# nLab replete subcategory

Replete subcategories

category theory

## Applications

#### Higher category theory

higher category theory

# Replete subcategories

## Idea

A subcategory $D$ of a 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.

Last revised on September 5, 2022 at 13:48:13. See the history of this page for a list of all contributions to it.