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I am wondering if anyone knows any more on the history of the term 'co-domain' as it relates to functions.

Two sources I found:

Russell and Whitehead, Principia Mathematica, 1915, page 34 :

the class of all terms to which something or other has the relation $R$ is called the converse domain of $R$; it is the same as the domain of the converse of $R$.

Cassius Keyser, Mathematical Philosophy, 1922, page 168:

A relation $R$ has what is called a domain, - the class of all the terms such that each of them has the relation to something or other, - and also a codomain - the class of all the terms such that, given any one of them, something has the relation to it.

It seems to me that when Keyser talks about a 'codomain', he is talking about the same thing as Russell and Whitehead's 'converse domain'. So, it looks like we went from 'converse domain' to 'codomain' .... to 'co-domain'? That would seem to make sense.

Also, both texts talk about relations, not functions. But, a function is of course a special kind of relation. So ... it still makes sense.

However! (and this is really why I am asking this question): the way these two texts talk about the 'converse domain' and 'codomain' is (when applied to functions) what we nowadays call the 'range' or 'image' of the function, and not what we nowadays call its 'co-domain'.

Concrete example:

Take a function $f$ whose domain is defined as $\mathbb{R} - \{ 0 \}$, whose co-domain is defined as $\mathbb{R}$, and whose mapping is defined as $f(x) =1/x$.

For this function, the range or image is $\mathbb{R} - \{ 0 \}$, and that is what (again, if we see this function as a relation) Russell & Whitehead would consider its 'converse domain' what Keyser would call its 'codomain'.

But the 'co-domain' of this function was defined as $\mathbb{R} - \{ 0 \}$

So I think there has been a shift in the use of the term ... That is, it seems like we got:

'converse domain' -> 'codomain' -> 'range'

... while 'co-domain' is something different!

This is weird! What happened? Does anyone have some insight into any of this?

Bram28
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    The domain and codomain are both part of what it means to define a function: using domain $\mathbf R-{0}$, the codomain could be $\mathbf R - {0}$ or $\mathbf R$ or $\mathbf C$ or ... With codomain $\mathbf R - {0}$ the function is invertible, but not in the other two cases. Therefore I disagree that in modern times we always would say the domain is $\mathbf R$. It depends on what you are doing. – KCd Aug 17 '20 at 20:20
  • Related: https://hsm.stackexchange.com/questions/4929/history-of-the-definition-of-injective-surjective-function – Spencer Aug 17 '20 at 22:28
  • @KcD You are quite right: Just by saying that $f(x)=1/x$ I can;t say what the co-domain is ... or even the domain. My mistake, and I'll fix my Post accordingly! So let's say that this function $f$ has as its domain $\mathbb{R} - {0 }$, and as its co-domain $\mathbb{R}$. I can do that, right? But this is weird. Because we now have a situation where the function is not invertible, i.e. there is no converse. And yet, there is a 'converse domain' (R&W), as well as a codomain (Keyser), which is $\mathbb{R} - { 0 }$ ... and yet this is not the 'co-domain'! – Bram28 Aug 18 '20 at 01:59
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    You can certainly take the domain of $1/x$ to be $\mathbf R - {0}$ and the co-domain to be $\mathbf R$, just as you can take the domain of $1/x$ to be $(0,1)$ and the co-domain to be $(0,5)$. There is nothing weird about this. It all depends on what it is you want to do with the functions. For example, it's natural to consider all rational functions with real coefficients to be "real-valued functions defined where they maximally make sense", so $1/(x^2-x)$ has domain $\mathbf R - {0,1}$ and co-domain $\mathbf R$. It is not every function's aspiration in life to be invertible. – KCd Aug 18 '20 at 03:08
  • @KcD Right ...but my question is not about the current use of domain and co-domain. If I had a question about that I would simply go to the Mathhematics Stack exchange. My question is a historical question about how the use of the terms evolved: clearly what Russell and Whitehead and Keyser described as 'converse domain' and 'codomain' doesn't match up with the modern usage of 'co-domain'. That to me is the weird part, and I am hoping that someone could tell me how or why that shift happened historically. – Bram28 Aug 18 '20 at 03:13
  • This is a set theory thing, for situations where a relation is defined as a set of ordered pairs rather than algebraically. In that situation, all functions are in effect "surjective" so the difference doesn't matter. "Converse domain" is the domain of the relation's converse, equivalent to the image of an algebraically-defined function. – Spencer Aug 18 '20 at 21:37
  • Also related: https://math.stackexchange.com/questions/719227/exactly-who-popularized-the-modern-definition-of-domain-and-codomain-of-function – Spencer Aug 18 '20 at 21:45
  • @Spencer But even for relations, the 'range' and 'co-domain' can be different ... and yet what R&W and Keyser call the 'converse domain' and 'codomain' respectively is what we nowawadays call the 'range', and not the 'co-domain'. No? – Bram28 Aug 18 '20 at 22:53
  • "converse domain" was used by Bernays in 1958 to mean essentially the image...it's more primitive than "codomain". – Spencer Aug 18 '20 at 23:09
  • @spencer Thanks! .... so it looks like Bernays uses 'converse domain' the same way Russell and Whitehead do? Also, what do you mean when you say that it is more 'primitive' than 'codomain'? – Bram28 Aug 19 '20 at 00:08
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    (Modern) codomain is an unnatural notion for relations in set theory, only image is intrinsically definable. But it changes in category theory, where the target object of a morphism is not its image unless it is an epimorphism. It is interesting that Eilenberg-MacLane's seminal paper (1945) uses "range" for modern codomain, while MacLane's 1971 book already calls it "codomain". The ambiguous use of "range" continues to this day. – Conifold Aug 19 '20 at 05:22
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    Lawvere's Elementary Theory of the Category of Sets (1964) also uses "codomain" in the modern sense. – Conifold Aug 19 '20 at 05:33
  • @Conifold Thanks! So both 'range' and 'codomain' have been ambiguous? – Bram28 Aug 19 '20 at 12:26
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    My guess is that when category theory people started transferring terminology they took the closest term to the target object, which was "codomain" or "range" at the time, and used it. Since the target does not always match the range from set theory pedantic authors started distinguishing codomain and range, but not everybody followed. – Conifold Aug 20 '20 at 09:31
  • (@Conifold, I have to say... it was a bit jarring to hear "MacLane's 1971 book already..." as though this were "history"... since I acquired a copy of that book when it was brand new. :) – paul garrett Jan 16 '22 at 04:46
  • Shouldn't the first reference be: Whitehead & Russell, Principia Mathematica, Vol. 1 (1910)? – Calum Gilhooley Jan 18 '22 at 17:57

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It's an early recognition of duality in set theory. Domain vs Codomain suggests a relationship that is missing from domain and range.

This is hidden in set theory as functions are biased in that they are not symmetrically defined. Nor is it easy to conceptualise one to many functions naturally, and dually to many to one functions, which they do naturally.

This is fixed in category theory where duality is made explicit, rather than in the secret, furtive way it's done in set theory. Moreover, category theory is the correct conceptualisation of covariance as in the notion of general covariance that Einstein used heuristically in his investigations into the general character of physical law.

Interestingly, one of the major discoveries of string theory is the role that dualities play in physics. (In ordinary physics, we see duality manifest itself in the duality between the electric and magnetic fields). It wouldn't surprise me if at bottom this had the same root as dualities in category theory.

Mozibur Ullah
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