nLab subcategory

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.

Related concepts

• subcategory

  • full, wide, dense, replete

  • reflective, coreflective, bireflective

  • abelian, Serre topologizing

  • internal subcategory

  • dense sub-site

• sub-2-category

  • full sub-2-category

  • locally full sub-2-category

  • reflective sub-2-category

• localizing subcategory

• sub-(∞,1)-category (internal formulation)

  • reflective sub-(∞,1)-category (internal formulation)

  • reflective product-preserving sub-(∞,1)-category (internal formulation)

• thick subcategory (of a triangulated category)

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• embedding of categories