# nLab 2-congruence

Contents

## In higher category theory

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Idea

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 $(n-1)$-category; exactness says that conversely every “internal $(n-1)$-category” is represented by an object. When $n=1$, 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.

## Definition

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

### 2-Congruences

###### Definition

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

• a discrete morphism $D_1\to D_0\times D_0$, together

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

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 $HDC(K)$ of hom-wise discrete 2-categories in $K$.

###### Remark

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

###### Remark

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

###### Remark

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 $HDC(K)$ rather than a 3-category.

###### Definition

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

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

###### Remark

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

###### Theorem

If $D_1\,\rightrightarrows\, D_0$ is a homwise-discrete category in $K$, the following are equivalent.

1. $D_0 \leftarrow D_1 \to D_0$ is a two-sided fibration in $K$.

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

###### Proof

We consider the case $K=Cat$; the general case follows because all the notions are defined representably. A homwise-discrete category in $Cat$ 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 $D_1\to D_0\times D_0$ 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 $D_0 \leftarrow D_1 \to D_0$ is a two-sided fibration. Then for any (vertical) arrow $f:x\to y$ in $D_0$ we have cartesian and opcartesian morphisms (squares) in $D_1$:

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

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

$\array{ x & \overset{f_1}{\to} & y'\\ \cong \downarrow & & \downarrow\cong \\ x' & \overset{f_2}{\to} & y,}$

induced by factoring the horizontal identity square of $f$ through these cartesian and opcartesian squares, must be an isomorphism. We can then show that $f_1$ (or equivalently $f_2$) is a companion for $f$ just as in (Shulman 07, theorem 4.1). Conversely, from a companion pair we can show that $D_0 \leftarrow D_1 \to D_0$ 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 $D_0$ is essentially found in (Fiore 06), although stated only for the “edge-symmetric” case. In their language, a kernel $ker(A)$ is the double category $\Box A$ of commutative squares in $A$, and a functor $ker(D_0)\to D$ which is the identity on $D_0$ is a thin structure on $D$. In one direction, clearly $ker(D_0)$ has companions, and this property is preserved by any functor $ker(D_0)\to D$. In the other direction, sending any vertical arrow to its horizontal companion is easily checked to define a functor $ker(D_0)\to D$.

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

###### Definition

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

###### Example

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

More generally, the kernel $ker(f)$ 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 $f$ to its kernel $ker(f)$. The kernel then becomes the universal 2-fork on $f$, while the quotient of a 2-congruence is the couniversal 2-fork constructed from it.

###### Definition

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

$\array{ D_1\times_{D_0} D_1 & \to & D_1 & = & D_1\\ \downarrow && \downarrow & \Downarrow_\phi & \downarrow\\ D_1 & \to & D_0 && D_0\\ || &\Downarrow_\phi && \searrow^f & \downarrow f\\ D_1 & \to & D_0 & \overset{f}{\to} & X } \qquad = \qquad \array{ D_1\times_{D_0} D_1 \\ & \searrow^c\\ && D_1 & \to & D_0\\ && \downarrow & \Downarrow_\phi & \downarrow f\\ && D_1 & \overset{f}{\to} & X. }$

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

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

It is easy to see that any other 2-fork

$D_1 \;\rightrightarrows\; D_0 = A \overset{f}{\to} B$

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

If $D_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X$ is a 2-fork, we say that it equips $f$ with an action by the 2-congruence $D$. If $g:D_0\to X$ also has an action by $D$, say $\psi:g s \to g t$, a 2-cell $\alpha:f\to g$ is called an action 2-cell if $(\alpha t).\phi= \psi . (\alpha s)$. There is an evident category $Act(D,X)$ of morphisms $D_0\to X$ equipped with actions.

###### Definition

A quotient for a 2-congruence $D_1\;\rightrightarrows\; D_0$ in a 2-category $K$ is a 2-fork $D_1 \;\rightrightarrows\; D_0 \overset{q}{\to} Q$ such that for any object $X$, composition with $q$ defines an equivalence of categories

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

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

###### Lemma

The quotient $q$ in any 2-congruence is eso.

###### Proof

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

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

But this just says that an action of $D$ on $A$ is the same as an action of $D$ on $B$ which happens to factor through $m$, and this follows directly from the assumption that $m$ is ff.

###### Definition

A 2-fork $D_1 \;\rightrightarrows\; D_0 \overset{f}{\to} X$ is called exact if $f$ is a quotient of $D$ and $D$ is a kernel of $f$.

