¹¹institutetext: SpringerNature,
¹¹email: fdilke@gmail.com,
WWW home page: http://github.com/fdilke/bewl

Injective Hulls In a Locally Finite Topos

Felix Dilke 11

Abstract

We show that in a locally finite topos, every object has an essential extension that is injective, and that this extension is unique up to isomorphism.

The construction was motivated by work on Bewl, a software project for doing topos-theoretic calculations.

Keywords:

topos, category theory, injective

1 Motivation

These results emerged from trying to construct injective hulls in software using Bewl (http://github.com/fdilke/bewl), a software domain-specific language for topos theory based on the Scala programming language.

It seemed straightforward to use iterative algorithms to construct a minimal injective extension for any object. The question was how this can be theoretically justified. Of course, because topoi in Bewl are modelled on a finite computer, it is more or less enforced that they all satisfy the condition of local finiteness.

So the results below show that this condition alone ensures that injective hulls exist in such a topos.

2 Preliminaries

2.1 Locally finite categories

Let $\mathcal{C}$ be a category which is locally finite, i.e. in which for any two objects A, B there are only finitely many arrows A $\rightarrow$ B.

Lemma 1

Let A $\in\mathcal{C}$ , $f:A\rightarrow A$ monic. Then $f$ is an automorphism of A.

Proof

Since all powers $f^{n}:A\rightarrow A$ lie in the finite set $Hom(A,A)$ , we can find $m,n\geq 0\in\bbbn$ with $f^{m+n+1}=f^{m}$ . Since $f^{m}$ is monic, $f^{n+1}=1_{A}$ and so ${f^{n}}$ is a two-sided inverse of $f$ . ∎

Applying this result to the dual category $\mathcal{C}^{op}$ , which is also locally finite, we note

Corollary 1

Any epic endomorphism of a $\mathcal{C}$ -object is iso. ∎

We also deduce a version of the Schröder-Bernstein theorem for $\mathcal{C}$ :

Theorem 2.1

Let A, B $\in\mathcal{C}$ with monics $f:A\rightarrow B$ , $g:B\rightarrow A$ . Then f, g are iso. In particular, $A\cong B$ .

Proof

By the lemma, the monics $fg$ and $gf$ are both iso, so $f$ is both left- and right-invertible, hence iso. Similarly for $g$ . ∎

Lemma 2

Let $A_{0}\rightarrow A_{1}\rightarrow\cdots\rightarrow A_{n}$ be a sequence of monics in $\mathcal{C}$ whose endpoints are isomorphic, $A_{0}\cong A_{n}$ . Then each arrow in the sequence is iso.

Proof

It is enough to prove this for $n=2$ .

So suppose $p:A_{0}\rightarrow A_{1},q:A_{1}\rightarrow A_{2}$ are monic, $r:A_{2}\rightarrow A_{0}$ an isomorphism. Then $qpr$ is an monic endomorphism of $A_{2}$ , hence an isomorphism by Lemma 1. Let $s$ be its inverse. Then $qprs=1_{A_{2}}$ . But then $qprsq=q$ , so cancelling $q$ from the left, $prsq=1_{A_{1}}$ . It follows that $q$ , $prs$ are mutual two-sided inverses. Hence $q$ is iso, and since $r$ , $s$ are both iso, it also follows that $p$ is iso. ∎

Dually, we have

Lemma 3

Let $A_{0}\rightarrow A_{1}\rightarrow\cdots\rightarrow A_{n}$ be a sequence of epimorphisms in $\mathcal{C}$ whose endpoints are isomorphic, $A_{0}\cong A_{n}$ . Then each arrow in the sequence is iso. ∎

2.2 Locally finite topoi

Now let $\mathcal{E}$ be a locally finite topos. We shall need some basic results about sequences of monics and epics in $\mathcal{E}$ .

