_i$ need not be distinct is via an example where they aren't. $\endgroup$ – Derek Elkins left SE Aug 6 '17 at 20:17

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