Given a category , a subcategory consists of a subcollection of the collection of objects of and a subcollection of the collection of morphisms of such that:
If the morphism is in , then so are and .
If and are in , then so is the composite .
If is in then so is the identity morphism .
These conditions ensure that is a category in its own right and the inclusion is a functor. Additionally, we say that is…
A full subcategory if for any and in , every morphism in is also in (that is, the inclusion functor is full).
A replete subcategory if for any in and any isomorphism in , both and are also in .
A wide subcategory if every object of is also an object of .
Just as subsets of a set can be identified with isomorphism classes of monic functions into , subcategories of a category can be identified with isomorphism classes of monic functors into . 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 is evil since being injective-on-objects refers to equality of objects. This raises the question: what is a good non-evil definition of subobject in a 2-category? It is the contention of the authors of this page that there are multiple such definitions. Two evident ones are:
The obvious generalizations (at least, obvious once you start thinking in terms of -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 -monomorphisms in any -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 in is equivalent to the inclusion of a full subcategory (here is the full image of ). 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.
The above discussion of non-evil variants is the result of the following discussion.
Mike says: I have a few objections to considering any faithful functor to be a “subcategory.”
If one really wants a notion of “non-full subcategory” that is invariant under equivalence of categories, I propose that pseudomonic functors are a better candidate than faithful ones.
Mathieu Dupont says: For 1., you can “enlarge” to an equivalent category by adding, for each group , an object and an isomorphism from to the underlying set of . Then the category of groups is a “naive” subcategory of (i.e. a subcategory as defined at the beginning of this entry). Perhaps is this cheating?
A problem with pseudomonic functors is that not all inclusions of “naive” subcategories are pseudomonic.
Mike replies: Yes, I’m aware that, as the original entry said, any faithful functor is a naive subcategory of some equivalent category. It doesn’t make me any happier about calling a “subcategory” of , or about calling a discrete large category a “subcategory” of the terminal category. I would be more inclined to exclude non-pseudomonic naive subcategories from the notion of “subcategory” than to include all faithful functors. Are there naturally occurring examples of non-pseudomonic naive subcategories (other than those obtained by doing violence to a naturally occurring faithful functor)?
Another way to phrase the question is, what is a good notion of “subobject” in a 2-category? Not that there has to be only one such notion; even in a 1-category there are many (monic, strong monic, regular monic, etc.). But it seems to me that to be worthy of the name “subobject” such a notion should be a conservative extension of some established notion of “subobject” for 1-categories; that is, it should reduce to something that deserves the name “subobject” when interpreted in locally-discrete 2-categories. Full-and-faithful functors and pseudomonic functors both reduce to monomorphisms in a locally-discrete 2-category, but every morphism in a locally-discrete 2-category is faithful.
I’m not disputing the importance of faithful functors, or more generally of faithful morphisms in a 2-category. I just think that calling them “subobjects” is stretching the meaning of the prefix “sub-” so far that it becomes counterintuitive and confusing.
Mathieu says: I agree with the fact that there is a terminological problem : full subcategories should be called “subcategories” and subcategories should be given a new name (“2-subcategories”?). Then a subcategory of a set will be a subset, so the prefix “sub” preserves its original meaning in this case.
An example of non-pseudomonic naive subcategory is given, for a non-skeletal internal category in , by the inclusion of the discrete category on the set of objects into the realisation of as a category.
My answer to the question “what is a good notion of “subobject” in a 2-category?”, at least in the case where every 2-cell is invertible, is that there are two such notions (plus their normal, regular, strong, etc., variations): fully faithful arrows with a given codomain, and faithful arrows with a given codomain. That’s the way I understood it when writing my PhD thesis. 2-dimensional kernels are in general faithful (and not pseudomonic), so I’m inclined to think of the faithful kind of subobjects as the 2-dimensional counterpart of 1-dimensional subobjects.
Toby Bartels: My opinion is more with Mathieu than Mike (although perhaps this is because I don't fully appreciate pseudomonic functors). In fact, I submit that Mathieu's description of how Grp becomes a naïve subcategory of Set is how ordinary mathematicians actually use the language, down to the idea that being a group is a property of a set. On the other hand, since there's a very important sense that being a group is not just a property of a set (that the forgetful functor, while faithful, is not full, not even on isomorphisms), probably we should fight against this abuse of language by not using the word ‘subcategory’ in this naïve sense (hence nor in its equivalence-invariant generalisation). Either Mathieu's version (‘subcategory’ means full subcategory, hence given by a full and faithful functor) or Mike's version (‘subcategory’ means pseudomic subcategory, hence given by a pseudomonic functor) will do this.
Mike: After thinking it over, while I would prefer defining “subcategory” to mean “pseudomonic functor” over defining it to mean “faithful functor,” I would actually most prefer that we not attempt to give “subcategory” an equivalence-invariant definition, because I don’t think there is one that is free of confusion. I would prefer we use some related, but different, terminology, which makes it clear that, as Mathieu says, there are a number of different notions of “subobject” in an -category. If, as Mathieu suggests, we use “2-subcategory” for “faithful functor,” then it would be natural to use “1-subcategory” for “full and faithful functor.” In the appendix of n-categories and cohomology I called faithful functors “1-monic” and full-and-faithful functors “0-monic,” but at the moment I don’t see any compelling reason for starting the numbering at one place or another.
Mathieu: If a “1-subcategory” of is a fully faithful functor with codomain , there is still room for “0-subcategories” (i.e. equivalences with codomain C) and of course there are also “-subcategories”, i.e. any functor with codomain C. But my principal reason for numbering in this way was that 1-monic = monic in the one-dimensional case, like 1-category = category. By the way, here is a family of simple examples of subcategories which are not pseudomonic: any inclusion of a proper subgroup into a group seen as a one-objet groupoid.
Mike: Re: subgroups, point taken; non-pseudomonic subcategories do happen. I think my motivation for starting the numbering as I did was that I was thinking that we should have 0-monic = monic for maps between 0-categories (sets). But at the moment I’m feeling more sympathetic towards your argument that we should have instead 1-monic = monic for maps in a 1-category. Have we reached a point in the discussion where we are willing to change the main entry?
Toby: For what it's worth, I would be happy with that.