Michael Shulman
exact completion of a 2-category

Regular completion

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

Lemma

Suppose that KK has finite limits. Then: 1. HDC(K)HDC(K) has finite limits. 1. nCong s(K)n Cong_s(K) is closed under finite limits in HDC(K)HDC(K). 1. Φ\Phi 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)\Phi(1) is a terminal object. If DD and EE are homwise-discrete categories, define P 0=D 0×E 0P_0 = D_0\times E_0 and P 1=D 1×E 1P_1 = D_1\times E_1; it is easy to check that then P 1P 0P_1 \;\rightrightarrows\; P_0 is a homwise-discrete category that is the product D×ED\times E in HDC(K)HDC(K). Since (D 0×E 0) 2(D 0) 2×(E 0) 2(D_0\times E_0) ^{\mathbf{2}} \simeq (D_0) ^{\mathbf{2}} \times (E_0) ^{\mathbf{2}}, and products preserve ffs, we see that PP is an nn-congruence if DD and EE are and that Φ\Phi preserves products.

For inserters, let f,g:CDf,g:C \;\rightrightarrows\; D be functors in HDC(K)HDC(K), define i 0:I 0C 0i_0:I_0\to C_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 1i_1:I_1 \to C_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 XX is the “object of commutative squares in DD.” Then I 1I 0I_1 \;\rightrightarrows\; I_0 is a homwise-discrete category and i:ICi:I\to C is an inserter of f,gf,g. Also, II is an nn-congruence if CC is, and Φ\Phi preserves inserters.

Finally, for equifiers, suppose we have functors f,g:CDf,g:C \;\rightrightarrows\; D and 2-cells α,β:fg\alpha,\beta:f \;\rightrightarrows\; g in HDC(K)HDC(K), represented by morphisms a,b:C 0D 1a,b:C_0 \;\rightrightarrows\; D_1 such that (s,t)a(f 0,g 0)(s,t)b(s,t) a \cong (f_0,g_0)\cong (s,t) b. Let e 0:E 0C 0e_0:E_0\to C_0 be the universal morphism equipped with an isomorphism ϕ:ae 0be 0\phi:a e_0 \cong b e_0 such that (s,t)ϕ(s,t)\phi is the given isomorphism (s,t)a(s,t)b(s,t) a\cong (s,t) b (this is a finite limit in KK.) Note that since (s,t):D 1D 0×D 0(s,t):D_1\to D_0\times D_0 is discrete, e 0e_0 is ff. Now let E 1=(e 0×e 0) *C 1E_1 = (e_0\times e_0)^*C_1; then E 1E 0E_1 \;\rightrightarrows\; E_0 is a homwise-discrete category and e:ECe:E\to C is an equifier of α\alpha and β\beta in HDC(K)HDC(K). Also EE is an nn-congruence if CC is, and Φ\Phi preserves equifiers.

For any morphism f:ABf:A\to B in KK, Φ(f)\Phi(f) is the functor ker(A)ker(B)ker(A)\to ker(B) that consists of f:ABf:A\to B and f 2:A 2B 2f^{\mathbf{2}}: A^{\mathbf{2}} \to B^{\mathbf{2}}. A transformation between Φ(f)\Phi(f) and Φ(g)\Phi(g) is a morphism AB 2A\to B ^{\mathbf{2}} whose composites AB 2BA\to B ^{\mathbf{2}} \;\rightrightarrows\; B are ff and gg; but this is just a transformation fgf\to g in KK. Thus, Φ\Phi 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 KK is an nn-category with finite limits, then nCong s(K)n Cong_s(K) is regular.

Proof

It is easy to see that a functor f:CDf:C\to D between nn-congruences is ff in nCong s(K)n Cong_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 KK.

We claim that if e:EDe:E\to D is a functor such that e 0:E 0D 0e_0:E_0\to D_0 is split (that is, e 0s1 D 0e_0 s\cong 1_{D_0} for some s:D 0E 0s:D_0\to E_0), then ee is eso in nCong s(K)n Cong_s(K). For if efge\cong f g for some ff f:CDf:C\to D as above, then we have g 0s:D 0C 0g_0 s:D_0 \to C_0 with f 0g 0se 0s1 D 0f_0 g_0 s \cong e_0 s \cong 1_{D_0}, and so the fact that C 1C_1 is a pullback induces a functor h:DCh:D\to C with h 0=g 0sh_0=g_0 s and fh1 Df h\cong 1_D. But this implies ff is an equivalence; thus ee is eso.

