Now that we’ve got monoidal categories as the category-level analogue of monoids we need the category-level analogue of monoid homomorphisms. Recall that a monoid homomorphism is a function from the underlying set of one monoid to that of another that preserves the monoid compositions and identities. Since functors are the category-level version of functions, the natural thing to do is to ask for functors that preserves the monoidal category structure.
Of course it can’t be just that simple. As usual, we want to only ask for things to work out “up to isomorphism”. That is, given monoidal categories and — I’ll write and for both monoidal products and both identities and trust context to make clear which is which — we don’t want to insist that a functor satisfies and and so on. Instead we want natural isomorphisms to replace these equations.
So there should be an isomorphism in . As specified objects of , both and are given by functors from the category to , and an isomorphism between the objects is automatically a natural isomorphism between these functors.
There should also be a natural isomorphism . That is, for every pair of objects and of there’s an isomorphism in , and these commute with arrows and in .
And, strictly speaking, we need such a natural isomorphism for every functor built from and . For example, there should be a natural isomorphism . And, of course, all these isomorphisms should fit together well. This makes for a lot of data, and how are we going to manage it all?
The Coherence Theorem will come to our rescue! There’s a unique natural isomorphism built from s, s, and s taking (say) to in (call it ) and another unique such isomorphism in (call it ). So we can build the above isomorphism as . Now all we need is the natural isomorphisms for completely right-parenthesized products with no slots filled with identity objects.
And we can build these from the natural isomorphisms and (and make sure all the isomorphisms fit together well) — as long as these satisfy a few coherence conditions to make sure they play well with the associator and other structural isomorphisms:
Similarly to the terminology for monoidal categories, if the natural isomorphisms and are actually identities (and thus so are all the others built from them) then we call a “strict monoidal functor”. Alternatively, some prefer to call what I’ve defined above a “weak monoidal functor”.
We aren’t quite done yet, though. At the level of sets we have sets and functions between them. Up at the category level we have categories, functors, and now also natural transformations, so we should also focus on the natural transformations that preserve monoidal structures. A monoidal natural transformation between monoidal functors is a natural transformation satisfying and .
Now we have all the analogous tools for monoidal categories as we used for just plain categories to define equivalence. We say that two monoidal categories and are monoidally equivalent if there are monoidal functors and with monoidal natural isomorphisms and .
It’s useful to verify at this point that the set of isomorphism classes of objects of a monoidal category form a monoid with as its composition and (the isomorphism class of) as identity, a monoidal functor induces a homomorphism of monoids , and an equivalence between and induces an isomorphism between the monoids and . These are all pretty straightforward, and they’re good for getting used to the feel of how equalities weaken to isomorphisms as we move from sets to categories.
Okay, as I promised yesterday, I’m going to prove the Coherence Theorem for monoidal categories today. That is, any two natural transformations built from , , and between any two functors built from and are equal once we know the pentagon and triangle identities are satisfied.
Here’s an example. Brace yourself for the diagram.
Okay, the outer pentagon is the pentagon identity for the quadruple . The triangles in the upper-right and on the left are the triangle identity. The quadrilaterals are all naturality squares for and . And the last triangle near the center is the identity we’re proving. It says that . There’s a similar diagram to show that , which you should find.
Now we’re ready to really get down to work. We can get any functor built from and by writing down a parenthesized sequence of multiplications like and filling some of the open slots with .
First, let’s get rid of those identity objects. Each of them shows up either to the right or to the left of some product symbol, so we can use either a or a to remove it, but there’s some choice involved here. For example, near the we’re trying to get rid of the functor might look like . We can either use a right away, or we can hit it with an associator and then use a . The triangle identity says that these two ways of removing the are the same. Similarly, if nearby the functor looks like or we have some leeway, but the two new triangles we proved above say we’ll get the same natural transformation to remove the either way. The upshot is that any two natural transformations we use to reach a functor without any slots filled with identity objects are the same.
