product operation: name and notation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:21:33Z http://mathoverflow.net/feeds/question/104494 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104494/product-operation-name-and-notation product operation: name and notation Wolfgang Jeltsch 2012-08-11T15:17:15Z 2012-08-11T18:53:48Z <p>For every product in a category, there exists an operation $\langle\cdot, \cdot\rangle$ that turns morphisms $f : C \to A$ and $g : C \to B$ into morphisms $\langle f, g\rangle : C \to A \times B$. How are this operation and its results usually called? Calling $\langle f, g\rangle$ the product of $f$ and $g$ does not make sense to me, since I would expect $f \times g$ to be the product of the two morphisms.</p> <p>Furthermore there is the generalisation of binary products in the form of arbitrary products $\prod_{i \in I} A_i$ for families <code>$\{A_i\}_{i \in I}$</code> of objects. These products have, of course, variants of the $\langle\cdot, \cdot\rangle$-operator. What is the usual notation for these variants? So far, I wrote <code>$\langle\{f_i\}_{i \in I}\rangle : C \to \prod_{i \in I} A_i$</code> for families of morphisms $f_i : C \to A_i$. Is this standard?</p> http://mathoverflow.net/questions/104494/product-operation-name-and-notation/104509#104509 Answer by Colin McLarty for product operation: name and notation Colin McLarty 2012-08-11T18:53:48Z 2012-08-11T18:53:48Z <p>Probably $\langle f, g\rangle : C \to A \times B$ is most often just called the arrow to the product. You are right it should not be called a product arrow. People who want a specific name for the operation have called it the "pairing arrow."</p> <p>I would write $\langle f_i\rangle_{i \in I} : C \to \prod_{i \in I} A_i$ without curly brackets in the angle brackets and i would call it the arrow to the product. A more specific name for the operation could be "tupling arrow."</p>