One way or another, enriched 1-category theory has held an important spot in the study of homological and homotopical phenomena practically since the very start of ordinary category theory. For many purposes, enriched 1-categories or their model 1-categorical counterparts are simply too rigid, or they might not even exist at all. In recent years various models of enriched (∞,1)-categories have been introduced, and some comparisons at differing levels have been made e.g. the underlying parameterizing ∞-operads or their ∞-categories (with a closed left action over Cat_∞). We are interested in a universal property that can compare these theories at a level which can detect pointwise Kan extensions for example. Part of one approach to this involves upgrading the underlying machinery appearing in Gepner and Haugseng to the scaled simplicial setting. This talk will be heavily focused on examples and justifying why we would want such theories anyway. The only prerequisite is some knowledge of enriched 1-category theory and an appetite for homotopy theory. Time permitting, we may discuss the situation with enriched (∞,1)-operads and (∞,1)-properads, or other possible uses of intermediate results.
