nLab
embedding of categories

Contents

Contents

Idea

In much of mathematics, certainly in the traditional treatment of mathematical structures from the point of view of logic and model theory, an “embedding” is typically a monomorphism between structures that preserves and reflects the structure specified within a given language or theory. Such a notion is often definable directly in terms of category theory, i.e., in terms of properties of morphisms in the designated category of structures and structure-preserving maps.

But for categories themselves, the situation and practice is more complicated, with no notion of “embedding of categories” that is universally accepted in the literature. A critical dividing line, leading to several distinct notions of embedding, lies in how one views the category of categories:

  • If we view categories as mathematical structures in the traditional sense, with central concepts being definable in the ordinary 1-category of categories and functors (which are the relevant structure-preserving maps), then notions of embedding tend to emphasize they should be injective maps on the classes of objects (and more besides).

  • If we view categories however as having a higher-dimensional aspect, i.e., if we view central concepts as being based rather on the 2-category of categories, functors, and natural transformations, then different notions of embedding come to the fore, where notably the assumption of injectivity on the object-classes clashes with the principle of equivalence (and is therefore rejected).

In this article we survey the various notions of categorical embedding that have appeared in the literature, and try to describe some of the underlying contexts and rationales.

Definitions

1-categorical definitions

Perhaps the most obvious 1-categorical definition of embedding is:

This definition is adopted in Adamek-Herrlich-Strecker and in Riehl, for example. In this definition we accept that categories are strict categories, where objects can be compared for equality. Embeddings in this sense are straightforwardly the same thing as monomorphisms in the 11-category Cat.

If we define categories with one collection of morphisms, then this definition is equivalent to saying that the action of FF on morphisms is an injective function Mor(C)Mor(D)Mor(C) \to Mor(D). In particular, if this latter condition holds and F(c)=F(c)F(c) = F(c') for objects c,cc, c' of CC, then F(1 c)=F(1 c)F(1_c) = F(1_{c'}) and thus 1 c=1 c1_c = 1_{c'} by injectivity on morphisms, whence c=cc = c'.

However, as noted above, in many cases an “embedding” means something stronger than a mere monomorphism, being required to reflect “all the structure”. For instance, an embedding of topological spaces is a subspace inclusion, not merely an injective continuous map (the latter being the monos in Top).

In particular, the above definition of embedding does not even reflect the property of objects being isomorphic, so that it is not necessarily injective on isomorphism classes of objects! This suggests that a (still 1-categorical) embedding of categories ought to be something stronger. Some possibilities include:

2-categorical definitions

Most of the above 1-categorical notions of embedding corresponds to a “pure” 2-categorical notion by simply removing the injectivity on objects. Thus we have the following candidates:

The choice between these is closely related to the question of giving an equivalence-invariant notion of subcategory.

Relationship

Every fully faithful functor is equivalent to one that is fully faithful and injective on objects. Let f:CDf:C\to D be fully faithful, and let EE be the “mapping cylinder” category, whose objects are the disjoint unions of those of CC and DD, with CC and DD embedded fully-faithfully (and of course injectively on objects), and E(c,d)=D(fc,d)E(c,d) = D(f c,d) and E(d,c)=D(d,fc)E(d,c) = D(d,f c). Full-faithfulness of ff allow us to compose all arrows in EE making it a category. There is a projection EDE\to D that is the identity on DD and acts by ff on CC, and which is an inverse equivalence to the inclusion DED\to E. Thus, f:CDf:C\to D is equivalent, in the 2-category whose objects are functors and whose morphisms are pseudo-commutative squares, to the inclusion functor CEC\to E, which is fully faithful and injective on objects.

Examples

The Yoneda embedding

Probably the most common embedding of categories encountered is the Yoneda embedding. This is fully faithful, but whether or not it is always injective on objects depends on set-theoretic details of how we define categories. If we use a definition of category with a family of collections of morphisms, then it might happen that two distinct (but necessarily isomorphic) objects xx and yy have C(z,x)=C(z,y)C(z,x) = C(z,y) as sets for all zCz\in C, so that the Yoneda embedding would take xx and yy to literally the same presheaf. (Such an equality of hom-sets is not as weird as it might sound; for instance, when regarding a preordered set as a category it’s natural to define every nonempty homset to be the same terminal set?.)

References

Last revised on July 20, 2017 at 14:15:06. See the history of this page for a list of all contributions to it.