Lemma 4

Each $A\in\mathcal{E}$ has only finitely many isomorphism classes of epimorphic images. More precisely, there is a finite set $\{I_{i}\in\mathcal{E}\}_{i=0}^{n}$ such that for any epi $f:A\rightarrow I$ , we have some $I_{i}\cong I$ .

Proof

We recall that in a topos every epi is the coequalizer of its kernel pair (theorem IV.7.8 in [1]), the kernel pair being a pair of jointly monic arrows $p,q:K\rightarrow A$ for some $K$ , or equivalently a subobject $K\rightarrow A\times A$ . But each such subobject is determined up to isomorphism by its characteristic arrow $A\times A\rightarrow\Omega$ , and there are only finitely many of these.

So the codomain $B$ of any epi $A\rightarrow B$ can be recovered up to isomorphism as the coequalizer of one of these finitely many possible kernel pairs. ∎

Theorem 2.2

Let $A_{n\in\bbbn}\in\mathcal{E}$ be a sequence of objects, $f_{n\in\bbbn}$ a sequence of epis with each $f_{n}:A_{n}\rightarrow A_{n+1}$ . Then all $f_{n}$ are iso from some point on.

Proof

By Lemma 4, we can find a finite set $\{I_{i}\}_{i=0}^{n}$ of representatives for isomorphism classes of epimorphic images of $A_{0}$ . But then every $A_{j}$ is isomorphic to some $I_{i}$ . By the pigeon-hole principle, we can find a sequence ${n_{0}<n_{1}<...\in\bbbn}$ such that all $A_{n_{k}}$ are isomorphic. But then each subsequence of epis $A_{n_{k}}\rightarrow\cdots\rightarrow A_{n_{k+1}}$ satisfies the hypotheses of Lemma 3, and so each of its component arrows is iso. This shows that all $f_{n}$ are isomorphisms for $n\geq n_{0}$ . ∎

Theorem 2.3

Let $E\in\mathcal{E}$ , $A_{0}\rightarrow A_{1}\rightarrow\cdots$ a sequence of monics where each $A_{n}$ is a subobject of $E$ . Then each arrow in the sequence is iso from some point on.

Proof

Since $E$ has only finitely many isomorphism classes of subobjects, we can again find a sequence ${n_{0}<n_{1}<...\in\bbbn}$ such that all $A_{n_{k}}$ are isomorphic. We can then argue as in the proof of Theorem 2.2. ∎

2.3 Injective objects in a topos

We recall that in any topos, the subobject classifier $\Omega$ is injective, and the exponential object $B^{A}$ is injective whenever $B$ is injective. Hence for any $A$ , $\Omega^{A}$ is injective, and the singleton map $\{\}:A\rightarrow\Omega^{A}$ therefore provides an embedding of $A$ into an injective object.

Note also that any retract of an injective object is injective.

2.4 Essential extensions in a topos

We recall that a monic $e:A\rightarrow B$ is essential, or an essential extension, if for every $C\in\mathcal{E}$ and $g:B\rightarrow C$ , $ge$ monic implies $g$ monic. Clearly a composite of essential extensions is essential.

We shall also need an alternative description in terms of epimorphisms, which relies on the fact that any arrow in a topos can be expressed as a product of a monic and an epic ([2] theorem 16.4).

Lemma 5

In a topos, a monic $e:A\rightarrow B$ is essential iff for any epic $g:B\rightarrow C$ , $ge$ monic $\Rightarrow$ $g$ iso.

Proof

$\Rightarrow$ : Clearly $g$ is monic; being epic as well, it must be iso ([2] theorem 13.1). $\Leftarrow$ : Take an arbitrary $g:B\rightarrow C$ with $ge$ monic. Factorize $g=pq$ with $p:X\rightarrow C$ , $q:B\rightarrow X$ epic. Then $ge=pqe$ is monic. Since $p$ is monic, $qe$ is monic, forcing $q$ to be monic and hence iso. This shows that $g=pq$ is monic. ∎

3 Injective hulls in $\mathcal{E}$

Theorem 3.1

An object $A\in\mathcal{E}$ is injective iff every essential extension of A is iso.

