Terminology question relating to magmoidal semicategories - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:55:21Z http://mathoverflow.net/feeds/question/109565 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109565/terminology-question-relating-to-magmoidal-semicategories Terminology question relating to magmoidal semicategories Salvo Tringali 2012-10-13T23:12:50Z 2012-10-13T23:17:51Z <p>Suppose that $\mathfrak C = (\mathbf C, \otimes)$ is a magmoidal <a href="http://ncatlab.org/nlab/show/semicategory" rel="nofollow">semicategory</a> and let <code>$\mathcal P := \{{\rm P_n}\}_{n=1}^\infty$</code> be a set of parenthesizations such that ${\rm P}_n$ has length $n$ for each $n$. It is then possible to recursively define <em>long tensor products parenthesized by $\mathcal P$</em>, simply mimicking what we would do in a magma. Thus, one can safely handle stuff like $\blacksquare_1 \otimes \blacksquare_2 \otimes \cdots \otimes \blacksquare_n$, the black squares being either objects or arrows from the lovely $\mathfrak C$, by implicitly referring to the <em>parenthesized tensor product</em> ${\rm P}_n(\blacksquare_1, \blacksquare_2, \ldots, \blacksquare_n)$. If $\mathfrak C$ is [weakly] associative, as in the case of monoidal (or simply semigroupal) categories, then we are freely given the existence of natural isomorphisms making all <em>parenthesized tensor products</em> look like each other. Thus, my question is:</p> <blockquote> <blockquote> <p><strong>Question.</strong> Is there any (possibly standard) naming for a <em>long parenthesized (tensor) product</em>? If not, how would you call it, were you in my place?</p> </blockquote> </blockquote> <p>Thank you in advance for any suggestion. If relevant, my motivations are linked to <a href="http://mathoverflow.net/questions/106898/references-for-semicategories" rel="nofollow">this</a>.</p>