Let’s look at an example of a presheaf topos, to see what various things I’ve been talking about actually look like—especially the subobject classifier.
Our example will illustrate the connection between topos theory and intuitionistic logic: that is, logic where the law of excluded middle, “p or not p”, fails. Intuitionistic logic goes back to Brouwer, who happens to have been born this very day in 1881. Topos theory formalizes intuitionistic logic in a way he might not have liked. But it does roughly capture some of his thoughts about how mathematics is an activity that happens in time.
The subobject classifier in the category of sets is the 2-element set
and it’s a Boolean algebra with the usual operations of logic. But the subobject classifier in a topos is usually more complicated. It can have more than 2 elements. Of course, it’s not just a set, it’s an object in a topos—but remember, an object of a topos still has a set of ‘elements’. The subobject classifier is usually not a Boolean algebra: it’s something more general called an internal Heyting algebra, where the law of excluded middle may not hold. When that happens, logic inside the topos is intuitionistic.
We’re not really ready to talk about internal Heyting algebras, but we are ready to see a topos with an interesting subobject classifier.
Let’s imagine a world where sets are ‘time-dependent’, but in the following special way. As time passes, a set can get new elements as we learn more about it. We can also learn new equations between elements, so elements that were previously distinct can merge. But we never think an element is in a set and then discover it’s not, and we never think two elements are equal and then discover they’re not.
You can think of this as a naive model of an infallible mathematician proving more and more as time goes by.
To keep things simple, let’s suppose that time starts at zero and proceeds in steps. So, our set of times will be the natural numbers. Let’s think of the the natural numbers with their usual ordering as a category. We get a category where the objects are natural numbers and there’s one morphism from to if and none otherwise. We’ll write
for the unique morphism from to when Note that all these morphisms are composites of those of the form
Our topos will consist of presheaves on Of course the definition of presheaf also also involves an ‘op’, so the two ops cancel and our topos is
The double ‘op’ is somewhat annoying when you’re first trying to find your footing, but such is life.
What’s an object of like? It consists of a set and a map
for each natural number As time passes, new elements can appear. That is, can have elements that aren’t in the image of Elements can merge. That is, can map two elements of to the same element of But elements can’t ‘disappear’ or ‘split’.
You should draw such a thing. In class I drew it as something like an ‘infinitely deep tree’ where we move down the tree as time passes. Newly appearing elements are leaves, and branches can merge as we move down the tree. It doesn’t have a root—it keeps going down forever! Also, it’s not necessarily connected. So, we could call it an ‘infinitely deep forest’: a disjoint union of infinitely deep trees. You can imagine a surreal sort of rainforest that keeps going down forever.
What’s a representable object of like? Thanks to the annoying ‘op’ business, a representable object is a functor of the form
So, is empty for and it has one element for We can say it's a time-dependent set with a single element, `discovered at time ’. In terms of our rainforest imagery, it looks like a very skinny infinitely deep tree with no branches—sort of like a telephone pole.
The Yoneda lemma, or one of its offshoots, says that every object in is a colimit of representables. See Proposition 1 in Section I.5 for details. In terms of our pictures, this means that any rainforest can be built from telephone poles.
How does that work, exactly? The colimit of any diagram can be built by first taking the coproduct of every object in the diagram and then taking a coequalizer that forces all the necessary triangles to commute. So, we should think about first taking a coproduct of representables, and then a coequalizer.
A coproduct acts like a 'disjoint union'. So, a coproduct of representables can be visualized as a disjoint union of telephone poles of various heights. Note that in this sort of time-dependent set elements can appear but never merge.
Puzzle. Show that is a coproduct of representables if and only if all the maps
We can take a coequalizer to ‘glue together’ telephone poles and get more interesting infinitely deep trees… or indeed, infinitely deep forests.
Puzzle. Let be a time-dependent set with 2 elements at time 0 and 1 element at all later times. Show how to build by taking a coproduct of two representables and then taking a coequalizer.
In terms of pictures, we are taking two telephone poles and gluing them together except at the very top, getting an infinitely deep tree that branches at the top.
What’s a subobject of an object like? By definition it’s an equivalence class of monomorphisms into I explained the equivalence relation last time. But we can work out what all this amounts to:
Puzzle. Show that a subobject of is the same as a subset for each natural number such that is mapped into by the map
Puzzle. Draw an object of together with a subobject.