Proof

$\Rightarrow$ : Let $e:A\rightarrow B$ be essential. Then by injectivity, the identity map $1_{A}:A\rightarrow A$ can be extended along $e$ to an arrow $f:B\rightarrow A$ , i.e. $fe=1_{A}$ . But then $fe$ is monic, so invoking the fact that $e$ is essential, $f$ is monic. We now have monics $e$ , $f$ in each direction between $A$ and $B$ . Invoking Theorem 2.2, in particular, $e$ is iso.

$\Leftarrow$ : We show that any monic $e:A\rightarrow E_{0}$ can be retracted onto A. This will prove the result because we can take $E_{0}$ to be injective.

The proof is by contradiction. Suppose $e$ has no left inverse. Then in particular it is not iso, hence not an essential extension.

Applying Lemma 5, we can therefore find a proper epic (i.e. non-iso) $q:E_{0}\rightarrow E_{1}$ with $qe$ monic.

If $qe$ were iso, we could now construct a left inverse to $e$ , contrary to hypothesis. The argument of the last paragraph can therefore be repeated with $qe$ in place of $e$ , yielding a sequence of proper epics $E_{0}\rightarrow E_{1}\rightarrow E_{2}$ whose composite with $e$ is monic. Continuing in this way, we find an infinite sequence of proper epics $E_{0}\rightarrow E_{1}\rightarrow\cdots$ , contradicting Theorem 2.2. ∎

Theorem 3.2 (Existence and uniqueness of injective hulls)

For any $A\in\mathcal{E}$ ,

1.

There is an essential extension $e:A\rightarrow E$ with $E$ injective.
2.

For any other essential extension $f:A\rightarrow F$ with $F$ injective, there is an isomorphism $h:E\cong F$ with $f=he$ .

Proof

1.

Find a monic $A\rightarrow I$ with $I$ injective. Up to isomorphism, there are only finitely many subobjects of $I$ which are essential extensions of $A$ ; we can therefore find a maximal one $E$ , say $A\rightarrow E\rightarrow I$ . We show that $E$ is injective. If not, by theorem 3.1 there is a nontrivial essential extension $e:E\rightarrow G$ . Since $I$ is injective, we can extend the embedding $E\rightarrow I$ to $G$ , and the resulting extension is monic, i.e. we have expressed $G$ as a subobject of $I$ . But then $A\rightarrow E\rightarrow G$ is a strictly larger essential extension of $A$ within $I$ , a contradiction. ∎
2.

Because $F$ is injective, we can find a map $h:E\rightarrow F$ with $f=he$ . Because $f$ is monic and $e$ essential, $h$ must be monic. Similarly we can find a monic $k:F\rightarrow E$ . But now, by theorem 1, $h$ and $k$ must both be isomorphisms. ∎

References

[1] Moerdijk, I., MacLane, S.: Sheaves in Geometry and Logic: a First Introduction to Topos Theory. Springer Verlag (1992)
[2] McLarty, C.: Elementary Categories, Elementary Toposes. Oxford University Press (1995)

Injective Hulls In a Locally Finite Topos

Abstract

Keywords:

1 Motivation

2 Preliminaries

2.1 Locally finite categories

Lemma 1

Proof

Corollary 1

Theorem 2.1

Proof

Lemma 2

Proof

Lemma 3

2.2 Locally finite topoi

Lemma 4

Proof

Theorem 2.2

Proof

Theorem 2.3

Proof

2.3 Injective objects in a topos

2.4 Essential extensions in a topos

Lemma 5

Proof

3 Injective hulls in ℰ\mathcal{E}

Theorem 3.1

Proof

Theorem 3.2 (Existence and uniqueness of injective hulls)

Proof

References

3 Injective hulls in $\mathcal{E}$