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Let $X_1$ and $X_2$ be two objects of a category with products and let $p_1: X_1 \times X_2 \to X_1$ and $p_2: X_1 \times X_2 \to X_2$ be the projections. Given two morphisms $f_1: X \to X_1$ and $f_2: X \to X_2$, the product of $f_1$ and $f_2$ is the unique morphism $f:X \to X_1 \times X_2$ such that $p_1 \circ f = p_2 \circ f$.

Question. In a category where this makes sense (like monoids, groups, etc.), is there a specific name for the restriction of $f$ to its image (which is a submonoid, subgroup, etc. of $X_1 \times X_2$)?

P.S. At the moment, I am using the term restricted product, but I wonder if there is a better term.

J.-E. Pin
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    Yes, I mean image, sorry. I will edit my question. – J.-E. Pin Mar 24 '20 at 18:42
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    You mean restricting the codomain of $f$? There are notions of factorisation systems, for example a (regular epi, mono) factorisation, which allows you to write $f$ as a regular epi ('surjective map onto its image') followed by a mono ('inclusion of its image'). – Mark Kamsma Mar 24 '20 at 19:10
  • Just to clarify terminology: the unique morphism into $X_1 \times X_2$ is usually called the pairing of $f_1$ and $f_2$ and denoted $(f_1, f_2)$. The product of $f_1$ and $f_2$ is the morphism $f_1 \times f_2 : X \times X \to X_1 \times X_2$ defined componentwise. – varkor Mar 24 '20 at 19:16
  • @varkor For the product of $f_1$ and $f_2$, I followed the terminology given in the wikipedia entry Product (category theory). – J.-E. Pin Mar 24 '20 at 19:54
  • @J.-E.Pin: ah, it seems this terminology is a little ambiguous, in that case: this issue is mentioned in this other math.stackexchange question. Something to be aware of. – varkor Mar 24 '20 at 19:59

1 Answers1

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No. This is a quite special construction to have a particular name, and there is none that would be widely recognized.

Kevin Carlson
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