to this situation. Such a set up will be called a *simplicially enriched category* or more simply *an $\mathcal{S}$-category*. Enriched category theory is a well established branch of category theory. It has many useful tools and not all of them have yet been exploited for the particular case of $\mathcal{S}$-categories and its applications in homotopy theory.

Some authors use the term simplicial category for what we have termed a simplicially enriched category. There is a close link with the notion of simplicial category that is consistent with usage in simplicial theory *per se*, since any simplicially enriched category can be thought of as a simplicial object in the ‘category of categories’, but a simplicially enriched category is not just a simplicial object in the ‘category of categories’ and not all such simplicial objects correspond to such enriched categories. That being said that usage need not cause problems provided the reader is aware of the usage in the paper to which reference is being made.

**Examples**

\(i) $\mathcal{S}$, the category of simplicial sets:\
here $$\mathcal{S}(K,L)_n:= S(\Delta[n] \times K,L);$$ Composition : for $f \in \mathcal{S}(K,L)_n$, $ g \in \mathcal{S}(L,M)_n$, so $f : \Delta[n] \times K \to L$, $g : \Delta[n] \times L 