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# Macquarie University  Department of Mathematics

## Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

### Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

#### July 18-21 2005, Australian National University, Canberra

Tue, 19 July:  17:00 - 17:40

##### On Lawvere-completeness for lax algebras
###### Clementino, Maria Manuel (Universidade de Coimbra)

The notion of reflexive and transitive $(T,V)$-algebra --- or $(T,V)$-category ---, for a symmetric monoidal-closed category $V$ and a lax monad $T$ in the category of $V$-matrices, has been recently introduced and studied \bibref{2,3}. It comprises $V$-categories, when $T$ is the identity monad, and Barr's relational algebras \bibref{1}, when $V=2$. Although it was this latter instance that raised the subject --- having in mind that several interesting results of convergence structures could be generalized to other settings --- the former one gives a new insight to these structures. Indeed, a considerable amount of knowledge of $V$-categories can be interpreted in the general setting of $(T,V)$-categories.

In this talk we will concentrate on Lawvere's notion of (Cauchy-)complete $V$-category. We will deal with a (possible) generalization of this concept, explore some examples and results.

\begin{references}
\bibitem M.\ Barr, {\em Relational algebras}, in: Springer Lecture Notes in Math.\ 137 (1970), 39--55.
\bibitem M.M.\ Clementino and D.\ Hofmann, {\em Topological features of lax algebras}, Appl.\ Categ.\ Struct.\ 11 (2003), 267--286.
\bibitem M.M.\ Clementino and W.\ Tholen, {\em Metric, Topology and Multicategory -- a Common Approach}, J.\ Pure Appl.\ Algebra 179 (2003), 13--47.
\bibitem F.W.\ Lawvere, {\em Metric spaces, generalized logic, and closed categories}, Rend.\ Sem.\ Mat.\ Fis.\ Milano 43 (1973), 135--166.
\end{references}

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