So now we just have to deal with functors built from monoidal products. Each of them is just a different way to parenthesize a multiplication of some number of variables, and the associator (or its inverse) lets us move between different ways of parenthesizing the same number of variables. One convenient way of parenthesizing is to have all pairs of parentheses at the end, like , and what we’re going to do is try to make every functor look like this.
Now imagine a big diagram with all the different ways of parenthesizing a product of terms as its vertices, and draw an arrow from one vertex to another if you can go from the first to the second by applying one associator. When , this is just the pentagon diagram. Try actually drawing the one for (there should be 14 vertices). We’re allowed to walk forwards or backwards along the arrows, since the associator is an isomorphism. Every path is a natural isomorphism between the two functors at its ends, since it’s a composition of natural isomorphisms. What the Coherence Theorem says is that any two paths between the same two points give the same natural isomorphism.
Let’s say that we have a path from a vertex in this diagram to a vertex , which might consist of some forward arrows and some reverse arrows. Each time we change directions from forward to reverse we can pick a path using only forward arrows from the turning point to the vertex , which has all its parentheses to the right. Here’s a sketch of what I’m talking about
Each arrow here is a path consisting of only forward arrows. We start with the path along the top. At the two points where we change from walking forwards along the arrows to walking backwards, we pick a path made up of only forward arrows to the point . Now the path we started with is the same as walking forwards a step from , down to , back the way we just came (undoing the isomorphism), forwards to the next juncture, down to and back, and on to .
It seems we’ve made our lives more complicated by adding in these side-trips to , just to undo them immediately, but something really nice happens. There are two paths here from the middle point of our original path down to , and each of them consists only of forward arrows. If these two paths are the same, I can cut out that whole middle loop. Then our path walks straight down from to along forward arrows, then back up to along backward arrows. If there are more changing points in the original path, then there are just more loops to cut out. So if any two paths from a vertex to consisting of only forward arrows give the same isomorphism, then every path from to is the same as one passing through . And even better, the paths from and to themselves consist entirely of forward arrows, so there’s only one of each of them. That is, if we can show that any two paths made of forward arrows from an arbitrary vertex to the fixed vertex give the same natural isomorphism, then we’re done.
Now here’s where it gets a bit messy. We’ll need to use inductions on both the number of terms in our functors and on how close they are to . We know the Coherence Theorem works up to terms, because that’s the pentagon identity. We’ll prove it for terms assuming it’s known for fewer than terms and work our way up. Similarly, for a fixed we’ll start by establishing the result for the vertices closest to and work our way out.
Let’s start with a functor , which is the product of functors and . Let and be two arrows out of to functors and , respectively, which them continue on to paths to . We need to find a and paths from and to to make the square commute. Since and are both closer than to , we can assume that both paths from to in the diagram give the same isomorphism, as do both paths from to . Then since is even closer to we can assume there’s only one transformation from to , so the result follows for . Here’s another sketch:
If and are already the same there’s nothing to be done. Pick and go home. Otherwise there are three possibilities for
- applies an associator inside
- applies an associator inside
- , and is an associator
and the same three for .
If and both act within or within then we’re done because of the induction on the number of terms. If acts in one factor and in the other then they commute because is a functor of two variables. All that’s left to see is if is an associator and acts in either or .
If acts inside , sending it to , then we get by the naturality of . A similar statement holds if acts inside or . Finally, if is itself the associator , then and are actually the first steps around the pentagon identity, so we can use the far vertex of the pentagon as our .
And after all these cases, we’re done. If two paths start out differently from we can guide them back together so that the square where they differed commutes, and we can keep doing this until we get to . Thus there’s only one natural isomorphism built from associators taking a functor the the fully right-parenthesized functor on the same number of terms.
Thus there’s only one natural isomorphism built from associators between any two functors built from monoidal products on the same number of terms.
Thus any two natural transformations built from , , and between any two functors built from and are equal, given that the pentagon and the triangle identities are satisfied.
We know that monoids are one of the most basic algebraic structures on which many others are built. Naturally, they’re one of the first concepts we want to categorify. That is, we want to consider a category with some extra structure making its objects behave like a monoid.
