Discrete Vertices in Simplicial Objects Internal to a Monoidal Category

We follow the work of Aguiar (Internal categories and quantum groups. PhD Thesis, Cornell University, 1997) on internal categories and introduce simplicial objects internal to a monoidal category as certain colax monoidal functors. Then we compare three approaches to equipping them with a discrete set of vertices. We introduce based colax monoidal functors and show that under suitable conditions they are equivalent to the templicial objects defined by Lowen and Mertens (Algebr Geom Topol, 2024). We also compare templicial objects to the enriched Segal precategories appearing in Lurie ((Infinity,2)-categories and the Goodwillie calculus I. Preprint at https://arxiv.org/abs/0905.0462v2), Simpson (Homotopy theory of higher categories. New mathematical monographs, Cambridge University Press, Cambridge, vol 19, p 634, 2012, https://doi.org/10.1017/CBO9780511978111) and Bacard (Theory Appl Categ 35:1227–1267, 2020), and show that they are equivalent for cartesian monoidal categories, but not in general.