Moreover, if e 0:E 0D 0e_0:E_0\to D_0 is split, then the same is true for any pullback of ee. For the pullback of e:EDe:E\to D along some k:CDk:C\to D is given by a PP where P 0=E 0× D 0D iso× D 0C 0P_0 = E_0 \times_{D_0} D_{iso} \times_{D_0} C_0; here D isoD 1D_{iso}\hookrightarrow D_1 is the “object of isomorphisms” in DD. What matters is that the projection P 0C 0P_0\to C_0 has a splitting given by combining the splitting of e 0e_0 with the “identities” morphism D 0D isoD_0\to D_{iso}.

Now suppose that f:DEf:D\to E is any functor in nCong s(K)n Cong_s(K). It is easy to see that if we define Q 0=D 0Q_0=D_0 and let Q 1Q_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 fmef \cong m e where e:DQe:D \to Q and m:QEm:Q\to E are the obvious functors. Moreover, clearly mm is ff, and ee satisfies the condition above, so any pullback of it is eso. It follows that if ff itself were eso, then it would be equivalent to ee, and thus any pullback of it would also be eso; hence esos are stable under pullback.

Since mm is ff, the kernel of ff is the same as the kernel of ee, so to prove KK regular it remains only to show that ee is a quotient of that kernel. If CDC \;\rightrightarrows\; D denotes ker(f)ker(f), then CC is the comma object (f/f)(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:DXg:D\to X is equipped with an action by ker(f)ker(f), then the action 2-cell is given by a morphism Q 1=C 0X 1Q_1=C_0\to X_1, and the action axioms evidently make this into a functor QXQ\to X. Thus, QQ is a quotient of ker(f)ker(f), as desired.

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

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

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

Theorem

For any finitely complete 2-category KK, the 2-category K reg/lexK_{reg/lex} is regular, and the functor Φ:KK reg/lex\Phi:K\to K_{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 KK.

Here Reg(,)Reg(-,-) denotes the 2-category of regular functors, transformations, and modifications between two regular 2-categories, and likewise Lex(,)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/lexK_{reg/lex} is closed under finite limits in 2Cong s(K)2 Cong_s(K), and also under the eso-ff factorization constructed in Theorem 1; thus it is regular. If F:KLF:K\to L is a lex functor where LL is regular, we extend it to K reg/lexK_{reg/lex} by sending ker(f)ker(f) to the quotient in LL of ker(Ff)ker(F f), which exists since LL is regular. It is easy to verify that this is regular and is the unique regular extension of FF.

In particular, if KK is a regular 1-category, K reg/lexK_{reg/lex} is the ordinary regular completion of KK. 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)n Cong_s(K), we need to modify its morphisms.

Exact completion

Recall that 2-congruences in CatCat 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 KK is a finitely complete 2-site and that DD, EE, and FF are 2-congruences in KK. A functor g:FDg:F\to D 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 0g_0:F_0\to D_0 is a cover (a one-element covering family). An anafunctor DED\to E is a span of functors Df sFf tED \overset{f^s}{\leftarrow} F \overset{f^t}{\to} E such that f sf^s is a weak equivalence.

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

Definition

If DFED \leftarrow F \to E and DGED \leftarrow G \to E are anafunctors between 2-congruences, then a transformation FGF\to G is a transformation between the two induced functors F× DGEF\times_D G\;\rightrightarrows\; E.

(Here F× DGF\times_D G denotes the pullback in 2Cong s(K)2 Cong_s(K).)

