nLab indexed (infinity, 1)-category

Idea

An indexed (∞,1)-category is the (∞,1)-category theoretic analogue of an indexed category. If $S$ is an (∞,1)-category, then an indexed (∞,1)-category is a functor from $S^{op}$ to (∞,1)Cat.

The (∞,1)-Grothendieck construction establishes an equivalence between (∞,1)-categories indexed by $S$ and Cartesian fibrations over $S$.

Variants

Varieties of indexed (∞,1)-category can be formed by requiring that the target be monoidal, symmetric, closed, and so on.

Not all kinds of fibration of $(\infty, 1)$-category can be formed in this way. In particular, exponentiable ∞-functors are treated in (AyalaFrancis by maps to $(\infty,1)Prof$, but with extra ‘flagged’ structure. An $(\infty, 1)$-version of proarrow equipments should work here.

In addition, for some purposes conditions may be placed on the indexing category, $S$, such as the atomic orbital $\infty$-categories of Parametrized Higher Category Theory and Higher Algebra.

Related pages

• fibrations of quasi-categories
• Conduché functor
• Parametrized Higher Category Theory and Higher Algebra

References

• David Ayala and John Francis, Fibrations of ∞-categories, arxiv