2-Category theory

Internal (,1)(\infty,1)-Categories



The notion of 2-congruence is the generalization of the notion of congruence from category theory to 2-category theory.

The correct notions of regularity and exactness for 2-categories is one of the subtler parts of the theory of first-order structure. In particular, we need a suitable replacement for the 1-categorical notion of equivalence relation. The (almost) correct definition was probably first written down in StreetCBS.

One way to express the idea is that in an n-category, every object is internally a (n1)(n-1)-category; exactness says that conversely every “internal (n1)(n-1)-category” is represented by an object. When n=1n=1, an “internal 0-category” means an internal equivalence relation; thus exactness for 1-categories says that every equivalence relation is a kernel (i.e. is represented by some object). Thus, we need to find a good notion of “internal 1-category” in a 2-category.

Of course, there is an obvious notion of an internal category in a 2-category, as a straightforward generalization of internal categories in a 1-category. But internal categories in Cat are double categories, so we need to somehow cut down the double categories to those that really represent honest 1-categories. These are the 2-congruences.


Before we define 2-congruences below in def. 3, we need some preliminaries.



If KK is a finitely complete 2-category, a homwise-discrete category in KK consists of

  • a discrete morphism D 1D 0×D 0D_1\to D_0\times D_0, together

  • with composition and identity maps D 0D 1D_0\to D_1 and D 1× D 0D 1D 1D_1\times_{D_0} D_1\to D_1 in K/(D 0×D 0)K/(D_0\times D_0),

which satisfy the usual axioms of an internal category up to isomorphism.

Together with the evident notions of internal functor and internal natural transformation there is a 2-category HDC(K)HDC(K) of hom-wise discrete 2-categories in KK.


Since D 1D 0×D 0D_1\to D_0\times D_0 is discrete, the structural isomorphisms will automatically satisfy any coherence axioms one might care to impose.


The transformations between functors DED\to E are a version of the notion for internal categories, thus given by a morphism D 0E 1D_0\to E_1 in KK. The 2-cells in K(D 0,E 0)K(D_0,E_0) play no explicit role, but we will recapture them below.


By homwise-discreteness, any “modification” between transformations is necessarily a unique isomorphism, so (after performing some quotienting, if we want to be pedantic) we really have a 2-category HDC(K)HDC(K) rather than a 3-category.


If f:ABf:A\to B is any morphism in KK, there is a canonical homwise-discrete category (f/f)A×A(f/f) \to A\times A, where (f/f)(f/f) is the comma object of ff with itself. We call this the kernel ker(f)ker(f) of ff (the “comma kernel pair” or “comma Cech nerve” of ff).

In particular, if f=1 Af=1_A then (1 A/1 A)=A 2(1_A/1_A) = A^{\mathbf{2}}, so we have a canonical homwise-discrete category A 2A×AA^{\mathbf{2}} \to A\times A called the kernel ker(A)ker(A) of AA.


It is easy to check that taking kernels of objects defines a functor Φ:KHDC(K)\Phi:K \to HDC(K); this might first have been noticed by Street. See prop. 1 below.


If D 1D 0D_1\,\rightrightarrows\, D_0 is a homwise-discrete category in KK, the following are equivalent.

  1. D 0D 1D 0D_0 \leftarrow D_1 \to D_0 is a two-sided fibration in KK.

  2. There is a functor ker(D 0)D\ker(D_0)\to D whose object-map D 0D 0D_0\to D_0 is the identity.

Actually, homwise-discreteness is not necessary for this result, but we include it to avoid worrying about coherence isomorphisms, and since that is the case we are most interested in here.


We consider the case K=CatK=Cat; the general case follows because all the notions are defined representably. A homwise-discrete category in CatCat is, essentially, a double category whose horizontal 2-category is homwise-discrete (hence equivalent to a 1-category). We say “essentially” because the pullbacks and diagrams only commute up to isomorphism, but up to equivalence we may replace D 1D 0×D 0D_1\to D_0\times D_0 by an isofibration, obtaining a (pseudo) double category in the usual sense. Now the key is to compare both properties to a third: the existence of a companion for any vertical arrow.

