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Topological semigroups/monoids: topological space endowed with a continuous semigroup/monoid structure, or, equivalently, semigroup/monoid endowed with a compatible topology.
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Pointed versus unpointed maps into a topological monoid
I've just stumbled on something that seems either too good to be true,
or else too good for me not to have heard of it before.
It has to do with the basepoint forgetting map
$$
u: [A, M] \to \langle A …