Is there some universal sense of -ification (eg, groupification) in category theory

I have three questions.

1:

Does the groupification of a semigroup always exist? I believe this should be yes because for every $x$ in the semigroup one could just define an element $x'$ that should work as its inverse. But what would then happen to the product $x'y$ for $x,y$ elements of the semigroup? It feels like we get choices (or maybe not) here that messes things up.

2:

When defining the groupification, $G$, of a semigroup $S$ one require it to come with a morphism (of semigroups) $S \rightarrow G$ such that any other morphism (of semigroups) from $S$ to another group $G'$ factorizes through the previous map. Exactly which type of objects can be groupified? I guess one cannot groupify a topological space.

3:

This is a broad question but is there some sense of -ification? In the example one could replace "group" by "topological space" and talk about topologyfication. Now, no such word seem to exist so I guess one could not "topologyfy".

We can (i think) consider the groupification functor from the category of semigroups to the category of groups and it should be adjoint to the forgetful functor from the category of groups to the category of semigroups. This would suggest that we need some sense of a forgetful functor in the first place to talk about a -ification.

Apologies for this bad question, sometimes asking the right question is just as hard as answering it.

I take the liberty of answering only the parts of your question, to which I think I can give a precise answer.

1: Yes, it is called the Grothendieck group $G(N)$ of the given semigroup $S$. This construction is functorial.

2: There is always a canonical semigroup homomorphism $S\to G(S)$, but this need not be injective in general. For example, the Grothendieck group corresponding to $\mathbb N\cup\{\infty\}$ with the obvious addition $(n+\infty=\infty)$ is trivial!

3: Concerning -ification in general, at the moment I have nothing to add to BBischof's great comments above.

As BBischof says, the standard notion of -ification is to take the left adjoint of a forgetful functor. This includes the following as special cases:

• The groupification of a monoid or semigroup,
• The free group on a set, the free vector space on a set, etc.
• The abelianization of a group,
• The group ring of a group (the forgetful functor here sends a ring to its group of units),

and many other examples. I do not think one can reasonably talk about universal -ification without a specific choice of forgetful functor; if no good choice exists, you won't get a good notion of -ification, and on the other hand there may be more than one choice.

(Notably, I remember reading that in at least one example the natural construction is to take the right adjoint, but I don't remember what this example was.)