replete subcategory


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Replete subcategories


A subcategory DD of a category CC is called replete if it respects isomorphism of morphisms in the arrow category of CC. It is a subcategory for which the property of (strictly) belonging to it respects the principle of equivalence of categories.


A subcategory DD of CC is replete if for any object xx in DD and any isomorphism f:xyf\colon x\cong y in CC, both yy and ff are also in DD. Equivalent ways to state this include:

  • If fDf \in D and fgf \cong g in the arrow category Arr(C)Arr(C), then gDg \in D.

  • The inclusion DCD\hookrightarrow C is an isofibration.

Repletion and replete images

Since repleteness is a “closure condition,” the intersection of any collection of replete subcategories is again replete. Therefore, any subcategory is contained in a smallest replete subcategory, called its repletion. We can construct the repletion repl(D)repl(D) of DCD\subset C explicitly as follows:

  • its objects are those objects of CC which admit an isomorphism to some object of DD, and
  • its morphisms are those morphisms of CC which can be written as a composite of morphisms in DD and isomorphisms in CC.

Repleteness and repletions are most often applied to full subcategories, in which case the repletion is simply the full subcategory of CC determined by those objects which are isomorphic to some object of DD. In particular, in this case, the repletion is equivalent to DD. More generally, we can say:


The inclusion Drepl(D)D\hookrightarrow repl(D) is an equivalence if and only if the inclusion DCD\hookrightarrow C is pseudomonic.


The inclusion of a replete subcategory is always pseudomonic, so if Drepl(D)D\hookrightarrow repl(D) is an equivalence, and in particular full and faithful, then DCD\hookrightarrow C must also be pseudomonic.

Conversely, if DCD\hookrightarrow C is pseudomonic, then every morphism of repl(D)repl(D) can be written as gfh 1g f h^{-1} for some morphism ff in DD and isomorphisms gg and hh in CC. If the domain and codomain are in DD, then since DCD\hookrightarrow C is pseudomonic, both gg and hh must also be in DD, hence such a morphism is itself necessarily in DD. Thus the inclusion Drepl(D)D\hookrightarrow repl(D) is full. It is clearly faithful and essentially surjective, so it is an equivalence.

The replete image of a functor is the repletion of its image. The replete full image of a functor is the repletion of its full image, i.e. the full subcategory of its target determined by those objects isomorphic to some object in its image. See also essential image.

Higher-categorical versions

More generally, for n0n\geq 0, a subcategory DD of an nn-category CC is replete if all equivalences in CC whose either source or target is in DD are themselves in DD. In particular, all kk-cells equivalent in CC to some kk-cell in DD are themselves in DD (because they are sources of some equivalence in DD). Then replete subcategories of 11-categories are those which are replete in this sense.

Here we are using the weakest notion of equivalence; one could also talk about kk-replete nn-subcategories if every kk-equivalences in CC belongs to DD if either its source or target does. Then using the usual definition of ‘category’ as a model for 11-categories, replete subcategories are already those which are 00-replete (indeed, 00-equivalence is isomorphism in this model); but using a model of 11-categories in which hom-sets are setoids (as is common, for example, in type-theoretic foundations), we must insist on 11-repleteness.

Note that a subcategory DD of an nn-category CC is (k1)(k-1)-replete if and only if: * for every (nk)(n-k)-cell in DD all (nk)(n-k)-cells (k1)(k-1)-equivalent to an (nk)(n-k)-cell in DD are themselves in CC, and * for any two (nk)(n-k)-cells x,yx,y the hom kk-category hom(x,y)hom(x,y) is (k2)(k-2)-replete.

Do we want this parametrised equivalence? I see that isomorphism is 00-equivalence and equality is (1)(-1)-equivalence, which is ugly enough but the sort of thing that does happen. But I've never actually heard anybody say that, for example, some objects of a 33-category are 22-equivalent. Why not just say that they're equivalent? (If anything, I'd rather say that they're 22-isomorphic and let ‘equivalent’ mean \infty-isomorphic always.) —Toby

Mike: Definitely, unqualified “equivalent” in an nn-category should mean (n1)(n-1)-equivalent, or equivalently (if you like your nn-categories to be secretly \infty-categories) it should mean \infty-equivalent. I think that is by far the most common usage, so it won’t have to change. However, there are occasionally reasons to speak about stronger sorts of equivalence; for instance in a strict 2-category it is occasionally important to distinguish between isomorphism and equivalence. People do also talk about “biequivalence” for bicategories and “2-equivalence” for 2-categories. I would venture that kk-equivalence in an nn-category for k<n1k\lt n-1 is not likely to be useful unless your nn-category is some sort of semistrict, but in that case it might occasionally turn out to be helpful.

Zoran Škoda: The 00-cells in nn-category are by default (n1)(n-1)-equivalent (what is of course maximal weakness and equals infty-equivalent), while it is possible that they are equivalent in more strict sense, for example isomorphic (there is a strictly invertible 1-arrow in between); for higher cells equivalence is the same as equivalence in lower dimension category which is its hom, thuis the number is not (n1)(n-1) by default for k cells k bigger than 0; of course you are right that all these maximal equivalences are in the same time equivalences in higher sense because there is no difference since higher dimensional cells do not exist different than identities. But regarding that sometimes stricter versions are needed, not only maximal the terminology can be kept precise. For example in Cat one can consider both isomorphic and equivalent categories and these are two different notions; and of course equal categories as the strictest possible notion of (-1)-equivalence. I do not recall if I first read this in print from Leinster’s book, but before that Igor explained me those after his readings of Australian school works.

Toby: Not every concept of nn-category even has a concept of equivalence of objects other than (n1)(n-1)-equivalence (or more generally a concept of equivalence for kk-morphisms other than (nk1)(n-k-1)-equivalence). I find it confusing to put in all of these prefixes to describe what even you, Zoran, agree is the default notion. So I have given this default first, and then added a note about using stricter levels of equivalence when available.

Zoran, you might also want to address these issues at equivalence.

Revised on April 3, 2017 02:20:17 by Urs Schreiber (