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\section*{completion}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{idempotents}{}\paragraph*{{Idempotents}}\label{idempotents}

[[!include idempotents - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{completions}{}\section*{{Completions}}\label{completions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{completion_and_idempotence}{Completion and idempotence}\dotfill \pageref*{completion_and_idempotence} \linebreak
\noindent\hyperlink{list_of_completions}{List of completions}\dotfill \pageref*{list_of_completions} \linebreak
\noindent\hyperlink{free_completion_and_laxidempotence}{Free completion and lax-idempotence}\dotfill \pageref*{free_completion_and_laxidempotence} \linebreak
\noindent\hyperlink{list_of_completions_in_category_theory}{List of completions in category theory}\dotfill \pageref*{list_of_completions_in_category_theory} \linebreak
\noindent\hyperlink{idempotent_completions}{Idempotent completions}\dotfill \pageref*{idempotent_completions} \linebreak
\noindent\hyperlink{nonidempotent_completions}{Non-idempotent completions}\dotfill \pageref*{nonidempotent_completions} \linebreak
\noindent\hyperlink{nonunique}{Non-unique completions}\dotfill \pageref*{nonunique} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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 \emph{completion} of $C$ with respect to these properties.

\hypertarget{completion_and_idempotence}{}\subsection*{{Completion and idempotence}}\label{completion_and_idempotence}

Outside of category theory (see below), ``completion'' is usually used mostly for \emph{idempotent} operations. That is, the completion of a thing with respect to some property is itself \emph{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 \emph{property}. Namely, according to the yoga of SSP, given some ambient category $K$, the subcategory of objects having a property $P$ should be a \emph{full} subcategory, whose inclusion [[functor]] forgets that property. A ``completion'' operation is then usually a left (or, occasionally, right) [[adjoint functor|adjoint]] to that inclusion functor, making the category of objects with property $P$ [[reflective subcategory|reflective]] (or coreflective). Note that a [[full and faithful functor]] with a left adjoint is always [[monadic functor|monadic]] and the monad is [[idempotent monad|idempotent]].

By contrast, when we add \emph{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 functor|faithful]] [[reflector]]. Note that the reflector is faithful if and only if the unit of the reflection is [[monomorphism|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''.

\hypertarget{list_of_completions}{}\subsection*{{List of completions}}\label{list_of_completions}

\begin{itemize}%
\item [[complete space|Cauchy completion]] of a [[metric space]], or more generally a [[uniform space]]


\item [[Dedekind completion]] of a [[linear order]] (or sometimes a more general [[preorder]] or [[quasiorder]])


\item [[completion of a ring]]


\item [[MacNeille completion]] of a [[partially ordered set]]


\item [[profinite completion of a group|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.


\item [[Grothendieck group]] of a cancellative commutative [[monoid]]; group completion of a cancellative monoid


\item Field of fractions of an [[integral domain]]


\item The [[Stone-∞ech compactification]] of a [[Tychonoff space]]


\item [[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.



\end{itemize}
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.

\hypertarget{free_completion_and_laxidempotence}{}\subsection*{{Free completion and lax-idempotence}}\label{free_completion_and_laxidempotence}

The general contrast between ``completion'' for adding a property and ``free'' for adding structure applies to operations on [[category|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 \emph{free completions}. The monads they induce are not idempotent, but they are often [[lax-idempotent monad|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 \emph{different} monoid (with a new identity); this shows the distinction between free completion and mere completion.

\hypertarget{list_of_completions_in_category_theory}{}\subsection*{{List of completions in category theory}}\label{list_of_completions_in_category_theory}

\hypertarget{idempotent_completions}{}\subsubsection*{{Idempotent completions}}\label{idempotent_completions}

The following completions add a \emph{property} in the strictest sense, and as such are idempotent.

\begin{itemize}%
\item [[Cauchy complete category|Cauchy completion]] of an enriched category: addition of all absolute limits.
\item [[exact completion]] of a [[regular category]] (the ``ex/reg'' completion)
\item [[extensive category|extensive]] completion
\item [[pretopos]] completion of a [[coherent category]]

\end{itemize}
\hypertarget{nonidempotent_completions}{}\subsubsection*{{Non-idempotent completions}}\label{nonidempotent_completions}

These completions add a property-like structure, are often lax-idempotent or colax-idempotent.

\begin{itemize}%
\item [[free cocompletion]]: the [[presheaf category]] of a [[small category]] is its free cocompletion under all small colimits.
\item [[free completion]]: the dual (the opposite of the free cocompletion of the opposite).
\item Ind-completion: the ``free cocompletion under [[filtered colimits]],'' consisting of all [[ind-object|ind-objects]].
\item Pro-completion: the dual (free completion under cofiltered limits, the opposite of the Ind-completion of the opposite), consisting of all [[pro-objects]].
\item [[ideal completion]]: the free cocompletion under filtered colimits of monomorphisms.

\end{itemize}
Note that these completions take small categories to large categories. Free completions and cocompletions of large categories can be obtained using categories of [[accessible functor|accessible presheaves]]. However, if we only want to add limits and/or colimits for a given \emph{set} of diagram shapes, the free completion process will preserve smallness. For example we have:

\begin{itemize}%
\item the free completion under finite products or coproducts
\item the free completions under finite limits or colimits

\end{itemize}
All these sorts of completions work just as well in an [[enriched category|enriched]] setting, as long as the enriching category $V$ is nice enough ([[locally presentable category|locally presentable]] is certainly enough).

Some other non-idempotent completions:

\begin{itemize}%
\item The [[reg/lex completion]] and [[ex/lex completion]] of a category with finite limits.

\end{itemize}
\hypertarget{nonunique}{}\subsection*{{Non-unique completions}}\label{nonunique}

The completions above are mostly (all?) [[the|unique in the relevant sense]]: unique up to unique isomorphism for objects of [[groupoids]] (or [[categories]]), more generally with a [[contractible space|contractible]] $\infty$-[[infinity-groupoid|groupoid]] of completions.

However, there are other sorts of completions in mathematics, such as:

\begin{itemize}%
\item `The' [[algebraic closure]] of a [[field]] is unique up to isomorphism, but the isomorphism is not unique. This is an example of an [[injective hull]].
\item Every [[model theory|model]] of a [[first-order theory]] may be interpreted as giving a completion of that theory (but not one with a recursively enumerable axiomatisation); every statement that is true in the model is an axiom of the complete theory. However, this is far from unique.
\item An [[ultrapower]] of a structure may be viewed as an elementary completion (see [[elementary embedding]]), as described by [[Terry Tao]] in \href{http://terrytao.wordpress.com/2010/11/27/nonstandard-analysis-as-a-completion-of-standard-analysis/}{a blog post}.

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[completion monad]]


\item [[completion of a ring]], [[completion of a module]]


\item [[analytic completion]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Dennis Sullivan]], pp.9 in \emph{Localization, Periodicity and Galois Symmetry} (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springer (\href{http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf}{pdf})


\item [[Marta Bunge]], \emph{Tightly Bounded Completions} , TAC \textbf{28} no. 8 (2013) pp.213-240. (\href{http://www.tac.mta.ca/tac/volumes/28/8/28-08.pdf}{pdf})



\end{itemize}
[[!redirects completion]] [[!redirects completions]]



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