Suppose first that D 0D 1D 0D_0 \leftarrow D_1 \to D_0 is a two-sided fibration. Then for any (vertical) arrow f:xyf:x\to y in D 0D_0 we have cartesian and opcartesian morphisms (squares) in D 1D_1:

(1)x id x x f 1 y opcart f f cart x f 2 y y id y \array{ x & \overset{id}{\to} & x & \qquad & x & \overset{f_1}{\to} & y' \\ {}^{\mathllap{\cong}}\downarrow & opcart & \downarrow^{\mathrlap{f}} & \qquad & {}^{\mathllap{f}}\downarrow & cart & \downarrow^{\mathrlap{\cong}} \\ x' & \overset{f_2}{\to} & y & \qquad & y & \overset{id}{\to} & y }

The vertical arrows marked as isomorphisms are so by one of the axioms for a two-sided fibration. Moreover, the final compatibility axiom for a 2-sided fibration says that the square

(2)x f 1 y x f 2 y, \array{ x & \overset{f_1}{\to} & y'\\ \cong \downarrow & & \downarrow\cong \\ x' & \overset{f_2}{\to} & y,}

induced by factoring the horizontal identity square of ff through these cartesian and opcartesian squares, must be an isomorphism. We can then show that f 1f_1 (or equivalently f 2f_2) is a companion for ff just as in (Shulman 07, theorem 4.1). Conversely, from a companion pair we can show that D 0D 1D 0D_0 \leftarrow D_1 \to D_0 is a two-sided fibration just as as in loc cit.

The equivalence between the existence of companions and the existence of a functor from the kernel of D 0D_0 is essentially found in (Fiore 06), although stated only for the “edge-symmetric” case. In their language, a kernel ker(A)ker(A) is the double category A\Box A of commutative squares in AA, and a functor ker(D 0)Dker(D_0)\to D which is the identity on D 0D_0 is a thin structure on DD. In one direction, clearly ker(D 0)ker(D_0) has companions, and this property is preserved by any functor ker(D 0)Dker(D_0)\to D. In the other direction, sending any vertical arrow to its horizontal companion is easily checked to define a functor ker(D 0)Dker(D_0)\to D.

In particular, we conclude that up to isomorphism, there can be at most one functor ker(D 0)Dker(D_0)\to D which is the identity on objects.


A 2-congruence in a finitely complete 2-category KK is a homwise-discrete category, def. 1 in KK satisfying the equivalent conditions of Theorem 1.


The kernel ker(A)ker(A), def. 2 of any object is a 2-congruence.

More generally, the kernel ker(f)ker(f) of any morphism is also a 2-congruence.

2-Forks and Quotients

The idea of a 2-fork is to characterize the structure that relates a morphism ff to its kernel ker(f)ker(f). The kernel then becomes the universal 2-fork on ff, while the quotient of a 2-congruence is the couniversal 2-fork constructed from it.


A 2-fork in a 2-category consists of a 2-congruence s,t:D 1D 0s,t:D_1\;\rightrightarrows\; D_0, i:D 0D 1i:D_0\to D_1, c:D 1× D 0D 1D 1c:D_1\times_{D_0} D_1\to D_1, and a morphism f:D 0Xf:D_0\to X, together with a 2-cell ϕ:fsft\phi:f s \to f t such that ϕi=f\phi i = f and such that

D 1× D 0D 1 D 1 = D 1 ϕ D 1 D 0 D 0 ϕ f f D 1 D 0 f X=D 1× D 0D 1 c D 1 D 0 ϕ f D 1 f X.\array{ D_1\times_{D_0} D_1 & \to & D_1 & = & D_1\\ \downarrow && \downarrow & \Downarrow_\phi & \downarrow\\ D_1 & \to & D_0 && D_0\\ || &\Downarrow_\phi && \searrow^f & \downarrow f\\ D_1 & \to & D_0 & \overset{f}{\to} & X } \qquad = \qquad \array{ D_1\times_{D_0} D_1 \\ & \searrow^c\\ && D_1 & \to & D_0\\ && \downarrow & \Downarrow_\phi & \downarrow f\\ && D_1 & \overset{f}{\to} & X. }

The comma square in the definition of the kernel of a morphism f:ABf:A\to B gives a canonical 2-fork

(f/f)AfB.(f/f) \;\rightrightarrows\; A \overset{f}{\to} B.

It is easy to see that any other 2-fork

D 1D 0=AfBD_1 \;\rightrightarrows\; D_0 = A \overset{f}{\to} B

factors through the kernel by an essentially unique functor Dker(f)D \to ker(f) that is the identity on AA.

If D 1D 0fXD_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X is a 2-fork, we say that it equips ff with an action by the 2-congruence DD. If g:D 0Xg:D_0\to X also has an action by DD, say ψ:gsgt\psi:g s \to g t, a 2-cell α:fg\alpha:f\to g is called an action 2-cell if (αt).ϕ=ψ.(αs)(\alpha t).\phi= \psi . (\alpha s). There is an evident category Act(D,X)Act(D,X) of morphisms D 0XD_0\to X equipped with actions.


