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In mathematics, it is very common to turn one object into another by removing some structure. For example, you may turn a field into a ring and then into an abelian group if you ignore division and multiplication. I believe forgetting is the standard term here.

It is also possible to add some structure (though mathematicians normally do not like this and only do it when there is a very good reason). For example, you may turn the euclidean plane $\mathbb{E}^2$ into the Cartesian product $\mathbb{R}^2$ if you promote a point into the origin and draw two lines through it. Or you can make a smooth manifold into a Riemannian one.

I have two questions about this stuff. Is there a standard name for this operation? And, what is the correct (coherent, conceptual, etc) treatment of it? I have no clue, for example, how to properly describe it in categorical terms (as opposed to the former case where it is a very well known forgetful functor).

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    $\begingroup$ In model theory it's called taking the expansion of a structure: en.wikipedia.org/wiki/Model_theory#Reducts_and_expansions $\endgroup$
    – Todd Trimble
    Commented Sep 15, 2017 at 11:51
  • $\begingroup$ I suppose there is some difference between a reduction (in model theory) and a forgetful functor (what I would want is rather the converse to the latter, not to the former). Am I wrong? $\endgroup$
    – Alex Gavrilov
    Commented Sep 15, 2017 at 12:26
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    $\begingroup$ Since there is no canonical choice involved, you are just picking an element of a fiber of a forgetful functor. Naturally, you want to parameterize your choices, and these are parameterized by the fibers, i.e. by the forgetful functor. So the appropriate concept might be just the forgetful functor. Every surjective functor is forgetful of something, so the right concept might be surjective functor. $\endgroup$
    – Ben McKay
    Commented Sep 15, 2017 at 13:10
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    $\begingroup$ In fact there's a general set of ideas in category theory whereby any functor $F: A \to B$ can be considered to be a "forgetful functor": ncatlab.org/nlab/show/stuff,+structure,+property Echoing what Ben just said, just as $F$ can be considered a way in which to forget some of the stuff/structure/properties adhering to (objects of) $A$, it could also be viewed as a way in which to add stuff/structure/properties adhering to (objects of) $B$, parallel to how we say that a structure $X$ (in the sense of model theory) is a reduct of another structure $Y$ iff $Y$ is an expansion of $X$. $\endgroup$
    – Todd Trimble
    Commented Sep 15, 2017 at 13:17

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The standard word used I believe is 'equip'. For example, one may equip a manifold with a metric. (Other obvious synonyms should work as well, e.g., 'furnish,' etc)

I think it is also slightly incorrect to say that mathematicians don't like adding structure to objects! Folks working on moduli spaces would balk at this sentiment. (Moduli spaces in some sense are for understanding different ways one can add structure to something.)

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  • $\begingroup$ And, of course, one also adds structure to the moduli space itself …. (I'd also like to take this-comment to re-register my objection to the term "moduli space" by anyone who doesn't also say/write "vectors space".) $\endgroup$
    – LSpice
    Commented Oct 22, 2017 at 20:42

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