  It is shown that the endomorphism monoids of the category $2\mathfrak{Cob}$
of all $2$-cobordisms do not have finitely axiomatizable equational theories.
The same holds for the {topological annular category} and various quotients of
the latter, like the affine Temperley--Lieb category. Analogous results are
obtained for versions with the algebraic signature enriched by involutions.
Von-Neumann-regular extensions of these categories are also considered.
