Michael Shulman
exact completion of a 2-category

Regular completion

Recall that there is a 2-category HDC(K) of homwise-discrete categories in any finitely complete 2-category K. We write nCong s(K) for its full sub-2-category spanned by the n-congruences (always we take n= 2, (2,1), (1,2), or 1). Recall that there is a functor Φ:K2Cong s(K) sending each object to its kernel; if K is an n-category then the image of Φ is contained in nCong(K).

Lemma

Suppose that K has finite limits. Then:

  1. HDC(K) has finite limits.
  2. nCong s(K) is closed under finite limits in HDC(K).
  3. Φ is 2-fully-faithful (that is, an equivalence on hom-categories) and preserves finite limits.
Proof

It suffices to deal with finite products, inserters, and equifiers. Evidently Φ(1) is a terminal object. If D and E are homwise-discrete categories, define P 0=D 0×E 0 and P 1=D 1×E 1; it is easy to check that then P 1P 0 is a homwise-discrete category that is the product D×E in HDC(K). Since (D 0×E 0) 2(D 0) 2×(E 0) 2, and products preserve ffs, we see that P is an n-congruence if D and E are and that Φ preserves products.

For inserters, let f,g:CD be functors in HDC(K), define i 0:I 0C 0 by the pullback

I 0 D 1 i 0 C 0 (f 0,g 0) D 0×D 0,\array{I_0 & \to & D_1\\ i_0 \downarrow && \downarrow \\ C_0 & \overset{(f_0,g_0)}{\to} & D_0\times D_0,}

and define i 1:I 1C 1 by the pullback

I 1 X i 1 C 1 (f 1,g 1) D 1×D 1\array{I_1 & \to & X\\ i_1\downarrow && \downarrow\\ C_1 & \overset{(f_1,g_1)}{\to} & D_1\times D_1}

where X is the “object of commutative squares in D.” Then I 1I 0 is a homwise-discrete category and i:IC is an inserter of f,g. Also, I is an n-congruence if C is, and Φ preserves inserters.

Finally, for equifiers, suppose we have functors f,g:CD and 2-cells α,β:fg in HDC(K), represented by morphisms a,b:C 0D 1 such that (s,t)a(f 0,g 0)(s,t)b. Let e 0:E 0C 0 be the universal morphism equipped with an isomorphism ϕ:ae 0be 0 such that (s,t)ϕ is the given isomorphism (s,t)a(s,t)b (this is a finite limit in K.) Note that since (s,t):D 1D 0×D 0 is discrete, e 0 is ff. Now let E 1=(e 0×e 0) *C 1; then E 1E 0 is a homwise-discrete category and e:EC is an equifier of α and β in HDC(K). Also E is an n-congruence if C is, and Φ preserves equifiers.

For any morphism f:AB in K, Φ(f) is the functor ker(A)ker(B) that consists of f:AB and f 2:A 2B 2. A transformation between Φ(f) and Φ(g) is a morphism AB 2 whose composites AB 2B are f and g; but this is just a transformation fg in K. Thus, Φ is homwise fully faithful. And homwise essential-surjectivity follows from the essential uniqueness of thin structures, or equivalently a version of Prop 6.4 in FBMF.

Moreover, we have:

Theorem

If K is an n-category with finite limits, then nCong s(K) is regular.

Proof

It is easy to see that a functor f:CD between n-congruences is ff in nCong s(K) iff the square

C 1 D 1 C 0×C 0 D 0×D 0\array{C_1 & \to & D_1\\ \downarrow && \downarrow\\ C_0\times C_0 & \to & D_0\times D_0}

is a pullback in K.

We claim that if e:ED is a functor such that e 0:E 0D 0 is split (that is, e 0s1 D 0 for some s:D 0E 0), then e is eso in nCong s(K). For if efg for some ff f:CD as above, then we have g 0s:D 0C 0 with f 0g 0se 0s1 D 0, and so the fact that C 1 is a pullback induces a functor h:DC with h 0=g 0s and fh1 D. But this implies f is an equivalence; thus e is eso.

Moreover, if e 0:E 0D 0 is split, then the same is true for any pullback of e. For the pullback of e:ED along some k:CD is given by a P where P 0=E 0× D 0D iso× D 0C 0; here D isoD 1 is the “object of isomorphisms” in D. What matters is that the projection P 0C 0 has a splitting given by combining the splitting of e 0 with the “identities” morphism D 0D iso.