So let’s charge ahead and try to write down what this means. We need some gadget that takes two objects and spits out another. The natural thing to use here is a functor . We’re using the same symbol we did for tensor products — and for a good reason — but we don’t need it to be that operation.
Now we need this functor to satisfy a couple rules to make it like a monoid multiplication. It should be associative, so for all objects , , and in . There should be an “identity” object so that for all objects .
We know that the natural numbers form a monoid under multiplication with as the identity, and we know that the category of finite sets categorifies the natural numbers with Cartesian products standing in for multiplication. So let’s look at it to verify that everything works out. We use as our monoidal structure and see that … but it doesn’t really. On the left we have the set , and on the right we have the set , and these are not the same set. What happened?
The problem is that the results are not the same, but are only isomorphic. The monoid conditions are equations
So when we categorify the concept we need to replace these by natural isomorphisms
These say that while the results of the two functors on either side of the arrow might not be the same, they are isomorphic. Even better, the isomorphism should commute with arrows in , as described by the naturality squares. For instance, if we have an arrow in then we can apply it before or after : as arrows from to .
As a side note, the isomorphism is often called the “associator”, but I don’t know of a similarly catchy name for the other two isomorphisms. When we’ve “weakened” the definition of a monoidal category like this we sometimes call the result a “weak monoidal category”. Alternatively — and this is the convention I prefer — we call these the monoidal categories, and the above definition with equalities instead of just isomorphisms gives “strict monoidal categories”.
Unfortunately, we’re not quite done with revising our definition yet. We’ll be taking our tensor products and identity objects and stringing them together to make new functors, and similarly we’ll be using these natural isomorphisms to relate these functors, but we need to make sure that the relationship doesn’t depend on how we build it from the basic natural isomorphisms. An example should help make this clearer.
This is the pentagon diagram. The vertices of the pentagon are the five different ways of parenthesizing a product of four different objects. The edges are single steps, each using one associator. Around the left, we apply the associator to the first three factors and leave alone (use the identity arrow ), then we apply the associator to , , and , and finally we apply the associator to the last three factors and leave alone. Around the right, we apply the associator twice, first to , , and , and then to , , and . So we have two different natural isomorphisms from to . And we have to insist that they’re the same.
Here’s another example:
This triangle diagram is read the same as the pentagon above: we have two different natural transformations from to , and we insist that they be the same.
What’s happened is we’ve replaced equations at the level of sets with (natural) isomorphisms at the level of the category, but these isomorphisms are now subject to new equations. We’ve seen two examples of these new equations, and it turns out that all the others follow from these two. I’ll defer the justification of this “coherence theorem” until later.
For now, let’s go back to We can use the universal property of the product to give an arrow , and we can verify that these form the components of a natural isomorphism. Similarly, we can use the singleton as an identity object and determine isomorphisms and . They do indeed satisfy the pentagon and triangle identities above, making into a monoidal category.
In fact, you could establish the associator and other isomorphisms for by looking at the elements of the sets and defining particular functions, but if we do it all by the universal properties of products and terminal objects we get a great generalization: any category with finite products (in particular, pairwise products and a terminal object) can use them as a monoidal structure. Dually, any category with finite coproducts can use them as a monoidal structure.
For any ring , the category of all bimodules has a monoidal structure given by , and because of this monoidal categories are often called “tensor categories” and the monoidal structure a tensor product.
I haven’t been flagging the new postings of >TWF for a while, though I really should have. Anyhow, this “week” Baez talks about what exceptional Lie algebras have to do with elementary particles.
Categorification is a big, scary word that refers to a much simpler basic idea than you’d expect, and it’s also huge business right now. There’s a lot of this going on at The n-Category Café, for instance. Both knot Floer homology and Khovanov homology are examples of categorifications that seem to be popular with the crew at the Secret Blogging Seminar. John Baez has a bunch of papers circling around “higher-dimensional algebra”, which is again about categorification.
So what is categorification? Well, as I said the core idea’s not really that hard to grasp, but it’s even easier to understand by first considering the reverse process: decategorification. And decategorification is best first approached through an example.
