## Centralisers, centres and all that

### Richard Garner – 22 February 2023

If S is a commutative monoid and f: S ---> T is a monoid map then there is an "obvious" commutative submonoid of T through which f factors: first take the centraliser in T of the image of f, and then the centre of that. Prima facie the construction is not actually that obvious, and certainly does not appear to be canonical. We explain why it is, really, in terms of the parametric left adjoint nature of the monad Sx(-) on the category of monoids.

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