Theorem

For any subcanonical and finitely complete 2-site KK (such as a regular nn-category with its regular coverage), there is a finitely complete 2-category 2Cong(K)2Cong(K) of 2-congruences, anafunctors, and transformations in KK. It contains 2Cong s(K)2Cong_s(K) as a homwise-full sub-2-category (that is, 2Cong s(K)(D,E)2Cong(K)(D,E)2Cong_s(K)(D,E)\hookrightarrow 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 sf^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)2 Cong_S(K) remain products in nCong(K)n Cong(K). Before dealing with inserters and equifiers, we observe that if AFBA\leftarrow F \to B is an anafunctor in 2Cong(K)2 Cong(K) and e:X 0F 0e:X_0\to F_0 is any eso, then pulling back F 1F_1 to X 0×X 0X_0\times X_0 defines a new congruence XX and an anafunctor AXBA \leftarrow X \to B which is isomorphic to the original in 2Cong(K)(A,B)2 Cong(K)(A,B). Thus, if AFBA\leftarrow F\to B and AGBA\leftarrow G\to B are parallel anafunctors in 2Cong(K)2 Cong(K), by pulling them both back to F× AGF\times_A G we may assume that they are defined by spans with the same first leg, i.e. we have AXBA\leftarrow X \;\rightrightarrows\; B.

Now, for the inserter of FF and GG as above, let EXE\to X be the inserter of XBX \;\rightrightarrows\; B in 2Cong s(K)2 Cong_s(K). It is easy to check that the composite EXAE\to X \to A is an inserter of F,GF,G in 2Cong(K)2 Cong(K). Likewise, given α,β:FG\alpha,\beta: F \;\rightrightarrows\; G with FF and GG as above, we have transformations between the two functors XBX \;\rightrightarrows\; B in 2Cong s(K)2 Cong_s(K), and it is again easy to check that their equifier in 2Cong s(K)2 Cong_s(K) is again the equifier in 2Cong(K)2 Cong(K) of the original 2-cells α,β\alpha,\beta. Thus, 2Cong(K)2 Cong(K) has finite limits. Finally, by construction clearly the inclusion of 2Cong s(K)2 Cong_s(K) preserves finite limits.

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

Theorem

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

Proof

Since Φ:KnCong s(K)\Phi:K \to n Cong_s(K) is 2-fully-faithful and nCong s(K)nCong(K)n Cong_s(K)\to n Cong(K) is homwise fully faithful, Φ:KnCong(K)\Phi:K \to n Cong(K) is homwise fully faithful. For homwise essential-surjectivity, suppose that ker(A)Fker(B)ker(A) \leftarrow F \to ker(B) is an anafunctor. Then h:F 0Ah:F_0 \to A is a cover and F 1F_1 is the pullback of A 2A ^{\mathbf{2}} along it; but this just says that F 1=(h/h)F_1 = (h/h). The functor FBF\to B consists of morphisms g:F 0Bg:F_0\to B and F 1=(h/h)B 2F_1 = (h/h) \to B ^{\mathbf{2}}, and functoriality says precisely that the resulting 2-cell equips gg with an action by the congruence FF. But since FF is precisely the kernel of h:F 0Ah:F_0\to A, which is a cover in a subcanonical 2-site and hence the quotient of this kernel, we have an induced morphism f:ABf:A\to B in KK. It is then easy to check that ff is isomorphic, as an anafunctor, to FF. Thus, Φ\Phi is homwise an equivalence.

Now suppose that KK is an nn-exact nn-category and that DD is an nn-congruence. Since KK is nn-exact, DD has a quotient q:D 0Qq:D_0\to Q, and since DD is the kernel of qq, we have a functor Dker(Q)D \to ker(Q) which is a weak equivalence. Thus, we can regard it either as an anafunctor Dker(Q)D\to ker(Q) or ker(Q)Dker(Q)\to D, and it is easy to see that these are inverse equivalences in nCong(K)n Cong(K). Thus, Φ\Phi 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 KK 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)n Cong(K) \simeq n Cong_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)n Cong_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 KK is a 2-exact 2-category with enough groupoids, then K2Cong(gpd(K))K\simeq 2 Cong(gpd(K)). Likewise, if KK is 2-exact and has enough discretes, then K2Cong(disc(K))K\simeq 2 Cong(disc(K)).

