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Let $C$ be a category with all small products, and $f_i: X_i \to Y_i$ be morphisms in $C$ for each $i \in I$.

There are two concepts of a product of these morphisms:

  • In the case that $X_i = X$ for each $i \in I$, then the product morphism is the unique morphism $\prod_C f : X \to \prod_C Y$ which satisfies $$f_i = [\textstyle\prod_C Y]_i \circ \prod_C f,$$ where $[\prod_C Y]_i: \prod_C Y \to Y_i$ is the projection to the i:th factor. These come from the definition of categorical product.

  • The parallel-product morphism (*) is $\overline{\prod}_C f : \prod_C X \to \prod_C Y$ which satisfies $$f_i \circ [\textstyle\prod_C X]_i = [\textstyle\prod_C Y]_i \circ \overline{\prod}_C f.$$

Problem

What standard notations and names are used for these products of functions?

Notes

  • This answer shows pages of Engelking's General Topology book. In Theorem 3.7.9 on page 184 he uses the notation $\prod_{s \in S} f_s$ for the parallel-product of $f_s: X_s \to Y_s$.
  • In contrast, above I use the notation $\prod_C f$ for the product morphism, because it appears together with the product object $\prod_C Y$ in the definition of the categorical product.
  • The notations for the two products should be different to avoid ambiguity. For example, if $X_i = X$ for each $i \in I$, then both are applicable, but $\prod^J_C f \neq \overline{\prod}^J_C f$.

On naming

Engelking defines his notation for the parallel-product on page 79 as the "Cartesian product of mappings". In this book the category is topological spaces. But "Cartesian" does not seem like a good name on a general category.

(*) I made up both the name and the symbol.

kaba
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1 Answers1

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The "parallel-product morphism" is the actual categorical product of the morphisms $f_i,\,i\in I$, so the notation $\prod_{i\in I}f_i$ should be reserved for that ($\prod_{i\in I}$ is a functor $\prod_{i\in I}C\rightarrow C$ if all the relevant products exist). The adjective "cartesian" can also be used for products in an arbitrary category (especially in contexts where you wanna distinguish it from a different product like a tensor product, but that's another story). I do not know of a particular naming convention for what you call the "product morphism". It is simply the morphism induced by the $f_i,\,i\in I$ via the universal property. Notationally, the component-wise notation $(f_i)_{i\in I}$ would be the most common one.

Thorgott
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  • Thanks. Also practically I noticed that I have lots of uses for the parallel-product, but not for the explicit use of the product morphism. So what I'll do is use $\prod f$ for the parallel-product, and have some other notation for the product morphism. – kaba May 20 '25 at 20:31