Abstract

The theory of monads on Set has been hampered by a lack of good counterexamples – for example, although at first it was believed that the continuations monad might be nasty enough that there would be some monad with which it would have no tensor product, it was shown by Goncharov and Schroeder that this monad is uniform, and so it does have tensor products with all other monads on Set. There is a serious lack of examples of non-uniform monads, though one such monad (the wellorder monad) has been examined and shown to have no tensor product with the nonempty list monad.

I’ll present a new technique for building badly behaved monads on Set, by making use of large algebraic structures which don’t have small generating sets (I’ll call such structures barren). I’ll use this technique to show how a couple of interesting counterexamples can be built: a monad with no tensor product with the finite power set monad, and an N-indexed sequence of monads whose colimit is universe-dependent.