Talk:PlanetPhysics/Algebra Formed From a Category
Original TeX Content from PlanetPhysics Archive[edit source]
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: algebra formed from a category %%% Primary Category Code: 02.10.-v %%% Filename: AlgebraFormedFromACategory.tex %%% Version: 1 %%% Owner: rspuzio %%% Author(s): rspuzio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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Given a \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{C}$ and a ring $R$, one can construct an
algebra $\mathcal{A}$ as follows. Let $\mathcal{A}$ be the set of all formal finite linear combinations of the form \[\sum_i c_i e_{a_i, b_i, \mu_i},\] where the coefficients $c_i$ lie in $R$ and, to every pair of \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $a$ and $b$ of $\mathcal{C}$ and every \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\mu$ from $a$ to $b$, there corresponds a basis element $e_{a,b,\mu}$. Addition and \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} multiplication are defined in the usual way. Multiplication of elements of $\mathcal{A}$ may be defined by specifying how to multiply basis elements. If $b \not= c$, then set $e_{a, b, \phi} \cdot e_{c, d, \psi} = 0$; otherwise set $e_{a, b, \phi} \cdot e_{b, c, \psi} = e_{a, c, \psi \circ \phi}$. Because of the associativity of \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} of morphisms, $\mathcal{A}$ will be an associative algebra over $R$.
Two instances of this construction are worth noting. If $G$ is a \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, we may regard $G$ as a category with one object. Then this construction gives us the group algebra of $G$. If $P$ is a partially ordered set, we may view $P$ as a category with at most one morphism between any two objects. Then this construction provides us with the incidence algebra of $P$.
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