July 14, 2021

Logical and Sublogical Functors

Posted by John Baez

I’m trying to understand sublogical functors, so I’m looking for examples of sublogical functors between presheaf categories, preferably direct images (if that’s possible).

Just to make this post a bit more interesting, I’ll explain that sentence! This will give beginners a chance to learn something, and experts a chance to catch mistakes in what I’m saying, so that beginners can learn something true.

Let’s write $\widehat{C}$ for the category of presheaves on $C$:

$\widehat{C} = Set^{C^op}$

This is an example of an elementary topos. Precomposition with a functor $f \colon C \to D$ gives a functor

$f^\ast \colon \widehat{D} \to \widehat{C}$

This is an example of the ‘inverse image’ part of a ‘geometric morphism’ because it has a left adjoint that preserves finite limits. But this particular example of a geometric morphism is better than average since it also has a right adjoint.

We say $f^\ast$ is a ‘logical functor’ if it preserves finite limits (which it does, since it’s a right adjoint) and also the natural map

$\phi \colon f^\ast P A \to P f^\ast A$

is an isomorphism for all $A \in \widehat{D}$, where $P A$ is the power object of $A$, namely $\Omega^A$ where $\Omega$ is the subobject classifier.

Johnstone says $f^\ast$ is a sublogical functor if it preserves finite limits and $\phi$ is a monomorphism.

So, I’m really looking for examples where $f^\ast$ is sublogical… and not logical.

Posted at July 14, 2021 3:03 AM UTC

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Re: Logical and Sublogical Functors

Hello, I’m not sure whether there’s a typo and you meant “inverse image” rather than “direct image”. Anyway, the Elephant has quite a lot: Lemma C3.1.2 characterizes sublogical inverse images between presheaf categories (induced by precomposition), and sublogical direct images are characterized in Proposition C3.1.8, and logical inverse images are in C3.5.

Posted by: Sam Staton on July 14, 2021 7:57 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Thanks, Sam! I was having trouble grokking Johnstone’s Lemma C3.1.2. I still don’t understand it, but at least I see a couple of examples now.

Johnstone gives a necessary and sufficient condition on a functor $f: C \to D$ for the induced functor $F : \mathrm{Set}^D \to \mathrm{Set}^C$ to be sublogical. (Note he uses copresheaves here, rather than presheaves.) It goes like this:

Lemma. $F$ is sublogical iff for any object $c \in C$ and any morphism $\alpha : f(c) \to d$, there is a morphism $\beta : c \to c'$ such that $\alpha$ is the composite $f(c) \xrightarrow{f(\beta)} f(d) \xrightarrow{r} d$ where $r$ is part of a section-retraction pair $r: f(c') \to d, \qquad s: d \to f(c'), \qquad r \circ s = 1_d$

I find this rather cryptic, but here are two nice examples, one due to Jens Hemelaer below.

1. The obvious forgetful functor from $\mathbb{Z}$-sets to $\mathbb{N}$-sets is sublogical but not logical.

2. The obvious forgetful functor from reflexive graphs to graphs is sublogical but not logical.

Posted by: John Baez on July 20, 2021 12:16 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Besides what Sam said, could you please fix the notation so that $Set$ appears in the base and not the exponent?

Posted by: Todd Trimble on July 14, 2021 10:18 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

…plus change $X$ to $C$ in the first line below the bar.

Posted by: Tim Porter on July 14, 2021 1:05 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Done, and done. Thanks!

Posted by: John Baez on July 14, 2021 10:44 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

I think you’re still not saying what you mean to say:

Precomposition with a functor $f: C \to D$ gives a functor

$f_\ast: \hat{D} \to \hat{C}$

This is an example of the ‘direct image’ part of a ‘geometric morphism’ because it has a left adjoint that preserves finite limits.

The functor you call $f_\ast$ is conventionally called $f^\ast$. But more importantly, whatever you call it, its left adjoint is left Kan extension along $f$, which doesn’t necessarily preserve finite limits. So your functor $\hat{D} \to \hat{C}$ is not the direct image part of a geometric morphism. It is the inverse image part of a geometric morphism, though.