Since this post will be going to this fortnight’s Carnival, I’ll be saying a fair amount that I’ve said before and linking back to definitions most regular readers should know offhand. It might seem slower, but I’ll be back up to the regular rhetorical pace tomorrow. Or you could take it as a bit of review, just to make sure you’re on top of things. Because of this, the post is going to get long, so I’ll put it behind a jump.
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Let’s look at universal arrows a bit more closely. If we have a functor , an object , and a universal arrow , then we can take any arrow and form the composition . The universal condition says that to every arrow corresponds a unique so that . That is, there is a bijection for every .
It’s even better than that, though: these bijections commute with arrows in . That is, if we have then the naturality condition holds for all . This makes the into the components of a natural isomorphism . That is, a universal arrow from to is a representation of the functor .
Conversely, let’s say we have a representable functor , with representation . Yoneda’s lemma tells us that this natural transformation corresponds to a unique element of , which as usual we think of as an arrow . So a representation of a functor is a universal arrow in .
So, representations of functors are the same as universal arrows. Colimits are special kinds of universal arrows, and since a universal arrow is an initial object in an apropriate comma category it is a colimit. Representations are universals are colimits.
If we swap the category for the dual category , we change limits into colimits, couniversals into universals, and contravariant representations into covariant ones. All the reasoning above applies just as well to . Thus, we see that in representations of contravariant functors are couniversals are limits.
There’s actually yet another concept yet to work into this story, but I’m going to take a break from it to flesh out another area of category theory for a while.
Remember that we defined a colimit as an initial object in the comma category , and a limit as a terminal object in the comma category . It turns out that initial and terminal objects in more general comma categories are also useful.
So, say we have a categories and , an object , and a functor , so we can set up the category . Recall that an object of this category consists of an object and an arrow , and a morphism from to is an arrow such that .
Now an initial object in this category is an object of and an arrow so that for any other arrow there is a unique arrow satisfying . We call such an object, when it exists, a “universal arrow from to “. For example, a colimit of a functor is a universal arrow (in ) from to .
Dually, a terminal object in consists of an object and an arrow so that for any other arrow there exists a unique arrow satisfying . We call this a “couniversal arrow from to “. A limit of a functor is a couniversal arrow from to .
Here’s another example of a universal arrow we’ve seen before: let’s say we have an integral domain and the forgetful functor from fields to integral domains. That is, takes a field and “forgets” that it’s a field, remembering only that it’s a ring with no zerodivisors. An object of consists of a field and a homomorphism of integral domains . The universal such arrow is the field of fractions of .
As a more theoretical example, we can consider a functor and the comma category . Since an arrow in is an element of the set , we call a universal arrow from to a “universal element” of . It consists of an object and an element so that for any other element we have a unique arrow with .
As an example, let’s take to be a normal subgroup of a group . For any group we can consider the set of group homomorphisms that send every element of to the identity element of . Verify for yourself that this gives a functor . Any such homomorphism factors through the projection , so the group and the projection constitute a universal element of . We used cosets of to show that such a universal element exists, but everything after that follows from the universal property.
Another example: let be a right module over a ring , and be a left -module. Now for any abelian group we can consider the set of all -middle-linear functions from to — functions which satisfy , , and . This gives a functor from to , and the universal element is the function to the tensor product.
The concept of a universal element is a special case of a universal arrow. Very interestingly, though, when is locally small (as it usually is for us)$ the reverse is true. Indeed, if we’re considering an object and a functor then we could instead consider the set and the functor . Universal arrows for the former setup are equivalent to universal elements in the latter.
It turns out that in that happy situation where a category has -limits for some small category , the assignment of a limit to a functor is itself a functor from to . That is, given functors and from to and a natural transformation we get an arrow , and composition of natural transformations goes to composition of the limit arrows.
Since the proof of this is so similar to how we established limits in categories of functors, I’ll just refer you back to that and suggest this as practice with those techniques.
Today I want to give a great example of creation of limits that shows how useful it can be. For motivation, take a set , a monoid , and consider the set of functions from to . Then inherits a monoid structure from that on . Just define and take the function sending every element to the identity of as the identity of . We’re going to do the exact same thing in categories, but with having limits instead of a monoid structure.