Now suppose that f:DE is any functor in nCong s(K). It is easy to see that if we define Q 0=D 0 and let Q 1 be the pullback

Q 1 E 1 Q 0×Q 0 f 0×f 0 E 0×E 0\array{ Q_1 & \to & E_1 \\ \downarrow && \downarrow\\ Q_0 \times Q_0 & \overset{f_0\times f_0}{\to} & E_0\times E_0}

then fme where e:DQ and m:QE are the obvious functors. Moreover, clearly m is ff, and e satisfies the condition above, so any pullback of it is eso. It follows that if f itself were eso, then it would be equivalent to e, and thus any pullback of it would also be eso; hence esos are stable under pullback.

Since m is ff, the kernel of f is the same as the kernel of e, so to prove K regular it remains only to show that e is a quotient of that kernel. If CD denotes ker(f), then C is the comma object (f/f) and thus we can calculate

C 0=D 0× E 0E 1× E 0D 0Q 1.C_0 = D_0\times_{E_0} E_1 \times_{E_0} D_0 \cong Q_1.

Therefore, if g:DX is equipped with an action by ker(f), then the action 2-cell is given by a morphism Q 1=C 0X 1, and the action axioms evidently make this into a functor QX. Thus, Q is a quotient of ker(f), as desired.

However, there are three problems with the 2-category nCong s(K).

  1. It is too big. It is not necessary to include every n-congruence in order to get a regular category containing K, only those that occur as kernels of morphisms in K.
  2. It is too small. While it is regular, it is not exact.
  3. It doesn’t remember information about K. If K is already regular, then passing to nCong s(K) destroys most of the esos and quotients already present in K.

The solution to the first problem is straightforward. If K is a 2-category with finite limits, define K reg/lex to be the sub-2-category of 2Cong s(K) spanned by the 2-congruences which occur as kernels of morphisms in K. If K is an n-category then any such kernel is an n-congruence, so in this case K reg/lex is contained in nCong s(K) and is an n-category. Also, clearly Φ factors through K reg/lex.

Theorem

For any finitely complete 2-category K, the 2-category K reg/lex is regular, and the functor Φ:KK reg/lex induces an equivalence

Reg(K reg/lex,L)Lex(K,L)Reg(K_{reg/lex},L) \simeq Lex(K,L)

for any regular 2-category K.

Here Reg(,) denotes the 2-category of regular functors, transformations, and modifications between two regular 2-categories, and likewise Lex(,) denotes the 2-category of finite-limit-preserving functors, transformations, and modifications between two finitely complete 2-categories.

Proof

It is easy to verify that K reg/lex is closed under finite limits in 2Cong s(K), and also under the eso-ff factorization constructed in Theorem 1; thus it is regular. If F:KL is a lex functor where L is regular, we extend it to K reg/lex by sending ker(f) to the quotient in L of ker(Ff), which exists since L is regular. It is easy to verify that this is regular and is the unique regular extension of F.

In particular, if K is a regular 1-category, K reg/lex is the ordinary regular completion of K. In this case our construction reduces to one of the usual constructions (see, for example, the Elephant).

To solve the second and third problems with nCong s(K), we need to modify its morphisms.

Exact completion

Recall that 2-congruences in Cat can be identified with certain double categories. As noted in PAPDC, edge-symmetric double categories with a thin structure are essentially the same as 2-categories, and homwise-discreteness makes them the same as 1-categories. Our lack of edge-symmetry means that we really have a 1-category with distinguished subclass of morphisms (the vertical ones), which must be preserved by functors between congruences. (Note that the transformations are “horizontal” and need not have distinguished components. Since every vertical arrow has a horizontal companion, any vertical transformation is represented by a horizontal one.) In order to eliminate the effect of the distinguished vertical morphisms, we can replace functors between congruences by anafunctors.

Definition

Suppose that K is a finitely complete 2-site and that D, E, and F are 2-congruences in K. A functor g:FD is a weak equivalence if 1. the square

F 1 g 1 D 1 F 0×F 0 g 0×g 0 D 0×D 0\array{F_1 &\overset{g_1}{\to} & D_1 \\ \downarrow && \downarrow\\ F_0\times F_0 & \overset{g_0\times g_0}{\to} & D_0\times D_0}

is a pullback, and 1. g 0:F 0D 0 is a cover (a one-element covering family). An anafunctor DE is a span of functors Df sFf tE such that f s is a weak equivalence.

The primary example we have in mind is when K is a regular 2-category with its regular coverage, but it is useful to consider the general case.

