Subcategories

category theory

# Subcategories

## Definition

Given a category $C$, a subcategory $D$ consists of a subcollection of the collection of objects of $C$ and a subcollection of the collection of morphisms of $C$ such that:

• If the morphism $f : x \to y$ is in $D$, then so are $x$ and $y$.

• If $f : x \to y$ and $g : y \to z$ are in $D$, then so is the composite $g f : x \to z$.

• If $x$ is in $D$ then so is the identity morphism $1_x$.

These conditions ensure that $D$ is a category in its own right and the inclusion $D\hookrightarrow C$ is a functor. Additionally, we say that $D$ is…

• A full subcategory if for any $x$ and $y$ in $D$, every morphism $f : x \to y$ in $C$ is also in $D$ (that is, the inclusion functor $D\hookrightarrow C$ is full).

• A replete subcategory if for any $x$ in $D$ and any isomorphism $f:x\cong y$ in $C$, both $y$ and $f$ are also in $D$.

• A wide subcategory if every object of $C$ is also an object of $D$.

## Variants in accord with the principle of equivalence

Just as subsets of a set $X$ can be identified with isomorphism classes of monic functions into $X$, subcategories of a category $C$ can be identified with isomorphism classes of monic functors into $C$. A functor is easily verified to be monic iff it is faithful and injective on objects. This can be generalized to monomorphisms in a strict 2-category.

However, this notion violates the principle of equivalence since being injective-on-objects refers to equality of objects. This raises the question: what is a good definition of subobject in a 2-category in accord with the principle of equivalence? It is the contention of the authors of this page that there are multiple such definitions. Two evident ones are:

• A morphism $f: A\to B$ in a 2-category $K$ is 1-monic if it is full and faithful, i.e. $K(X,A) \to K(X,B)$ is full and faithful for all $X$. A 1-subobject of $B$ is an equivalence class of 1-monomorphisms into $B$, and a 1-subcategory is a 1-subobject in $Cat$.
• Likewise, $f$ is 2-monic if $K(X,A) \to K(X,B)$ is faithful for all $X$. A 2-subobject of $B$ is an equivalence class of 2-monomorphisms into $B$, and a 2-subcategory is a 2-subobject in $Cat$.

The obvious generalizations (at least, obvious once you start thinking in terms of $k$-surjectivity) are that every morphism is 3-monic, while the 0-monic morphisms are the equivalences. (Note that this numbering is offset by one from that used in Baez and Shulman.) There is likewise an evident generalization to $k$-monomorphisms in any $n$-category.

It is fairly undisputed that 1-subobjects, as defined above, are a good notion of subobject in a 2-category. In particular, any full and faithful functor $C\to D$ in $Cat$ is equivalent to the inclusion of a full subcategory $C'\to D$ (here $C'$ is the full image of $C$). Also, in a 1-category considered as a locally discrete 2-category, the 1-monomorphisms are precisely the usual sort of monomorphism.

In fact, any faithful functor is likewise equivalent to the inclusion of a (non-full) subcategory, but in this case the codomain must be modified as well as the domain. It is somewhat more disputable whether 2-subcategories all deserve to be called “subcategories;” for instance, is Grp a “subcategory” of Set? Note also that any functor between discrete categories is faithful, so that the terminal category has a proper class of inequivalent 2-subcategories, and similarly every morphism in a locally discrete 2-category is 2-monic. However, kernels of morphisms between 2-groups are 2-subobjects, not 1-subobjects, and likewise for any subgroup of a group (considered as a 1-object category). This motivates the term “2-subobject,” to make it clear that there is some relationship with the sort of subobjects we are used to in 1-categories, but also some notable generalization.

Other types of morphism in a 2-category which have some claim to be considered “subobjects” include pseudomonic morphisms and conservative morphisms. Pseudomonic morphisms might merit a name such as (2,1)-subcategory, since a functor is pseudomonic iff it is faithful (a 2-subcategory) and its induced functor between underlying groupoids is fully faithful (a 1-subcategory). See also stuff, structure, property.

### As Displayed Categories

A different way to describe subcategories is as displayed categories satisfying certain properties. This is analogous to defining a subset by its characteristic function, which allows to define subsets without a notion of global membership.

A subcategory of a category $C$ is a displayed category $D$ over $C$ such that all $D(c)$ and $D_f(A,B)$ are propositions. We can recover a subcategory in the original sense by taking the Grothendieck construction of $D$. The identity and composition in the displayed category correspond precisely to the inclusion of the subcategory being closed under identity and composition.

We can also capture the above flavors of subcategory by adding further conditions to $D$. We say $D$ is…

• A full subcategory if all $D_f(A,B)$ are true.
• A wide subcategory if all $D(c)$ are true.
• A replete subcategory if $D$ is an isofibration.

Last revised on February 9, 2024 at 17:35:28. See the history of this page for a list of all contributions to it.