As a preliminary result we need to note that if we have a set of categories for each of which has -limits, then the product category has -limits. Indeed, a functor from to the product consists of a list of functors from to each category , and each of these has a limiting cone. These clearly assemble into a limiting cone for the overall functor.
The special case we’re interested here is when all are the same category. Then the product category is equivalent to the functor category , where we consider as a discrete category. If has -limits, then so does for any set .
Now, any small category has a discrete subcategory : its set of objects. There is an inclusion functor . This gives rise to a functor . A functor gets sent to the functor . I claim that creates all limits.
Before I prove this, let’s expand a bit to understand what it means. Given a functor and an object we can get a functor that takes an object and evaluates at . This is an -indexed family of functors to , which is a functor to . A limit of this functor consists of a limit for each of the family of functors. The assertion is that if we have such a limit — a -limit in for each object of — then these limits over each object assemble into a functor in , which is the limit of our original .
We have a limiting cone for each object . What we need is an arrow for each arrow in and a natural transformation for each . Here’s the diagram we need:
We consider an arrow in . The outer triangle is the limiting cone for the object , and the inner triangle is the limiting cone for the object . The bottom square commutes because is functorial in and separately. The two diagonal arrows towards the bottom are the functors and applied to the arrow . Now for each we get a composite arrow , which is a cone on . Since is a limiting cone on this functor we get a unique arrow .
We now know how must act on arrows of , but we need to know that it’s a functor — that it preserves compositions. To do this, try to see the diagram above as a triangular prism viewed down the end. We get one such prism for each arrow , and for composable arrows we can stack the prisms end-to-end to get a prism for the composite. The uniqueness from the universal property now tells us that such a prism is unique, so the composition must be preserved.
Finally, for the natural transformations required to make this a cone, notice that the sides of the prism are exactly the naturality squares for a transformation from to and , so the arrows in the cones give us the components of the natural transformations we need. The proof that this is a limiting cone is straightforward, and a good exercise.
The upshot of all this is that if has -limits, then so does . Furthermore, we can evaluate such limits “pointwise”: .
As another exercise, see what needs to be dualized in the above argument (particularly in the diagram) to replace “limits” with “colimits”.
Yesterday I talked about functors that create limits. It’s theoretically interesting, but a different condition that’s often more useful in practice is that a functor “preserve” limits.
Given a small category , a functor preserves -limits if whenever is a limiting cone for a functor then is a limiting cone for the functor . A functor is called “continuous” if it preserves all small limits. By the existence theorem, a functor is continuous if and only if it preserves all (small) products and pairwise equalizers.
As stated above, this condition is different than creating limits, but the two are related. If is a category which has -limits and is a functor which creates -limits, then we know also has -limits. I claim that preserves these limits as well. More generally, if is complete and creates all small limits then is complete and is continuous.
Indeed, let be a functor and let be a limiting cone for . Then since creates limits we know there is a unique cone in with , and this is a limiting cone on . But limiting cones are unique up to isomorphism, so this is the limit of and preserves it.
Now it’s useful to have good examples of continuous functors at hand, and we know one great family of such functors: representable functors. It should be clear that two naturally isomorphic functors are either both continuous or both not, so we just need to consider functors of the form .
First let’s note a few things down. A cone in is a family of arrows (into a family of different objects) indexed by , so we can instead think of it as a family of elements in the family of sets . That is, it’s the same thing as a cone , with vertex a set with one element. Also, remember from our proof of the completeness of that if we have a cone then we get a function from to the set of cones on with vertex . That is, . Finally, we can read the definition of a limit as saying that the set of cones is in bijection with the set of arrows
Now we see that
In the first bijection we use the equivalence of cones with vertex and functions from to the set of cones with vertex . In the second we use the fact that a cone in from to a family of hom-sets is equivalent to a cone in . In the third bijection we use the definition of limits to replace a cone from to by an arrow from to .
But now we can use the definition of limits again, now saying that
and since this holds for any set we must have . Therefore preserves the limit.