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.

###### Definition

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

$2 Cong_s(K) \hookrightarrow HDC(K) \,.$
###### Proposition

There is a 2-functor

$\Phi : K\to 2Cong_s(K)$

sending each object to its kernel, def. .

###### Definition

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

$2 Cong(K)$

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

###### Remark

The evident inclusion

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

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

## Properties

### Opposites

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

### Regularity

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

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

$n = (0,1), (1,1), (2,1), (2,2) \,.$
###### Lemma

Suppose that $K$ has finite 2-limits. Then:

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

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

3. The 2-functor $\Phi : K \to 2 Cong_s(K)$, prop. , 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 $\Phi(1)$ is a terminal object. If $D$ and $E$ are homwise-discrete categories, define $P_0 = D_0\times E_0$ and $P_1 = D_1\times E_1$; it is easy to check that then $P_1 \;\rightrightarrows\; P_0$ is a homwise-discrete category that is the product $D\times E$ in $HDC(K)$. Since $(D_0\times E_0) ^{\mathbf{2}} \simeq (D_0) ^{\mathbf{2}} \times (E_0) ^{\mathbf{2}}$, and products preserve ffs, we see that $P$ is an $n$-congruence if $D$ and $E$ are and that $\Phi$ preserves products.

For inserters, let $f,g:C \;\rightrightarrows\; D$ be functors in $HDC(K)$, define $i_0:I_0\to C_0$ by the pullback

$\array{I_0 & \to & D_1\\ i_0 \downarrow && \downarrow \\ C_0 & \overset{(f_0,g_0)}{\to} & D_0\times D_0,}$

and define $i_1:I_1 \to C_1$ by the pullback

$\array{I_1 & \to & X\\ i_1\downarrow && \downarrow\\ C_1 & \overset{(f_1,g_1)}{\to} & D_1\times D_1}$

where $X$ is the “object of commutative squares in $D$.” Then $I_1 \;\rightrightarrows\; I_0$ is a homwise-discrete category and $i:I\to C$ is an inserter of $f,g$. Also, $I$ is an $n$-congruence if $C$ is, and $\Phi$ preserves inserters.

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

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

###### Proof

It is easy to see that a functor $f:C\to D$ between $n$-congruences is ff in $n Cong_s(K)$ iff the square

$\array{C_1 & \to & D_1\\ \downarrow && \downarrow\\ C_0\times C_0 & \to & D_0\times D_0}$

is a pullback in $K$.

We claim that if $e:E\to D$ is a functor such that $e_0:E_0\to D_0$ is split (that is, $e_0 s\cong 1_{D_0}$ for some $s:D_0\to E_0$), then $e$ is eso in $n Cong_s(K)$. For if $e\cong f g$ for some ff $f:C\to D$ as above, then we have $g_0 s:D_0 \to C_0$ with $f_0 g_0 s \cong e_0 s \cong 1_{D_0}$, and so the fact that $C_1$ is a pullback induces a functor $h:D\to C$ with $h_0=g_0 s$ and $f h\cong 1_D$. But this implies $f$ is an equivalence; thus $e$ is eso.

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

Now suppose that $f:D\to E$ is any functor in $n Cong_s(K)$. It is easy to see that if we define $Q_0=D_0$ and let $Q_1$ be the pullback

$\array{ Q_1 & \to & E_1 \\ \downarrow && \downarrow\\ Q_0 \times Q_0 & \overset{f_0\times f_0}{\to} & E_0\times E_0}$

then $f \cong m e$ where $e:D \to Q$ and $m:Q\to E$ 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 $C \;\rightrightarrows\; D$ denotes $ker(f)$, then $C$ is the comma object $(f/f)$ and thus we can calculate

$C_0 = D_0\times_{E_0} E_1 \times_{E_0} D_0 \cong Q_1.$

Therefore, if $g:D\to X$ is equipped with an action by $ker(f)$, then the action 2-cell is given by a morphism $Q_1=C_0\to X_1$, and the action axioms evidently make this into a functor $Q\to X$. Thus, $Q$ is a quotient of $ker(f)$, as desired.

###### Remark

There are three “problems” with the 2-category $n Cong_s(K)$.

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 $n Cong_s(K)$ destroys most of the esos and quotients already present in $K$.

The solution to the first problem is straightforward.

