the Departamento de Mathematica Fundamental in the Universidad de La Laguna, Tenerife, simlarly and also to Carlos Simpson, Bertrand Toen, André Joyal, Clemens Berger, André Hirschowitz and others at the Nice meeting in May 2003, since that is where bits of ideas that I had gleaned over a longish period of time fitted together so that I think I begin to understand the way that a lot of things interlock in this area better than I did before!

These notes have also benefitted from comments by Jim Stasheff and some of his colleagues on an earlier version.

$\mathcal{S}$-categories
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Categories with simplicial ‘hom-sets’
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We assume we have a category $\mathcal{A}$ whose objects will be denoted by lower case letter, $x$,$y$,$z$, …, at least in the generic case, and for each pair of such objects, $(x,y)$, a simplicial set $\mathcal{A}(x,y)$ is given; for each triple $x, y, z$ of objects of $\mathcal{A}$, we have a simplicial map, called *composition* $$\mathcal{A}(x,y)\times \mathcal{A}(y,z)\longrightarrow \mathcal{A}(x,z);$$ and for each object $x$ a map $$\Delta[0] \to \mathcal{A}(x,x)$$ that ‘names’ or ‘picks out’ the ‘identity arrow’ in the set of 0-simplices of $\mathcal{A}(x,x)$. This data is to satisfy the obvious axioms, associativity and identity, suitably adapted 