Higher categories and semirings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:37:06Z http://mathoverflow.net/feeds/question/65594 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65594/higher-categories-and-semirings Higher categories and semirings Mikola 2011-05-20T21:19:59Z 2011-05-21T06:41:15Z <p>Maybe my thinking here is completely wrong headed, but this seems like something that ought to have been answered before. Here is my question:</p> <blockquote> <p>What is the (n - )categorical analogue of a semiring?</p> </blockquote> <p>As a starting point here is a common bit of folklore: categories act as a generalization of monoids, where the latter is simply a special case consisting of the collection of endomorphisms for a single object. Similarly, semirings can be thought of as the collection of all endomorphisms of a commutative monoid.</p> <p>Now there is an unmistakable resemblence between 2-categories and semirings. Specifically, in a 2-category, we get 2 operations for composing 2-morphisms, vertical and horizontal composition, and these operations distribute over each other via the interchange law; just like the addition and multiplication within a semiring. However, I am having difficulty making this correspondence sharp (if it is even possible). </p> <p>Here is a sketch of something that doesn't work. First, take all 2-endomorphisms of the identity morphism of any object in a 2-category to be the elements of the semiring. Then map vertical composition to addition and horizontal composition to multiplication. However, this creates a problem since there identity element for addition and multiplication correspond to the same 2-morphism, so the semiring must be trivial.</p> <p>Is there a way to salvage this idea or am I just not understanding something obvious?</p> http://mathoverflow.net/questions/65594/higher-categories-and-semirings/65620#65620 Answer by Neil Strickland for Higher categories and semirings Neil Strickland 2011-05-21T04:49:02Z 2011-05-21T04:49:02Z <p>A category with two symmetric monoidal structures, one distributing over the other, is called a symmetric bimonoidal category. These have been studied extensively in stable homotopy theory, because one can apply a K-theory construction to such a category to get an $E_\infty$ ring spectrum. There are complicated coherence conditions for the interaction between the distributivity isomorphisms $A\otimes (B\oplus C)\to(A\otimes B)\oplus(A\otimes C)$ with the rest of the structure; these are best encoded using some kind of operadic formalism, or more recently, using Lawvere-type theories in an $\infty$-category setting. This paper would be one entry point into the literature:</p> <pre><code> \bib{MR1361586}{article}{ author={Dunn, Gerald}, title={$K$-theory of braided tensor ring categories with higher commutativity}, journal={$K$-Theory}, volume={9}, date={1995}, number={6}, pages={591--605}, issn={0920-3036}, review={\MR{1361586 (97b:18003)}}, doi={10.1007/BF00998132}, } </code></pre> <p>Things are easier if you do not want $\otimes$ to be symmetric, but I think that the coherence conditions are still fairly complex.</p> http://mathoverflow.net/questions/65594/higher-categories-and-semirings/65625#65625 Answer by Tilman for Higher categories and semirings Tilman 2011-05-21T06:41:15Z 2011-05-21T06:41:15Z <p>I'm afraid your "unmistakable resemblence between 2-categories and semirings" is mistaken. Distributivity (in the usual sense) means $(a+b)\ast(c+d)=a\ast c+a\ast d+b\ast c+b\ast d$, but you're looking for two structures $+$ and $\ast$ and a comparison map between $(a+b)\ast (c+d)$ and $(a\ast c)+(b\ast d)$, is that correct? Such a thing does not really have a $1$-analog because by the standard argument, if two operations on a set are unital and behave like this, then they are equal to each other and commute. In category theory, however, this is captured in the notion of a "$2$-monoidal category". It is the universal setting in which to define a bimonoid (a.k.a. bialgebra). This is explained very clearly, and with references to earlier work, in Chapter 6 of</p> <p>Marcelo Aguiar and Swapneel Mahajan. <i>Monoidal functors, species and Hopf algebras</i>, volume 29 of <i>CRM Monograph Series</i>. American Mathematical Society, Providence, RI, 2010.</p></p> <p>On the other hand, if you're really interested in the $2$-analog of semirings, go for Neil's suggestions.</p>