Proof

Define a functor K2Cong(gpd(K))K\to 2Cong(gpd(K)) by taking each object AA to the kernel of j:JAj:J\to A where jj is eso and JJ is groupoidal (for example, it might be the core of AA). Note that this kernel lives in 2Cong(gpd(K))2Cong(gpd(K)) since (j/j)J×J(j/j)\to J\times J is discrete, hence (j/j)(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 KK with enough groupoids, and essentially-surjective when KK 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 nn-exact completion of a regular nn-category, and a first step towards that is the following.

Theorem

If KK is a regular nn-category, so is nCong(K)n Cong(K). The functor Φ:KnCong(K)\Phi:K\to n Cong(K) is regular, and moreover for any nn-exact 2-category LL 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)n Cong(K) has finite limits and Φ\Phi preserves finite limits. The rest is very similar to Theorem 1. We first observe that an anafunctor AFBA \leftarrow F \to B is an equivalence as soon as FBF\to B is also a weak equivalence (its reverse span BFAB\leftarrow F \to A then provides an inverse.) Also, AFBA \leftarrow F \to B 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 AFBA\leftarrow F \to B is an anafunctor such that F 0B 0F_0\to B_0 is eso, then FF 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 MM is ff, then F 0B 0F_0\to B_0 being eso implies that M 0B 0M_0\to B_0 is also eso; thus MBM\to B is a weak equivalence and so MM is an equivalence. Moreover, by the construction of pullbacks in nCong(K)n Cong(K), anafunctors with this property are stable under pullback.

Now suppose that AFBA \leftarrow F \to B is any anafunctor, and define C 0=F 0C_0=F_0 and let C 1C_1 be the pullback of B 1B_1 to C 0×C 0C_0\times C_0 along C 0=F 0toB 0C_0 = F_0 to B_0. Then CC is an nn-congruence, CBC\to B is ff in nCong s(K)n Cong_s(K) and thus also in nCong(K)n Cong(K), and AFBA \leftarrow F \to B factors through CC. (In fact, CC is the image of FBF\to B in nCong s(K)n Cong_s(K).) The kernel of AFBA\leftarrow F\to B can equally well be calculated as the kernel of FBF\to B, which is the same as the kernel of FCF\to C.

Finally, given any AGDA\leftarrow G \to D with an action by this kernel, we may as well assume (by pullbacks) that F=GF=G (which leaves CC unchanged up to equivalence). Then since the kernel acting is the same as the kernel of FCF\to C, regularity of nCong s(K)n Cong_s(K) gives a descended functor CDC\to D. Thus, AFCA\leftarrow F \to C is the quotient of its kernel; so nCong(K)n Cong(K) is regular.

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

When KK is a regular 1-category, it is well-known that 1Cong(K)1 Cong(K) (which, in that case, is the category of internal equivalence relations and functional relations) is the 1-exact completion of KK (the reflection of KK from regular 1-categories into 1-exact 1-categories). Theorem 6 shows that in general, nCong(K)n Cong(K) will be the nn-exact completion of KK whenver it is nn-exact. However, in general for n>1n\gt 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))n Cong^r(K) = n Cong(n Cong^{r-1}(K)) and let K nex/reg=colim rnCong r(K)K_{n ex/reg} = colim_r n Cong^r(K).

Theorem

For any regular nn-category KK, K nex/regK_{n ex/reg} is an nn-exact nn-category and there is a 2-fully-faithful regular functor Φ:KK nex/reg\Phi:K\to K_{n ex/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 nn-exact 2-category LL.

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/regK_{n ex/reg} is nn-exact. But for any nn-congruence D 1D 0D_1 \;\rightrightarrows\; D_0 in K nex/regK_{n ex/reg}, there is some rr such that D 0D_0 and D 1D_1 both live in nCong r(K)n Cong^r(K), and thus so does the congruence since nCong r(K)n Cong^r(K) sits 2-fully-faithfully in K nex/regK_{n ex/reg} preserving finite limits. This congruence in nCong r(K)n Cong^r(K) is then an object of nCong r+1(K)n Cong^{r+1}(K) which supplies a quotient there, and thus also in K nex/regK_{n ex/reg}.

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