Consider the situation described in the following diagram, namely:

  • $A$, $A'$, $B$, and $B'$ are sets.
  • $\alpha:A\rightarrow A'$ and $\beta:B\rightarrow B'$ are bijections.
  • $f:A\rightarrow B$ and $\ f':A'\rightarrow B'$.
  • The following equations are satisfied. $$ f = \beta^{-1} \circ f' \circ \alpha\\ f' = \beta\circ f \circ \alpha^{-1} $$

A commutative diagram

In a sense $f$ and $f'$ are the same function, in that each can be computed in terms of the other, with no information being lost or gained.

Is there an accepted terminology for this sameness of $f$ and $f'$?

  • 1
    $\begingroup$ factorisation of $f$, $f$ being a multiple of $f'$? Although that'd be more general, $f$ being a multiple of $f'$ if there exists some map $\alpha $ such that $f = f'\alpha$ or something along those lines $\endgroup$ – Alvin Lepik Nov 15 '18 at 16:58
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    $\begingroup$ Not sure if this is relevant, but it might help at least. In differential geometry, one considers $f:S_1 \to S_2$ where $S_1, S_2 \subset \mathbb{R}^3$ are regular surfaces. In order to work with such functions, one usually takes local parametrizations $X_1$ and $X_2$ of and $S_2$ respectively, and then considers the function $X_2^{-1} \circ f \circ X_1:U_1 \to U_2.$ However, it would be inconvenient to write $X_2^{-1} \circ f \circ X_1$ every time the function is being discussed, so it is usual to just write $f$ and say that this is the expression of $f$ in the coordinates $X_1$ and $X_2$. $\endgroup$ – MisterRiemann Nov 15 '18 at 17:28

You could say that $f$ and $f'$ are isomorphic, since the functions $\alpha$ and $\beta$ define an isomorphism from $f$ to $f'$ in the arrow category $\mathbf{Set}^{\to}$.

Here's some details:

Given a category $\mathcal{C}$, the arrow category $\mathcal{C}^{\to}$ has the morphisms of $\mathcal{C}$ as its objects and commutative squares in $\mathcal{C}$ as its morphisms.

That is, a morphism from $(f : A \to B)$ to $(f' : A' \to B')$ is a pair $(\alpha,\beta)$ consisting of a morphism $\alpha : A \to A'$ and a morphism $\beta : B \to B'$, such that $\beta \circ f = f' \circ \alpha$. An isomorphism in $\mathcal{C}^{\to}$ is simply a pair $(\alpha,\beta)$ of isomorphisms in $\mathcal{C}$.

When $\mathcal{C} = \mathbf{Set}$, this says that an isomorphism from a function $f : A \to B$ to a function $f' : A' \to B'$ in $\mathbf{Set}^{\to}$ is a pair of bijections $\alpha : A \to A'$ and $\beta : B \to B'$ such that $\beta \circ f = f' \circ \alpha$—this is exactly your situation.

  • $\begingroup$ Thanks. I've accepted your answer, because it's correct, but I wish there was another name for this sameness, because when the sets possess an algebraic structure, e.g. if they are vector spaces, the term isomorphism is commonly understood as "structure preserving bijection", but what if I wish to describe isomorphic functions (in the sense described in your answer) that are not structure preserving? $\endgroup$ – Evan Aad Nov 15 '18 at 17:40
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    $\begingroup$ A more satisfactory answer to your question might be that there simply isn't a short, snappy word or phrase that you can use. If you want to talk about this concept, you're better off defining your terms as the need arises in order to avoid confusion. The word 'isomorphic' is fine so long as you can expect whoever your audience is can be expected to undestand that you mean 'isomorphic in the arrow category'—if not, you could first make a definition to say what you mean by 'isomorphic' and proceed from there. $\endgroup$ – Clive Newstead Nov 15 '18 at 17:47
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    $\begingroup$ You want a term for this "sameness," but that's what "isomorphic" means, so it is a perfectly suitable term. $\endgroup$ – Randall Nov 15 '18 at 18:22
  • $\begingroup$ I suppose you could use "equivalent" if you don't like "isomorphic", since this mirrors the terminology for matrices... but still, you'd need to define your terms, since it's not standard terminology. $\endgroup$ – Clive Newstead Nov 15 '18 at 20:12

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