Posted by: Tom Leinster on July 14, 2021 11:33 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

I hadn’t yet attempted to fix that bunch of mistakes, just the ones Todd and Tim pointed out. [Note added later: I have now tried to fix those mistakes.]

I am constantly confused about what people call $f^\ast$ and what people call $f_\ast$ in topos theory, in part because the ‘geometric’ way of saying which way geometric morphisms point is the opposite of the (to me more natural) ‘algebraic’ way.

If I have a functor $f: C \to D$, to me the most obvious map between presheaf categories is precomposition with $f$, which gives a functor $\widehat{D} \to \widehat{C}$. I would naturally call this $f^\ast$ because of the contravariance. But precisely because that’s what I want to do, I immediately suspect I should do the opposite!

(This is why I almost got run over in England: my mnemonic for traffic directions was “the cars are coming the opposite way from what you expect”, which worked until my expectations changed.)

And when I look at the nLab, my fears seem to be confirmed. It seems to say that this “precomposition with $f$” functor is called $f_\ast$:

Given a morphisms of sites $f : X \to Y$ coming from a functor $f^t : S_Y \to S_X$ of the underlying categories, the direct image operation $f_* : PSh(X) \to PSh(Y)$ on presheaves is just precomposition with $f^t$.

Admittedly, they are talking about precomposition with something called $f^t$, not $f$. They seem to have stuck in an algebraic/geometric reversal here by having arrows in some category of sites point the opposite direction from functors between the underlying categories. So I thought their $f^t$ is what I’d call $f$, while their $f$ is some clever formal trick. (Is it really helpful to introduce both $f$ and $f^t$ here?)

As I read this passage in the nLab, I could see my chances of getting the conventions right dwindling before my very eyes. But it does seem to say that precomposition with a functor gives a functor between presheaf categories called “direct image”, not “inverse image”. But you’re saying it’s called inverse image? What’s going on?

Posted by: John Baez on July 15, 2021 1:25 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

When I think about presheaf topoi and sheaf topoi, very often I am in a situation where the categories $C$ and $D$ have finite limits, and the functor $f: C\rightarrow D$ preserves finite limits. In that case, precomposition does define the pushforward part $f_\ast$ of a geometric morphism; indeed, in that case its left adjoint $f^\ast$ preserves finite limits. In particular, this is the case when $C$ and $D$ are, say, the categories of open subsets of topological spaces $X$ and $Y$ and the functor is given by taking the preimage along a continuous map $g: Y\to X$. In that case you do want to think about this precomposition functor as the direct image functor.

This is also the situation described in the nLab article you cite.

Actually, I must admit that I have trouble imagining this precomposition-with-$f$-functor as the inverse image part of a geometric morphism (even if I can easily verify that it satisfies the axioms). Somehow I do not yet see the relevant geometry in front of me. (And I have no intuition at all about logical or sublogical functors, sorry… but one question: doesn’t the natural map $\phi$ go the other way?)

Posted by: Peter Scholze on July 15, 2021 12:01 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

You’re right, the natural map $\phi$ goes the other way, like this:

$\phi \colon f^\ast P A \to P f^\ast A$

Thanks for catching that. I seem to have made all possible mistakes in this post.

Posted by: John Baez on July 19, 2021 7:01 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

My intuition about topoi mostly breaks down when I think about presheaf topoi. But here is an example of sublogical direct images in the world of sheaves on topological spaces:

If $f: Y\to X$ is an immersion of topological spaces, then the pushforward functor $f_\ast$ commutes with finite limits (as always) and the natural map $f_\ast P(A)\to P(f_\ast A)$ is injective for any sheaf $A$ on $Y$. Indeed, the claim is that, after replacing $X$ by any open subset, the map $f_\ast$ from subsheaves of $A$ to subsheaves of $f_\ast A$ is injective. But the subsheaf can be recovered by taking $f^\ast$. More generally, this argument shows that a direct image functor of topoi is sublogical as soon as it is fully faithful.

I hope I’m not screwing up.

Posted by: Peter Scholze on July 15, 2021 12:33 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

The notion of “morphism of sites” is a mess, with multiple conventions and definitions. I suggest ignoring it while you are looking at presheaves, and stick to functors.