###### Definition

If $K$ is a 2-category with finite limits, define

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

to be the sub-2-category of $2 Cong_s(K)$ spanned by the 2-congruences which occur as kernels of morphisms in $K$.

###### Remark

If $K$ is an $n$-category then any such kernel is an $n$-congruence, so in this case $K_{reg/lex}$ is contained in $n Cong_s(K)$ and is an $n$-category. Also, clearly $\Phi$ factors through $K_{reg/lex}$.

###### Theorem

For any finitely complete 2-category $K$, the 2-category $K_{reg/lex}$ is regular?, and the functor $\Phi:K\to K_{reg/lex}$ induces an equivalence

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

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 $2 Cong_s(K)$, and also under the eso-ff factorization constructed in Theorem ; thus it is regular. If $F:K\to L$ 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 $ker(F f)$, 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 $n Cong_s(K)$, we need to modify its morphisms.

### Exactness

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

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

###### Theorem

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

• is finitely complete;

• contains $2Cong_s(K)$, def. as a homwise-full sub-2-category (that is, $2Cong_s(K)(D,E)\hookrightarrow 2Cong(K)(D,E)$ is ff) closed under finite limits.

###### Proof

It is easy to see that products in $2 Cong_S(K)$ remain products in $n Cong(K)$. Before dealing with inserters and equifiers, we observe that if $A\leftarrow F \to B$ is an anafunctor in $2 Cong(K)$ and $e:X_0\to F_0$ is any eso, then pulling back $F_1$ to $X_0\times X_0$ defines a new congruence $X$ and an anafunctor $A \leftarrow X \to B$ which is isomorphic to the original in $2 Cong(K)(A,B)$. Thus, if $A\leftarrow F\to B$ and $A\leftarrow G\to B$ are parallel anafunctors in $2 Cong(K)$, by pulling them both back to $F\times_A G$ we may assume that they are defined by spans with the same first leg, i.e. we have $A\leftarrow X \;\rightrightarrows\; B$.

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

###### Theorem

If $K$ is a subcanonical finitely complete $n$-site, then the functor $\Phi:K\to n Cong(K)$, prop. , is 2-fully-faithful.
If $K$ is an $n$-exact $n$-category equipped with its regular coverage, then

$\Phi : K \to n Cong(K)$
###### Proof

Since $\Phi:K \to n Cong_s(K)$ is 2-fully-faithful and $n Cong_s(K)\to n Cong(K)$ is homwise fully faithful, $\Phi:K \to n Cong(K)$ is homwise fully faithful. For homwise essential-surjectivity, suppose that $ker(A) \leftarrow F \to ker(B)$ is an anafunctor. Then $h:F_0 \to A$ is a cover and $F_1$ is the pullback of $A ^{\mathbf{2}}$ along it; but this just says that $F_1 = (h/h)$. The functor $F\to B$ consists of morphisms $g:F_0\to B$ and $F_1 = (h/h) \to B ^{\mathbf{2}}$, 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 $h:F_0\to A$, which is a cover in a subcanonical 2-site and hence the quotient of this kernel, we have an induced morphism $f:A\to B$ in $K$. It is then easy to check that $f$ is isomorphic, as an anafunctor, to $F$. Thus, $\Phi$ 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 $q:D_0\to Q$, and since $D$ is the kernel of $q$, we have a functor $D \to ker(Q)$ which is a weak equivalence. Thus, we can regard it either as an anafunctor $D\to ker(Q)$ or $ker(Q)\to D$, and it is easy to see that these are inverse equivalences in $n Cong(K)$. Thus, $\Phi$ is essentially surjective, and hence an equivalence.

Note that by working in the generality of 2-sites, this construction includes the previous one.

###### Remark

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

$n Cong(K) \simeq n Cong_s(K) \,.$
###### Proof

This is immediate from the proof of Theorem , which implies that the first leg of any anafunctor relative to this coverage is both eso and ff in $n Cong_s(K)$, and hence an equivalence.

