nLab familial 2-monad

Idea

Just like parametric right adjoint functors are a generalization of polynomial functors, familial 2-monad are a generalization of polynomial 2-monads? with applications to higher category theory (Weber ‘07 and Shapiro ‘21).

Definition

The following is Definition 5.2 in Weber ‘07.

Definition

Let $T : \mathcal{A} \to \mathcal{B}$ be a 2-functor between finitely complete 2-categories. Then $T$ is familial when it is p.r.a and the 2-functor

$$T_1 : \mathcal{A} \to \mathcal{B}/T1$$

factors through $U_{T1} : \mathrm{Spl}(T1) \to \mathcal{B}/T1$, where $\mathrm{Spl}(T1)$ are split fibrations over $T1$.

A 2-monad is familial when its underlying 2-functor is familial, and its unit and multiplication are cartesian.

See also

• operad
• parametric right adjoint

References

The concept is defined in

• Mark Weber, Familial 2-functors and parametric right adjoints, Theory Appl. Categ. 18 (2007), No. 22, 665–732, (tac))

and generalised in:

• Charles Walker, Lax Familial Representability and Lax Generic Factorizations, Theory and Applications of Categories 35.37 (2020): 1424-1475 (url)

A closely related notion is contained in:

• Jason Brown?, 2-categorical Fam constructions, PhD thesis, Macquarie University, 2025 (url)

As a way to encode the shape of higher categories:

• Brandon Shapiro?, Familial Monads as Higher Category Theories (arXiv:2111.14796)