A quotient for a 2-congruence D 1D 0D_1\;\rightrightarrows\; D_0 in a 2-category KK is a 2-fork D 1D 0qQD_1 \;\rightrightarrows\; D_0 \overset{q}{\to} Q such that for any object XX, composition with qq defines an equivalence of categories

K(Q,X)Act(D,X).K(Q,X) \simeq Act(D,X).

A quotient can also, of course, be defined as a suitable 2-categorical limit.


The quotient qq in any 2-congruence is eso.


If m:ABm\colon A\to B is ff, then the square we must show to be a pullback is

Act(D,A) Act(D,B) K(D 0,A) K(D 0,B)\array{Act(D,A) & \overset{}{\to} & Act(D,B)\\ \downarrow && \downarrow\\ K(D_0,A)& \underset{}{\to} & K(D_0,B)}

But this just says that an action of DD on AA is the same as an action of DD on BB which happens to factor through mm, and this follows directly from the assumption that mm is ff.


A 2-fork D 1D 0fXD_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X is called exact if ff is a quotient of DD and DD is a kernel of ff.

This is the 2-categorical analogue of the notion of exact fork in a 1-category, and plays an analogous role in the definition of a regular 2-category and an exact 2-category.

The 2-category of 2-conguences

There is an evident but naive 2-category of 2-congruences in any 2-category. And there is a refined version where internal functors are replaced by internal anafunctors.


For KK a 2-category, write 2Cong s(K)2Cong_s(K) for the full sub-2-category of that of hom-wise discrete internal categories, def. 1 on the 2-congruences, def. 3

2Cong s(K)HDC(K). 2 Cong_s(K) \hookrightarrow HDC(K) \,.

There is a 2-functor

Φ:K2Cong s(K) \Phi : K\to 2Cong_s(K)

sending each object to its kernel, def. 2.


Let the 2-category KK be equipped with the structure of a 2-site. With this understood, write

2Cong(K) 2 Cong(K)

for the 2-category of 2-congruences with morphisms the anafunctors between them.


The evident inclusion

2Cong s(K)2Cong(K) 2 Cong_s(K) \hookrightarrow 2 Cong(K)

is a homwise-full sub-2-category closed under finite limits.



The opposite of a homwise-discrete category is again a homwise-discrete category. However, the opposite of a 2-congruence in KK is a 2-congruence in K coK^{co}, since 2-cell duals interchange fibrations and opfibrations. Likewise, passage to opposites takes 2-forks in KK to 2-forks in K coK^{co}, and preserves and reflects kernels, quotients, and exactness.


We discuss that when the ambient 2-category KK has finite 2-limits, then its 2-category 2Cong s(K)2 Cong_s(K) of 2-congruences, def. 7 is a regular 2-category. This is theorem 2 below. A sub-2-category of Cong s(K)Cong_s(K) is the regular completion of KK.

In the following and throughout, ”nn” denotes either of (see (n,r)-category)

n=(0,1),(1,1),(2,1),(2,2). n = (0,1), (1,1), (2,1), (2,2) \,.

Suppose that KK has finite 2-limits. Then:

  1. HDC(K)HDC(K) (def. 1) has finite limits.

  2. nCong s(K)n Cong_s(K) is closed under finite limits in HDC(K)HDC(K).

  3. The 2-functor Φ:K2Cong s(K)\Phi : K \to 2 Cong_s(K), prop. 1, is 2-fully-faithful (that is, an equivalence on hom-categories) and preserves finite limits.


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:


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


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.


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/lex2Cong s(K) K_{reg/lex} \hookrightarrow 2 Cong_s(K)

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}.


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.


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 2; 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.


Let now the ambient 2-category KK be equipped with the structure of a 2-site. Recall from def. 8 the 2-category 2Cong(K)2Cong(K) whose objects are 2-congruences in KK, and whose morpisms are internal anafunctors between these, with respect to the given 2-site structure.

Notice that when KK is a regular 2-category it comes with a canonical structure of a 2-site: its regular coverage.


For any subcanonical and finitely complete 2-site KK (such as a regular coverage), the 2-category 2Cong(K)2Cong(K) from def. 8

  • is finitely complete;

  • contains 2Cong s(K)2Cong_s(K), def. 7 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.


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.


If KK is a subcanonical finitely complete nn-site, then the functor Φ:KnCong(K)\Phi:K\to n Cong(K), prop. 1, 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.


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.


If KK is a finitely complete 2-category equipped with its minimal coverage, in which the covering families are those that contain a split epimorphism, then

nCong(K)nCong s(K). n Cong(K) \simeq n Cong_s(K) \,.

This is immediate from the proof of Theorem 2, 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.


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)) \,.

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 5 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.


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).

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 2. 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 7 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).