Definition

If DFE and DGE are anafunctors between 2-congruences, then a transformation FG is a transformation between the two induced functors F× DGE.

(Here F× DG denotes the pullback in 2Cong s(K).)

Theorem

For any subcanonical and finitely complete 2-site K (such as a regular n-category with its regular coverage), there is a finitely complete 2-category 2Cong(K) of 2-congruences, anafunctors, and transformations in K. It contains 2Cong s(K) as a homwise-full sub-2-category (that is, 2Cong s(K)(D,E)2Cong(K)(D,E) is ff) closed under finite limits.

Proof

Composition is, of course, by pullback. Since covers are stable under pullback and composition, the composite of anafunctors is again an anafunctor. The coverage must be subcanonical in order to define the vertical composite of natural transformations. We regard a functor as an anafunctor by taking f s to be the identity; it is then clear that a transformation between functors is the same as a transformation between their corresponding anafunctors.

It is easy to see that products in 2Cong S(K) remain products in nCong(K). Before dealing with inserters and equifiers, we observe that if AFB is an anafunctor in 2Cong(K) and e:X 0F 0 is any eso, then pulling back F 1 to X 0×X 0 defines a new congruence X and an anafunctor AXB which is isomorphic to the original in 2Cong(K)(A,B). Thus, if AFB and AGB are parallel anafunctors in 2Cong(K), by pulling them both back to F× AG we may assume that they are defined by spans with the same first leg, i.e. we have AXB.

Now, for the inserter of F and G as above, let EX be the inserter of XB in 2Cong s(K). It is easy to check that the composite EXA is an inserter of F,G in 2Cong(K). Likewise, given α,β:FG with F and G as above, we have transformations between the two functors XB in 2Cong s(K), and it is again easy to check that their equifier in 2Cong s(K) is again the equifier in 2Cong(K) of the original 2-cells α,β. Thus, 2Cong(K) has finite limits. Finally, by construction clearly the inclusion of 2Cong s(K) preserves finite limits.

We write nCong(K) for the full sub-2-category of 2Cong(K) on the n-congruences, which is a finitely complete n-category. Of course, it contains nCong s(K) as a homwise-full sub-n-category closed under finite limits, and when K is an n-category we have Φ:KnCong(K).

Theorem

If K is a subcanonical finitely complete n-site, then the functor Φ:KnCong(K) is 2-fully-faithful. If K is an n-exact n-category equipped with its regular coverage, then Φ:KnCong(K) is an equivalence of 2-categories.

Proof

Since Φ:KnCong s(K) is 2-fully-faithful and nCong s(K)nCong(K) is homwise fully faithful, Φ:KnCong(K) is homwise fully faithful. For homwise essential-surjectivity, suppose that ker(A)Fker(B) is an anafunctor. Then h:F 0A is a cover and F 1 is the pullback of A 2 along it; but this just says that F 1=(h/h). The functor FB consists of morphisms g:F 0B and F 1=(h/h)B 2, and functoriality says precisely that the resulting 2-cell equips g with an action by the congruence F. But since F is precisely the kernel of h:F 0A, which is a cover in a subcanonical 2-site and hence the quotient of this kernel, we have an induced morphism f:AB in K. It is then easy to check that f is isomorphic, as an anafunctor, to F. Thus, Φ is homwise an equivalence.

Now suppose that K is an n-exact n-category and that D is an n-congruence. Since K is n-exact, D has a quotient q:D 0Q, and since D is the kernel of q, we have a functor Dker(Q) which is a weak equivalence. Thus, we can regard it either as an anafunctor Dker(Q) or ker(Q)D, and it is easy to see that these are inverse equivalences in nCong(K). Thus, Φ is essentially surjective, and hence an equivalence.

Note that by working in the generality of 2-sites, this construction includes the previous one. Specifically, if K is a finitely complete 2-category equipped with its minimal coverage, in which the covering families are those that contain a split epic, then nCong(K)nCong s(K). This is immediate from the proof of Theorem 1, which implies that the first leg of any anafunctor relative to this coverage is both eso and ff in nCong s(K), and hence an equivalence.

We also remark in passing that this allows us to reconstruct 2-exact 2-categories with enough groupoids or discretes from their subcategories of such.

Theorem

If K is a 2-exact 2-category with enough groupoids, then K2Cong(gpd(K)). Likewise, if K is 2-exact and has enough discretes, then K2Cong(disc(K)).

