nLab (2,1)-functor

Idea

The concept of $(2,1)$-functors is that of the natural kind of morphisms between (2,1)-categories.

If (2,1)-categories are regarded as special cases of 2-categories, then $(2,1)$-functors are equivalently the 2-functors between (2,1)-categories.

If (2,1)-categories are regarded as special cases of (∞,1)-categories, then $(2,1)$-functors are equivalently the (∞,1)-functors between (2,1)-categories.

Examples

• The construction of groupoid convolution algebras constitutes a $(2,1)$-functor from differentiable stacks with proper maps between them and the opposite $(2,1)$-category of C*-algebras with Hilbert bimodules between them. For details see there.