The main point (and emphasised by Olivia Caramello and Riccardo Zanfa recently), is that a functor induces an essential geometric morphism between presheaf toposes, pointing in the same direction (where “essential” means it has an extra left adjoint).

Posted by: David Roberts on July 15, 2021 8:23 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

(This is why I almost got run over in England: my mnemonic for traffic directions was “the cars are coming the opposite way from what you expect”, which worked until my expectations changed.)

Yes, I know the phenomenon you mean with “it’s the opposite of what you think” mnemonics.

I also have a problem with traffic in England. When I go there, I tend to have the sensation that I’ve travelled abroad. This means that as I step off the kerb, I instinctively look the opposite way from the way I normally would. Disaster.

In phases of my life when I’m travelling a lot, I try to drill myself to look both ways both at home and away. Apart from the England phenomenon, I think I’m the greatest risk to myself just after I’ve got home.

Not that the analogous move would help mathematically… Anyway, the convention is that for a functor $f: C \to D$, precomposition with $f$ is written as $f^\ast: \hat{D} \to \hat{C}$, its right adjoint (right Kan extension along $f$) is written as $f_\ast$, and its left adjoint is written as $f_!$. So $f_! \dashv f^\ast \dashv f_\ast$. The adjunction $f^\ast \dashv f_\ast$ is a geometric morphism $\hat{C} \to \hat{D}$, with $f^\ast$ being the inverse image part and $f_\ast$ the direct image part.

Posted by: Tom Leinster on July 15, 2021 10:17 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Apropos of nothing mathematical, I have always been grateful for the London authorities’ solution to this perpetual confusion: giant signs on the pavement with a pointing arrow saying “LOOK RIGHT”.

Posted by: L Spice on July 18, 2021 7:28 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

In Barcelona, major crossings are decorated with grim reminders of pedestrian death statistics. It’s enough to make topos theory seem like a safe activity.

(Text: “In Barcelona, one person in three killed in traffic accidents was on foot. Attention! We are all pedestrians.”)

Posted by: Tom Leinster on July 19, 2021 12:39 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

I have a “falling star” mnemonic, so reading left to right the star falls to the ground: $f^\ast \dashv f_\ast$.

Posted by: Simon Willerton on July 15, 2021 3:01 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

I’m out of town at the moment, so I don’t have access to the Elephant, but maybe the most famous example of logical inverse image is by precomposing with functors between discrete small categories. This might not be what you want, or maybe you thought it too trivial to mention, but every student of topos theory ought to know this.

Posted by: Todd Trimble on July 14, 2021 11:19 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Maybe I sort of knew that example of a logical functor and wanted something a bit more exciting, but thanks for pointing it out: I need all the help I can get!

I also imagine that if starting from the canonical inclusion of categories $C \hookrightarrow C + D$, at least one of the various induced functors between $\widehat{C}$ and $\widehat{C + D}$ is logical. Right?

(I usually hate it when people say “a functor between $X$ and $Y$” since the word “between” doesn’t clearly indicate which way the functor is going. But here I’m talking that way deliberately to avoid saying which way the induced functor is going: I’m too confused to risk saying anything so precise. Clearly we can restrict presheaves on $C + D$ to presheaves on $C$, and this restriction has a left and a right adjoint, and I’m hoping that one of these three is logical. Okay, I’ll guess which one: the restriction functor itself.)

Of course what I really want are examples of sublogical functors that aren’t logical.

Posted by: John Baez on July 15, 2021 1:34 AM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Yes, the geometric morphism $F: \widehat{C} \to \widehat{C+D}$ is atomic (or in other words, its inverse image functor, given by restriction, is logical).

Here is a “geometric” kind of proof.

We can first write $\widehat{C + D} = \widehat{C} \sqcup \widehat{D}$ where $\sqcup$ denotes the coproduct in the category of toposes. Then we can write $\widehat{C} \to \widehat{C} \sqcup \widehat{D}$ as the pullback of $\widehat{1} \to \widehat{1} \sqcup \widehat{1}$ along $\widehat{C} \sqcup \widehat{D} \to \widehat{1} \sqcup \widehat{1}$.

