nLab replete subcategory

Idea

A subcategory $D$ of a strict category $C$ is called replete if it respects isomorphism of morphisms in the arrow category of $C$. It is a subcategory for which the property of (strictly) belonging to it respects the principle of equivalence of categories.

Definition

A subcategory $D$ of $C$ is replete if for any object $x$ in $D$ and any isomorphism $f\colon x\cong y$ in $C$, both $y$ and $f$ are also in $D$. Equivalent ways to state this include:

• If $f \in D$ and $f \cong g$ in the arrow category $Arr(C)$, then $g \in D$.

• The inclusion $D\hookrightarrow C$ is an isofibration.

Repletion and replete images

Since repleteness is a “closure condition,” the intersection of any collection of replete subcategories is again replete. Therefore, any subcategory is contained in a smallest replete subcategory, called its repletion. We can construct the repletion $repl(D)$ of $D\subset C$ explicitly as follows:

• its objects are those objects of $C$ which admit an isomorphism to some object of $D$, and
• its morphisms are those morphisms of $C$ which can be written as a composite of morphisms in $D$ and isomorphisms in $C$.

Repleteness and repletions are most often applied to full subcategories, in which case the repletion is simply the full subcategory of $C$ determined by those objects which are isomorphic to some object of $D$. In particular, in this case, the repletion is equivalent to $D$. More generally, we can say:

Proposition

The inclusion $D\hookrightarrow repl(D)$ is an equivalence if and only if the inclusion $D\hookrightarrow C$ is pseudomonic.

Proof

The inclusion of a replete subcategory is always pseudomonic, so if $D\hookrightarrow repl(D)$ is an equivalence, and in particular full and faithful, then $D\hookrightarrow C$ must also be pseudomonic.

Conversely, if $D\hookrightarrow C$ is pseudomonic, then every morphism of $repl(D)$ can be written as $g f h^{-1}$ for some morphism $f$ in $D$ and isomorphisms $g$ and $h$ in $C$. If the domain and codomain are in $D$, then since $D\hookrightarrow C$ is pseudomonic, both $g$ and $h$ must also be in $D$, hence such a morphism is itself necessarily in $D$. Thus the inclusion $D\hookrightarrow repl(D)$ is full. It is clearly faithful and essentially surjective, so it is an equivalence.

The replete image of a functor is the repletion of its image. The replete full image of a functor is the repletion of its full image, i.e. the full subcategory of its target determined by those objects isomorphic to some object in its image. See also essential image.

Higher-categorical versions

More generally, for $n\geq 0$, a subcategory $D$ of an $n$-category $C$ is replete if all equivalences in $C$ whose either source or target is in $D$ are themselves in $D$. In particular, all $k$-cells equivalent in $C$ to some $k$-cell in $D$ are themselves in $D$ (because they are sources of some equivalence in $D$). Then replete subcategories of $1$-categories are those which are replete in this sense.

Here we are using the weakest notion of equivalence; one could also talk about $k$-replete $n$-subcategories if every $k$-equivalences in $C$ belongs to $D$ if either its source or target does. Then using the usual definition of ‘category’ as a model for $1$-categories, replete subcategories are already those which are $0$-replete (indeed, $0$-equivalence is isomorphism in this model); but using a model of $1$-categories in which hom-sets are setoids (as is common, for example, in type-theoretic foundations), we must insist on $1$-repleteness.

Note that a subcategory $D$ of an $n$-category $C$ is $(k-1)$-replete if and only if:

• for every $(n-k)$-cell in $D$ all $(n-k)$-cells $(k-1)$-equivalent to an $(n-k)$-cell in $D$ are themselves in $C$, and
• for any two $(n-k)$-cells $x,y$ the hom $k$-category $hom(x,y)$ is $(k-2)$-replete.