Proof

Define a functor K2Cong(gpd(K)) by taking each object A to the kernel of j:JA where j is eso and J is groupoidal (for example, it might be the core of A). Note that this kernel lives in 2Cong(gpd(K)) since (j/j)J×J is discrete, hence (j/j) is also groupoidal. The same argument as in Theorem 4 shows that this functor is 2-fully-faithful for any regular 2-category K with enough groupoids, and essentially-surjective when K is 2-exact; thus it is an equivalence. The same argument works for discrete objects.

In particular, the 2-exact 2-categories having enough discretes are precisely the 2-categories of internal categories and anafunctors in 1-exact 1-categories.

Our final goal is to construct the n-exact completion of a regular n-category, and a first step towards that is the following.

Theorem

If K is a regular n-category, so is nCong(K). The functor Φ:KnCong(K) is regular, and moreover for any n-exact 2-category L it induces an equivalence

Reg(nCong(K),L)Reg(K,L).Reg(n Cong(K), L) \to Reg(K,L).
Proof

We already know that nCong(K) has finite limits and Φ preserves finite limits. The rest is very similar to Theorem 1. We first observe that an anafunctor AFB is an equivalence as soon as FB is also a weak equivalence (its reverse span BFA then provides an inverse.) Also, AFB is ff if and only if

F 1 B 1 F 0×F 0 B 0×B 0\array{F_1 & \to & B_1\\ \downarrow && \downarrow \\ F_0\times F_0 & \to & B_0\times B_0}

is a pullback.

Now we claim that if AFB is an anafunctor such that F 0B 0 is eso, then F is eso. For if we have a composition

F G M A C B\array{ &&&& F \\ &&& \swarrow && \searrow\\ && G &&&& M\\ & \swarrow && \searrow && \swarrow && \searrow\\ A &&&& C &&&& B}

such that M is ff, then F 0B 0 being eso implies that M 0B 0 is also eso; thus MB is a weak equivalence and so M is an equivalence. Moreover, by the construction of pullbacks in nCong(K), anafunctors with this property are stable under pullback.

Now suppose that AFB is any anafunctor, and define C 0=F 0 and let C 1 be the pullback of B 1 to C 0×C 0 along C 0=F 0toB 0. Then C is an n-congruence, CB is ff in nCong s(K) and thus also in nCong(K), and AFB factors through C. (In fact, C is the image of FB in nCong s(K).) The kernel of AFB can equally well be calculated as the kernel of FB, which is the same as the kernel of FC.

Finally, given any AGD with an action by this kernel, we may as well assume (by pullbacks) that F=G (which leaves C unchanged up to equivalence). Then since the kernel acting is the same as the kernel of FC, regularity of nCong s(K) gives a descended functor CD. Thus, AFC is the quotient of its kernel; so nCong(K) is regular.

Finally, if L is n-exact, then any functor KL induces one nCong(K)nCong(L), but nCong(L)L, so we have our extension, which it can be shown is unique up to equivalence.

When K is a regular 1-category, it is well-known that 1Cong(K) (which, in that case, is the category of internal equivalence relations and functional relations) is the 1-exact completion of K (the reflection of K from regular 1-categories into 1-exact 1-categories). Theorem 6 shows that in general, nCong(K) will be the n-exact completion of K whenver it is n-exact. However, in general for n>1 we need to “build up exactness” in stages by iterating this construction.

It is possible that the iteration will converge at some finite stage, but for now, define nCong r(K)=nCong(nCong r1(K)) and let K nex/reg=colim rnCong r(K).

Theorem

For any regular n-category K, K nex/reg is an n-exact n-category and there is a 2-fully-faithful regular functor Φ:KK nex/reg that induces an equivalence

Reg(K nex/reg,L)Reg(K,L)Reg(K_{n ex/reg},L) \simeq Reg(K,L)

for any n-exact 2-category L.

Proof

Sequential colimits preserve 2-fully-faithful functors as well as functors that preserve finite limits and quotients, and the final statement follows easily from Theorem 6. Thus it remains only to show that K nex/reg is n-exact. But for any n-congruence D 1D 0 in K nex/reg, there is some r such that D 0 and D 1 both live in nCong r(K), and thus so does the congruence since nCong r(K) sits 2-fully-faithfully in K nex/reg preserving finite limits. This congruence in nCong r(K) is then an object of nCong r+1(K) which supplies a quotient there, and thus also in K nex/reg.

Revised on February 6, 2009 05:34:31 by Mike Shulman (75.3.140.11)