not met ‘weak Kan complexes’ before, you will soon meet them in earnest!)

The theory is ‘geometrically’ nicer to work with if $\mathcal{C}$ is *tensored* or *cotensored*.

Tensoring and Cotensoring
-------------------------

**Tensored**

If for all $K \in \mathcal{S}$, $X, Y,  \in \mathcal{C}$, there is an object $K\bar{\otimes} X$ in $\mathcal{C}$ such that $$\mathcal{C}(K\bar{\otimes}X,Y) \cong
\mathcal{S}(K,\mathcal{C}(X,Y)$$naturally in $K$, $X$ and $Y$, then $\mathcal{C}$ is said to be *tensored* over $\mathcal{S}$.

**Cotensored**

Dually, if we require objects $\bar{\mathcal{C}}(K,Y)$ such that $$\mathcal{C}(X,\bar{\mathcal{C}}(K,Y)) \cong
\mathcal{S}(K,\mathcal{C}(X,Y)$$then we say $\mathcal{C}$ is *cotensored* over $\mathcal{S}$.

To gain some intuitive feeling for these, think of $K\bar{\otimes} X$ as being $K$ product with $X$, and $\bar{\mathcal{C}}(K,Y)$ as the object of functions from $K$ to $Y$. These words do not, as such, make sense in general, but do tell one the sort of tasks these constructions will be set to do. They will not be much used explicitly here however.

(cf. Kamps and Porter, )\
If $\mathcal{C}$ is a locally Kan $\mathcal{S}$-category tensored over $\mathcal{S}$ then taking $I\times X = \Delta[1]\bar{\otimes}X$, we get a good cylinder functor such that for the cofibrations relative to $I$ and weak equivalences taken to be homotopy equivalences, the category $\mathcal{C}$ has a cofibration category 