A Double (∞,1)-Categorical Nerve for Double Categories

A 2-category can be seen as an internal category to categories with discrete category of objects, i.e., a horizontal double category with only trivial vertical morphisms. Some aspects of 2-category theory, such as 2-limits, benefit from a passage to double categories. Going to the ∞-world, we expect to have a similar picture, which would allow one to develop aspects of (∞,2)-category theory, such as (∞,2)-limits, using double (∞,1)-categories.

A double (∞,1)-category was defined by Haugseng as a Segal object in complete Segal spaces, and then an (∞,2)-category in the form of a 2-fold complete Segal space can be interpreted as a ``horizontal'' double (∞,1)-category. In this talk, I will consider a slightly modified version of these double (∞,1)-categories and give a nerve construction from double categories to double (∞,1)-categories. This nerve is right Quillen and homotopically fully faithful from the category of double categories endowed with a model structure constructed in a joint work with Maru Sarazola and Paula Verdugo. By restricting along a ``homotopical'' horizontal embedding of 2-categories into double categories, we get a nerve from 2-categories into 2-fold complete Segal spaces, which is also right Quillen and homotopically fully faithful. I will show that these nerves are further compatible in a precise sense with the horizontal embedding of 2-categories into double categories, and this says that the ∞-setting indeed extends the strict setting.