Because atomic morphisms are stable under pullback, it is enough to show that $\widehat{1} \to \widehat{1} \sqcup \widehat{1}$ is atomic. But this is the geometric morphism induced by the inclusion of a point in the discrete two-point space, which is a local homeomorphism, in particular atomic.

Alternatively, you can show that $\widehat{C}$ can be written as the slice topos $\widehat{C+D}/Y$ for some object $Y$ in $\widehat{C+D}$. (Take $Y(c)=1$ for $c \in C$ and $Y(d)=\varnothing$ for $d \in D$.) The projection from the slice topos to the base topos is a local homeomorphism, so in particular atomic.

Posted by: Jens Hemelaer on July 15, 2021 4:46 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Isn’t the inverse image in this case just a projection? I.e., we have

$\widehat{C + D} \simeq \widehat{C} \times \widehat{D}$

and the restriction along the inclusion $C \to C + D$ is thereby just the projection

$\widehat{C} \times \widehat{D} \to \widehat{C}.$

But it’s pretty automatic that the projection will be logical, since limits, exponentials, and the subobject classifier are computed componentwise.

Posted by: Todd Trimble on July 16, 2021 12:07 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Yes, I agree that it is the projection. For me, it wasn’t intuitively clear that exponentials are computed componentwise, I had to write down the universal property $\mathrm{Hom}((A_1,A_2),({C_1}^{B_1},{C_2}^{B_2})) \simeq \mathrm{Hom}((A_1 \times B_1,A_2 \times B_2),(C_1,C_2))$ to see it, and similarly for the subobject classifier. But it is a very elegant proof, and it needs less terminology than the geometrical proof that I gave.

Posted by: Jens Hemelaer on July 16, 2021 5:46 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

I should have added that I’m glad you gave your proof, because I learned something by reading it. Namely, I never knew the connection between atomic sites and atomic morphisms; that’s interesting.

Posted by: Todd Trimble on July 16, 2021 7:14 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

Above Todd wrote “I’m out of town at the moment, so I don’t have access to the Elephant … ” which brings up (for me) a question: Does anyone know why, per the following comment, PTJ opposes ebook editions for his seminal works?

It is very unlikely that Johnstone will release [preliminary versions of volume 3 of the Elephant] electronically - he has not authorised an electronic copy of volumes 1 and 2 and has said that he won’t. – theHigherGeometer Apr 7 ‘14 at 5:47

https://math.stackexchange.com/questions/573118/volume-3-of-johnstones-sketches-of-an-elephant#comment1549060_573118

Stone Spaces is also unavailable electronically.

Posted by: Keith Harbaugh on July 15, 2021 1:41 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

I’ll write $F : \hat{C} \to \hat{D}$ for the essential geometric morphism induced by $f : C \to D$. Then the inverse image functor $F^*$ is the functor given by precomposition with $f$.

The geometric morphism $F$ is called atomic if $F^\ast$ is logical, and open if $F^\ast$ is sublogical.

This gives some geometric intuition: if $X$ and $Y$ are topological spaces, then $F : \mathbf{Sh}(X) \to \mathbf{Sh}(Y)$ is atomic if and only if the corresponding map $X \to Y$ is a local homeomorphism. Moreover, if $Y$ is a $T_D$-space, then $F : \mathbf{Sh}(X) \to \mathbf{Sh}(Y)$ is open if and only if the corresponding map $X \to Y$ is open.

In particular, the geometric morphism $\mathbf{Sh}(X) \to \mathbf{Sets}$ induced by the map $X \to 1$ is always open, but it is not atomic unless $X$ is discrete. To turn this into an example with presheaf toposes, we have to look at presheaves on posets. For every poset $P$, we have $\hat{P} \simeq \mathbf{Sh}(X)$ for some topological space $X$. So for any poset $P$, the essential geometric morphism $\hat{P} \to \hat{1}$ induced by $P \to 1$ is open but not atomic.