I would draw an ‘infinitely deep forest’ in green and then draw a ‘subforest’, maybe shading it in brown. As we follow any branch down, it can enter the subforest and change from green to brown. It can then never turn green again as you go further down.
In other words: at any time you can learn that an element of your time-dependent set is in some time-dependent subset But once you learn it’s in, you know this for the rest of time!
This suggests that in this topos, the truth values are not merely ‘true’ and ‘false’. For each there should be a truth value ‘becomes true at time ‘. And there should be a truth value ‘never becomes true’—or in other words, ‘false’.
The subobject classifier
Now let’s systematically study the subobject classifier of We saw last time how to determine the subobject classifier in any presheaf category, though I didn’t explain why this procedure actually works. Let’s apply the procedure in our example here to get some intuition for it.
Last time I claimed that the subobject classifier in a presheaf topos is a presheaf
that sends any object to the set of all sieves on Remember, a sieve on is a collection of morphisms with target that’s closed under precomposition.
So, let’s start by seeing what this means in our example. The annoying ‘op’ comes in now, because our category is the category of presheaves on By definition, a sieve on is a collection of morphisms in with target that is closed under precomposition. So, in terms of the category a sieve is a collection of morphisms with source that’s closed under postcomposition.
For each natural number there’s a sieve on consisting of all the morphisms from to the objects Clearly this is closed under postcomposition. Let’s call this sieve
There’s also another sieve on namely the empty sieve: the sieve with no morphisms at all. Let’s call this sieve
Puzzle. Show that all the sieves on are those of the form together with
Thus, we know this about the subobject classifier:
We’ll see that means ‘true now’, means ‘true tomorrow’, means ‘true the next day’, and so on… while means ‘never true’— or in other words, false. Mac Lane and Moerdijk call the subscript in the ‘time to truth’. It’s how long you have to wait for something to become true.
So we’ve seen what does to objects of But what does it do to morphisms?
We can guess: if something is true in days now, tomorrow it will be true in days, or in 0 days if And of course if it’s never true now, it will never be true tomorrow.
Let’s check this.
Suppose is any sieve on Then is a sieve on and a morphism is in this sieve iff its composite with is in This follows from the general description I gave last time.
Puzzle. Show that
maps to if , and to otherwise. (Here we say that infinity minus any natural number is still infinity.)
So, our idea is working!
Now you know enough to draw the subobject classifier. If you draw it as an infinitely deep forest, it has infinitely many leaves on top, where These are As we follow these down to the branch merges with the branch, while the other branches get renamed: gets renamed gets renamed and so on. But not all the branches get renamed! The branch is still called
As we continue to go down, the same thing keeps happening forever.
Puzzle. How many connected components does this infinitely deep forest have?
Next: a subobject classifier in a topos needs to be equipped with an element
What is this in our example? We can guess. ‘True’ should mean true now. The terminal time-dependent set has one element for each time So, should map this one element to
Now you can check this guess:
Puzzle. Show that for any time-dependent set morphisms
correspond in a one-to-one way with subobjects of
One nice way to start tackling this problem is to draw a time-dependent set and a subobject of it, say Next to these draw Then figure out which choice of sends precisely the elements of to
That will help you see how subobjects determine morphisms to But morphisms to should also determine subobjects! For this, draw a time-dependent set and a morphism from it to Show that the elements that get mapped to form a subobject of
Once you’ve done this, you’ll be ready to give a more formal argument:
Puzzle. Show that for any there is a bijection between morphisms
and subobjects of given as follows: for any such morphism , we pull back along obtaining a mono
and then take the subobject of corresponding to this.
When you do this, it means you understand the subobject classifier in the topos of time-dependent sets! You may understand it now… but if not, maybe you’ll understand it tomorrow, or the next day.
The series so far:
• Part 1: sheaves, elementary topoi, Grothendieck topoi and geometric morphisms.
• Part 2: turning presheaves into bundles and vice versa; turning sheaves into etale spaces and vice versa.
• Part 3: sheafification; the adjunction between presheaves and bundles.
• Part 4: direct and inverse images of sheaves.
• Part 5: why presheaf categories are elementary topoi: colimits and limits in presheaf categories.
• Part 6: why presheaf categories are elementary topoi: cartesian closed categories and why presheaf categories are cartesian closed.
• Part 7: why presheaf categories are elementary topoi: subobjects and subobject classifiers, and why presheaf categories have a subobject classifier.
• Part 8: an example: the topos of time-dependent sets, and its subobject classifier.