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Concrete categories $A$ carry a forgetful functor $U:A\rightarrow Set$, whose left adjoint if it exists is the free functor.

There are other forgetful functors such as $U':Ass\rightarrow Vect$ whose left adjoint is the free fucntor which produces the free tensor algebra.

From this, one might characterise forgetfulness in the context of an algebraic category as dropping one or more operations, so one gets a commuting graph of functors whose top node is that algebraic category and whose bottom node is $Set$ where all operations have been forgotten.

However, a functor $Top\rightarrow Set$ is also considered as forgetful, even though its a 'geometric' category rather than an algrebraic one. But one notes that it does have operations - meet, union, complement - so these operations can be forgotten - forgetting meet and union gives the forgetful functor $Top\rightarrow SLat$, from topological spaces to (union) semilattices.

But, one doesn't usually think of $Top$ as an algebraic category - its considered as geometric.

Further we have a forgetfull functor $Diff\rightarrow Top$ which appears to be essentially geometric in that we aren't forgetting 'algebraic' structure.

Is there a general and specific notion of forgetfullness that covers both algebraic & geometric senses? Presumably this can happen when we have a notion of geometrical structure so that metric, topological & differentiable structure are instances of these; and then we have the same general pattern alluded to above - the commuting graph of forgetfullness.

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You can define forgetful functors as faithful functors. This covers all the examples you have mentioned.

At the nlab you find another perspective: Every functor is a forgetful functor. But these functors are distinguished by what they forget, namely stuff, structure, property.

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