###### Theorem

If $K$ is a 2-exact 2-category with enough groupoids, then

$K\simeq 2 Cong(gpd(K)) \,.$

Likewise, if $K$ is 2-exact and has enough discretes, then

$K\simeq 2 Cong(disc(K)) \,.$
###### Proof

Define a functor $K\to 2Cong(gpd(K))$ by taking each object $A$ to the kernel of $j:J\to A$ 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 $(j/j)\to J\times J$ is discrete, hence $(j/j)$ is also groupoidal. The same argument as in Theorem 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 $n Cong(K)$. The functor $\Phi:K\to n Cong(K)$ is regular, and moreover for any $n$-exact 2-category $L$ it induces an equivalence

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

We already know that $n Cong(K)$ has finite limits and $\Phi$ preserves finite limits. The rest is very similar to Theorem . We first observe that an anafunctor $A \leftarrow F \to B$ is an equivalence as soon as $F\to B$ is also a weak equivalence (its reverse span $B\leftarrow F \to A$ then provides an inverse.) Also, $A \leftarrow F \to B$ is ff if and only if

$\array{F_1 & \to & B_1\\ \downarrow && \downarrow \\ F_0\times F_0 & \to & B_0\times B_0}$

is a pullback.

Now we claim that if $A\leftarrow F \to B$ is an anafunctor such that $F_0\to B_0$ is eso, then $F$ is eso. For if we have a composition

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

such that $M$ is ff, then $F_0\to B_0$ being eso implies that $M_0\to B_0$ is also eso; thus $M\to B$ is a weak equivalence and so $M$ is an equivalence. Moreover, by the construction of pullbacks in $n Cong(K)$, anafunctors with this property are stable under pullback.

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

Finally, given any $A\leftarrow G \to D$ 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 $F\to C$, regularity of $n Cong_s(K)$ gives a descended functor $C\to D$. Thus, $A\leftarrow F \to C$ is the quotient of its kernel; so $n Cong(K)$ is regular.

Finally, if $L$ is $n$-exact, then any functor $K\to L$ induces one $n Cong(K) \to n Cong(L)$, but $n Cong(L)\simeq L$, 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 $1 Cong(K)$ (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 shows that in general, $n Cong(K)$ will be the $n$-exact completion of $K$ whenver it is $n$-exact. However, in general for $n\gt 1$ 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 $n Cong^r(K) = n Cong(n Cong^{r-1}(K))$ and let $K_{n ex/reg} = colim_r n Cong^r(K)$.

###### Theorem

For any regular $n$-category $K$, $K_{n ex/reg}$ is an $n$-exact $n$-category and there is a 2-fully-faithful regular functor $\Phi:K\to K_{n ex/reg}$ that induces an equivalence

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

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 . Thus it remains only to show that $K_{n ex/reg}$ is $n$-exact. But for any $n$-congruence $D_1 \;\rightrightarrows\; D_0$ in $K_{n ex/reg}$, there is some $r$ such that $D_0$ and $D_1$ both live in $n Cong^r(K)$, and thus so does the congruence since $n Cong^r(K)$ sits 2-fully-faithfully in $K_{n ex/reg}$ preserving finite limits. This congruence in $n Cong^r(K)$ is then an object of $n Cong^{r+1}(K)$ which supplies a quotient there, and thus also in $K_{n ex/reg}$.

## Examples

### In $Grpd$

Under construction.

Let $K :=$ 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 $Grpd$ is equivalent to the 2-category Cat of small categories.

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

Let’s check:

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

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

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

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

$C^{\Delta \coprod \Delta \to \Delta} : C^{\Delta} \to C^{\Delta \coprod \Delta} = C \times C \,.$

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

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

Since Grpd is a (2,1)-category, the 2-pullbacks in Grpd are homotopy pullbacks. Using that $(s,t)$ is (under the right adjoint nerve embedding $N : Grpd \hookrightarrow sSet$) 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

$\mathbb{C}_1 \times_{\mathbb{C}_0} \mathbb{C}_1 \simeq Core(C^{\Delta}) \,.$

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

$\mathbb{C}_1 \times_{\mathbb{C}_0} \mathbb{C}_1 \simeq Core(C^{\Delta}) \stackrel{}{\to} Core(C^{\Delta}) \simeq \mathbb{C}_1 \,.$

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

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

$ker(\mathbb{C}_0) = ( \mathbb{C}_0^{\Delta} \stackrel{\to}{\to} \mathbb{C}_0) \,.$

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

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

$ker(\mathbb{C}_0) \simeq \mathbb{C}_0^{\Delta} \simeq Core(C)^{\Delta} \hookrightarrow Core(C^{\Delta}) \,,$

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. .

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

(…)

###### Remark

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

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

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

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

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

(…)

The above material is taken from

and

Some lemmas are taken from

and