The general idea of completion is that given an object of some sort $C$, we construct from it a larger object, containing $C$ as a subobject, which moreover has certain properties that $C$ might have lacked. This larger object is then called the completion of $C$ with respect to these properties.
Outside of category theory (see below), “completion” is usually used mostly for idempotent operations. That is, the completion of a thing with respect to some property is itself complete with respect to that property, and the completion process doesn’t change an object that is already complete. From the perspective of stuff, structure, property this makes sense because we complete with respect to a property. Namely, according to the yoga of SSP, given some ambient category $K$, the subcategory of objects having a property $P$ should be a full subcategory, whose inclusion functor forgets that property. A “completion” operation is then usually a left (or, occasionally, right) adjoint to that inclusion functor, making the category of objects with property $P$ reflective (or coreflective). Note that a full and faithful functor with a left adjoint is always monadic and the monad is idempotent.
By contrast, when we add structure to objects, i.e. we consider an adjoint to a forgetful functor which is faithful but not necessarily full, we usually do not use the word “completion” but rather the word “free”. For example, the category of complete metric spaces is a full reflective subcategory of the category of metric spaces, and the reflection is called “completion.” By contrast, the forgetful functor from the category of groups to the category of sets has a left adjoint, but we call that left adjoint the “free group on a set” rather than the “completion of a set into a group.”
Additionally, we tend to use the term ‘completion’ only for a faithful reflector. Note that the reflector is faithful if and only if the unit of the reflection is monic; we might even want to require it to be a regular monomorphism. For example, the left adjoint of the forgetful functor from abelian groups to groups is called ‘abelianisation’ and may even be called ‘free abelian group on a group’ but is not normally called ‘abelian completion’ or anything like that. (The tendency in such cases is rather to resort to suffixes like “-ification” and “-ization”: sheafification, abelianization, localization.) This fits with the intuition that a completion “adds more stuff” to the object we started with in order to “make it complete”.
Cauchy completion of a metric space, or more generally a uniform space
Dedekind completion of a linear order (or sometimes a more general preorder or quasiorder)
profinite completion of a discrete group
Is this a case of profinite completion of a category, i.e., adding cofiltered limits?
Mike Shulman: No, I don’t think so. Are you thinking of viewing a group as a one-object groupoid? The profinite completion of a group is not an ordinary group but a profinite group, which is thus not a category, at least not in the same sense.
Mike Shulman: Also, is the unit of this adjunction monic? Off the top of my head I see no reason for it to be.
Grothendieck group of a cancellative commutative monoid; group completion of a cancellative monoid
Field of fractions of an integral domain
Moore closure operators (monads) on posets or preorders, e.g., the subgroup generated by a subset of a group, the ideal generated by a subset of a ring, etc.
David: It’s not clear to me when in the ‘List of completions’ we have examples of enriched category completions: Cauchy completion of a metric space (yes) of a uniform space (no?), Dedekind completion of a linear order (yes for 2-enriched categories?).
Mike Shulman: Although orders are 2-enriched categories, Dedekind completion of an order is not a categorical completion, at least not in the sense of adding limits or colimits. That would be the construction of downsets or ideals.
Cauchy completion of a metric space is, of course, an instance of Cauchy completion of enriched categories. I believe that Cauchy completion of a uniform space is actually also an instance of a general categorical notion of Cauchy completion, but in the more general setting of an equipment (namely, the equipment of sets and filters). See “Categorical interpretation” at uniform space for a too-brief summary of this point of view.
David Corfield: so is there are clear-cut distinction of level in all these cases, where we can separate completions of (enriched) categories (equipment) from completions of sets (with properties and structures)?
Mike Shulman: I think the reason that completion of metric spaces and uniform spaces feels more like the completion of sets (than categories) is that they are basically (enriched) (0,1)-categories. If the question is where to separate this list from the one below, I would put completions of anything enriched in a preorder up here, and completions of things enriched non-posetally below.
The general contrast between “completion” for adding a property and “free” for adding structure applies to operations on categories as well. For instance, since split idempotents are preserved by any functor, the 2-category of categories with split idempotents is a full sub-2-category of Cat. Therefore, it makes perfect sense to call the left adjoint of its inclusion a “completion,” in this case the (Set-enriched) Cauchy completion or Karoubi envelope. By contrast, the forgetful functor from the 2-category of monoidal categories to $Cat$ forgets structure, rather than properties, so its left adjoint should be called a “free” construction rather than a “completion.”
However, in this case there is an intermediate notion between properties and structure, namely property-like structure. Intuitively, property-like structure is “structure which, when it exists, is essentially unique” or “a property which is not necessarily preserved by morphisms.” The canonical example is the existence of limits and colimits in a category: when they exist they are unique up to unique canonical isomorphism (so that “having limits” is intuitively a property of a category), but not every functor preserves limits, so the forgetful functor from categories-with-limits to categories is not full. (Left) adjoints to functors which forget property-like structure are usually called free completions. The monads they induce are not idempotent, but they are often lax-idempotent or colax-idempotent. (For example, when we freely add limits to a category that already had limits, the old limits are no longer limits and the new ones take their place.)
Property-like structure is most common in category theory, but it does occur elsewhere as well. For instance, a monoid is a semigroup with property-like structure: a semigroup has at most one identity element, but that identity element is not necessarily preserved by semigroup homomorphisms. Thus the adjoining of a new formal identity element to a semigroup (which is not an idempotent operation) might be called its “free completion into a monoid”. Notice that if the original semigroup is already a monoid, then its free completion into a monoid will be a different monoid (with a new identity); this shows the distinction between free completion and mere completion.
The following completions add a property in the strictest sense, and as such are idempotent.
These completions add a property-like structure, are often lax-idempotent or colax-idempotent.
Note that these completions take small categories to large categories. Free completions and cocompletions of large categories can be obtained using categories of accessible presheaves. However, if we only want to add limits and/or colimits for a given set of diagram shapes, the free completion process will preserve smallness. For example we have:
All these sorts of completions work just as well in an enriched setting, as long as the enriching category $V$ is nice enough (locally presentable is certainly enough).
Some other non-idempotent completions:
The completions above are mostly (all?) unique in the relevant sense?: unique up to unique isomorphism for objects of groupoids (or categories), more generally with a contractible $\infty$-groupoid of completions.
However, there are other sorts of completions in mathematics, such as: