nLab
subcategory

Subcategories

Definition

Given a category CC, a subcategory DD consists of a subcollection of the collection of objects of CC and a subcollection of the collection of morphisms of DD such that:

  • If the morphism f:xyf : x \to y is in DD, then so are xx and yy.

  • If f:xyf : x \to y and g:yzg : y \to z are in DD, then so is the composite gf:xzg f : x \to z.

  • If xx is in DD then so is the identity morphism 1 x1_x.

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

  • A full subcategory if for any xx and yy in DD, every morphism f:xyf : x \to y in CC is also in DD (that is, the inclusion functor DCD\hookrightarrow C is full).

  • A replete subcategory if for any xx in DD and any isomorphism f:xyf:x\cong y in CC, both yy and ff are also in DD.

  • A wide subcategory if every object of CC is also an object of DD.

Non-evil variants

Just as subsets of a set XX can be identified with isomorphism classes of monic functions into XX, subcategories of a category CC can be identified with isomorphism classes of monic functors into CC. 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:

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

The obvious generalizations (at least, obvious once you start thinking in terms of kk-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 kk-monomorphisms in any nn-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 CDC\to D in CatCat is equivalent to the inclusion of a full subcategory CDC'\to D (here CC' is the full image of CC). 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.

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Revised on November 18, 2011 10:50:53 by Urs Schreiber (217.232.18.193)