Algebra and Topology Seminar 

The symmetric topos $boldmathM(ℰ)$ arose as the classifier of the Lawvere distributions on a topos $ℰ$ and resulted in a unification of ideas in various fields in mathematics.

1. The algebraic construction of the symmetric algebra is done without using tensor products and provides an alternative to the usual construction in Algebra.
2. Mediating support there is a connection between the symmetric topos of a topos of sheaves on a locale $X$ and the topos of sheaves on the lower power locale of $X$, of interest in Theoretical Computer Science.
3. The association of the symmetric topos is part of a Kock-Z

Seja $G$ um grafo com conjunto de vértices $V=\{x_1,\dots ,x_n\}$ e $R=k[x_1,\dots ,x_n]$ um anel de polinómios sobre um corpo $k$. O ideal associado ao grafo $G$, $I(G)$, é o ideal de $R$ gerado pelos monómios $x_i{x}_j$ sempre que $x_i$ e $x_j$ são adjacentes em $G$. Dizemos que $G$ é um grafo Cohen-Macaulay (CM), se $R/I(G)$ for um anel Cohen-Macaulay. Muitas propriedades algébricas de $I(G)$ podem ser obtidas no grafo $G$, através do uso de coberturas de vértices para $G$. Serão apresentadas algumas classes de grafos Cohen-Macaulay, alguns processos para construir grafos CM e propriedades dos grafos bipartidos CM.

One of the principal reasons for the introduction of the concept of quantale was to find a categorical context, generalising to the non-commutative case that of locales, within which the insights of Giles and Kummer into the spectral representation of $C*$-algebras might be placed. Consideration of their ideas led to the concept of the spectrum $\mathrm{Max}A$ of any $C*$-algebra $A$, functorial on the category of $C*$-algebras and admitting a Gelfand-Naimark representation exactly generalising that in the commutative case.

The question remained of whether this spectrum $\mathrm{Max}A$ could be considered in some sense to be a quantal space. Approaching from the classical belief that its points should be the equivalence classes of irreducible representations, one may obtain a concept of a point of an involutive quantale which satisfies this criterion, yet extends that of a point of a locale. Generalising again the situation for locales, one may characterise those involutive quantales that are spatial as those having enough points, arriving finally at the concept of a quantal space.