This is the analog of a filtered category in the context of (∞,1)-categories.
The main purpose of considering filtered (∞,1)-categories is to define filtered (∞,1)-colimits, which are the colimits that commute with finite (∞,1)-limits.
Let be a regular cardinal, and let be an (∞,1)-category, incarnated as a quasicategory.
is called -filtered if for all -small and every morphism there is a morphism extending , where denotes the (right) cone of the simplicial set . is called filtered if it is -filtered.
This is HTT, prop. 188.8.131.52.
This appears as (Lurie, prop. 184.108.40.206). Since sifted (∞,1)-colimits are precisely those that commute with finite products, this is a direct reflection of the fact that finite products are a special kind of finite (∞,1)-limits.
Section 5.3.1 of