structure.$\blacksquare$

A cofibration category structure is just one of many variants of the abstract homotopy theory structure introduced to be able to push through homotopy type arguments in particular settings. There are variants of this result, due to Kamps, see references in , where $\mathcal{C}$ is both tensored and cotensored over $\mathcal{S}$ and the conclusion is that $\mathcal{C}$ has a Quillen model category structure. The examples of locally Kan $\mathcal{S}$-categories include $\mathcal{T}op$, $\mathcal{K}an$, $\mathcal{G}rpd$ and $\mathcal{C}rs$, but not $\mathcal{C}at$ or $\mathcal{S}$ itself.

Nerves and Homotopy Coherent Nerves.
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Kan and weak Kan conditions
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Before we get going on this section, it will be a good idea to bring to the fore the definitions of *Kan complex* and *weak Kan complex* (or *quasi-category*).

As usual we set $\Delta[n] = {\mathbf \Delta}( - , [n]) \in \mathcal{S}$, then for each $i$, $0 \leq i \leq n$, we can form a subsimplicial set, $\Lambda^i[n]$, of $\Delta[n] $ by discarding the top dimensional $n$-simplex (given by the identity map on $[n]$) and its $i^{th}$ face. We must also discard all the degeneracies of these simplices. This informal definition does not give a ‘picture’ of what we have, so we will list the various 