Where in algebraic geometry one considers spaces which are formally dual to commutative rings, in “2-algebraic geometry” one considers spaces formally dual to 2-rigs, namely to certain tensor categories or more generally to tensor (∞,1)-categories.

References

Jacob Lurie?, Tannaka duality for geometric stacks?.

Alexandru Chirvasitu?, Theo Johnson-Freyd?, The fundamental pro-groupoid of an affine 2-scheme?, Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. (DOI, arXiv:1105.3104)

Martin Brandenburg?, Tensor categorical foundations of algebraic geometry? (arXiv:1410.1716)