Abstract: In this lecture we are going to talk about an equivalence between the category of compact Hausdorff topological spaces and the category of commutative unital C*-algebras. We also introduce spectrum of an element in a unital C*-algebra and give some examples. Then we present GNS construction.

Abstract: For a symmetric monoidal model category $\mathscr{M} = (\underline{M};\otimes; I)$, we develop a theory of weakly enriched categories over $\mathscr{M}$, called co-Segal $\mathscr{M}$-categories. By ‘weakly enriched’ we mean a sort of enriched category where there is no prescription of a composition; but rather we allow many possible compositions, where each of them is associative up-to homotopies. Their definition derives from the philosophy of classical Segal categories; and just like for Segal categories they give rise to higher categorical structures. In this talk I will present the theory of co-Segal categories and will give the first results on their homotopy theory.