REU Meeting - 2026-07-13

This following is a brief summary of our research meeting on 2026-07-13.

Meeting summary

We talked all about group objects in Cartesian monoidal categories. We spent a fair chunk studying the question of whether one can define group objects in the (non-Cartesian) monoidal category $(\mathbf{S}\mathbf{e}\mathbf{t},\bigsqcup ,\mathrm{\varnothing })$, ultimately deciding that the diagrams encoding the action of the identity element, inverses, etc. don't make sense.

We also decided to backtrack and prove the Eckmann-Hilton result, which states if a set has two binary operations that share the same identity element and satisfy the interchange axiom, then those two operations must be the same and must be commutative.

Tasks for next meeting

• Investigate whether the category of Lie groups has (or does not have) each of the following categorical properties:
  • Has a terminal object?
  • Has an initial object?
  • Has a null object?
  • Has finite products?
  • Has all products?
  • Has pullbacks?
  • Has finite coproducts?
  • Has all coproducts?
  • Has pushfowards?
  • Has equalizers?
  • Has kernels?
  • Has quotients?
  • Has an analogue of the First Isomorphism Theorem?
  • Is an additive category?
  • Is an abelian category?

References

nLab