  We lay the foundations for a theory of quasi-categories in a monoidal
category $\mathcal{V}$ replacing $\mathrm{Set}$, aimed at realising weak
enrichment in the category $S\mathcal{V}$ of simplicial objects in
$\mathcal{V}$. To accomodate non-cartesian monoidal products, we make use of an
ambient category $S_{\otimes}\mathcal{V}$ of templicial - or
'tensor-simplicial' - objects in $\mathcal{V}$, which are certain colax
monoidal functors following Leinster. Inspired by the description of the
categorification functor due to Dugger and Spivak, we construct a templicial
analogue of the homotopy coherent nerve functor which goes from
$S\mathcal{V}$-enriched categories to templicial objects. We show that an
$S\mathcal{V}$-enriched category whose underlying simplicial category is
locally Kan, is turned into a quasi-category in $\mathcal{V}$ by this nerve
functor.