The criterion for openness that Sam Staton mentions (Lemma C3.1.2), reduces to something interesting for monoids (that Morgan Rogers and I use in our joint work). The essential geometric morphism $\hat{M} \to \hat{N}$ induced by a monoid map $\phi : M \to N$ is open if and only if for every $n \in N$ there is some $u \in N^\ltimes$ such that $nu \in \phi(M)$, where $N^\ltimes$ is the set of elements in $N$ that admit a right inverse.  For example if $\mathbb{N}$ and $\mathbb{Z}$ are the monoids of natural numbers resp. integers under addition, then the inclusion $\mathbb{N} \to \mathbb{Z}$ induces an open geometric morphism $F: \hat{\mathbb{N}} \to \hat{\mathbb{Z}}$. Toposes of presheaves on a monoid $M$ are atomic if and only if $M$ is a group. This shows that $F$ is not atomic, because the codomain is atomic and the domain is not.

More generally, if $F: \hat{M} \to \hat{G}$ is an essential geometric morphism induced by a monoid map $M \to G$, with $M$ a monoid and $G$ a group, then $F$ is always open, but only atomic if $M$ is a group as well.

Posted by: Jens Hemelaer on July 15, 2021 4:16 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

One of the examples above is the geometric morphism $F : \widehat{\mathbb{N}} \to \widehat{\mathbb{Z}}$ which has an inverse image functor $F^\ast$ that is sublogical but not logical.

If you are interested in a sublogical direct image functor, then the good news is that $F^\ast = G_\ast$ for a geometric morphism $G$ in the other direction. The reason is that in this case the left adjoint $F_!$ of $F^\ast$ preserves finite limits (because $\mathbb{Z}$ is flat as left $\mathbb{N}$-set). The adjoint functors $F_! \dashv F^\ast$ together determine a geometric morphism $G : \widehat{\mathbb{Z}} \to \widehat{\mathbb{N}}$. The geometric morphism $G$ has direct image functor $G_\ast = F^\ast$, so it is sublogical but not logical. However, $G$ is not induced by a monoid map $\mathbb{Z} \to \mathbb{N}$.

As another example, we can look at the monoid map $N \to 1$, where $N$ is the monoid of natural numbers (with zero) under multiplication, and $1$ is the trivial monoid. The induced functor $F : \widehat{N} \to \widehat{1}$ has the property that $F^\ast$ is sublogical and not logical (i.e. $F$ is open and not atomic). Further, $F_!$ and $F_\ast$ are isomorphic in this case (Morgan Rogers and I showed that this happens precisely because $N$ has a zero element). In particular $F_! = F_\ast$ preserves (finite) limits. So there is a geometric morphism $G : \widehat{1} \to \widehat{N}$ in the other direction, with $G^\ast = F_! = F_\ast$ and $G_\ast = F^\ast$, with now $G_\ast$ sublogical but not logical. This time, $G$ is an essential geometric morphism, i.e. $G^\ast$ has a further left adjoint (namely $G_\ast$). Essential geometric morphisms are induced by a functor between the idempotent completions of the original categories.

The corresponding functor in this case is the functor $f : 1 \to C$, where $1$ is the trivial category with one object, and $C$ is the full subcategory of the category of right $N$-sets consisting of two objects: the right $N$-set $N$ (under multiplication) and the terminal right $N$-set 1. The functor $f$ sends the unique object of $1$ to the terminal right $N$-set $1$.

Posted by: Jens Hemelaer on July 15, 2021 6:30 PM | Permalink | Reply to this

Re: Logical and Sublogical Functors

I just noticed that the direct image functor $F_*$ is sublogical if and only if the geometric morphism $F$ is an inclusion… (C3.1.8 in the Elephant, as Sam Staton pointed out). This agrees with what Peter Scholze mentioned for topological spaces.

So my examples above are a bit far-fetched. Any fully faithful functor $f: C \to D$ between small categories induces an inclusion $F : \widehat{C} \to \widehat{D}$, and these then all have sublogical direct image functor. On the other hand, if the direct image functor is logical, then in particular it preserves the subobject classifier, so $F$ is hyperconnected (A.4.6.6 in the Elephant). For inclusions this is only possible if $F$ is an equivalence (because hyperconnected geometric morphisms are surjective).

So any fully faithful functor $f : C \to D$ should work, as long as the induced geometric morphism $F : \widehat{C} \to \widehat{D}$ is not an equivalence.

Posted by: Jens Hemelaer on July 15, 2021 7:12 PM | Permalink | Reply to this

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