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.


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 7. 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}.


In GrpdGrpd

Under construction.

Let K:=K := Grpd be the 2-category of groupoids.

We would like to see that the following statement is true:

The 2-category of 2-congruences in GrpdGrpd is equivalent to the 2-category Cat of small categories.

2Cong(Grpd)Cat. 2Cong(Grpd) \simeq Cat \,.

Let’s check:

For CC a small category, construct a 2-congruence \mathbb{C} in GrpdGrpd as follows.

  • let 0:=Core(C)Grpd\mathbb{C}_0 := Core(C) \in Grpd be the core of CC;

  • let 1:=Core(C Δ[1])Grpd\mathbb{C}_1 := Core(C^{\Delta[1]}) \in Grpd be the core of the arrow category of CC;

  • let (s,t): 1 0(s,t) : \mathbb{C}_1 \to \mathbb{C}_0 be image under Core:CatGrpdCore : Cat \to Grpd of the endpoint evaluation functor

C Δ[0]Δ[0]Δ[1]:C Δ[1]C Δ[0]Δ[0]=C×C. C^{\Delta[0] \coprod \Delta[0] \to \Delta[1]} : C^{\Delta[1]} \to C^{\Delta[0] \coprod \Delta[0]} = C \times C \,.

(Here we are using the canonical embedding ΔCat\Delta \hookrightarrow Cat of the simplex category.)

This is clearly a faithful functor. Moreover, every morphism in Grpd is trivially a conservative morphism. So 1 0× 0\mathbb{C}_1 \to \mathbb{C}_0 \times \mathbb{C}_0 is a discrete morphism in Grpd.

Since Grpd is a (2,1)-category, the 2-pullbacks in Grpd are homotopy pullbacks. Using that (s,t)(s,t) is (under the right adjoint nerve embedding N:GrpdsSetN : Grpd \hookrightarrow sSet) a Kan fibration (by direct inspection, but also as a special case of standard facts about the model structure on simplicial sets), the object of composable morphisms is found to be

1× 0 1Core(C Δ[2]). \mathbb{C}_1 \times_{\mathbb{C}_0} \mathbb{C}_1 \simeq Core(C^{\Delta[2]}) \,.

Accordingly, let the internal composition in \mathbb{C} be induced by the given composition in CC:

1× 0 1Core(C Δ[2])Core(C Δ[1]) 1. \mathbb{C}_1 \times_{\mathbb{C}_0} \mathbb{C}_1 \simeq Core(C^{\Delta[2]}) \stackrel{}{\to} Core(C^{\Delta[1]}) \simeq \mathbb{C}_1 \,.

This is clearly associative and unital and hence makes \mathbb{C} a hom-wise discrete category, def. 1, internal to GrpdGrpd.

Observe next (for instance using the discussion and examples at homotopy pullback, see also path object) that

ker( 0)=( 0 Δ[1] 0). ker(\mathbb{C}_0) = ( \mathbb{C}_0^{\Delta[1]} \stackrel{\to}{\to} \mathbb{C}_0) \,.

Notice that up to equivalence of groupoids, this is just the diagonal Δ: 0 0× 0\Delta : \mathbb{C}_0 \to \mathbb{C}_0 \times \mathbb{C}_0.

Therefore there is an evident internal functor ker( 0)ker(\mathbb{C}_0) \to \mathbb{C}, which on the first equivalent incarnation of ker( 0)ker(\mathbb{C}_0) given by the inclusion

ker( 0) 0 Δ[1]Core(C) Δ[1]Core(C Δ[1]), ker(\mathbb{C}_0) \simeq \mathbb{C}_0^{\Delta[1]} \simeq Core(C)^{\Delta[1]} \hookrightarrow Core(C^{\Delta[1]}) \,,

but which in the second version above simply reproduces the identity-assigning morphism of the internal category \mathbb{C}.

It follows that \mathbb{C} is indeed a 2-congruence, def. 3.

Conversely, given a 2-congruence \mathbb{C} in GrpdGrpd, define a category CC as follows:



In the notation of the above proof, we can also form internally the core of \mathbb{C}. This is evidently the internally discrete category 0idid 0\mathbb{C}_0 \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbb{C}_0.

This means that the 2-congruences \mathbb{C} in the above proof are complete Segal spaces

:[n]Core(C Δ[n]), \mathbb{C} : [n] \mapsto Core(C^{\Delta[n]}) \,,

hence are internal categories in an (∞,1)-category in the (2,1)-category Grpd.

In a general (2,1)(2,1)-category



The above material is taken from


Some lemmas are taken from


Revised on November 27, 2012 16:58:23 by Urs Schreiber (