The Simplicial Category
There’s another approach to the theory of monoids which finds more direct application in topology and homology theory (which, yes, I’ll get to eventually) — the “simplicial category” . Really it’s an isomorphic category to , but some people think better in these other terms. I personally like the direct focus on the algebra, coupled with the diagrammatics so reminiscent of knot theory, but for thoroughness’ sake I’ll describe the other approach.
Note that the objects of correspond exactly with the natural numbers. Each object is the monoidal product of some number of copies of the generating object . We’re going to focus here on the model of given by the ordinal numbers. That is, the object corresponds to the ordinal number , which is a set of elements with its unique (up to isomorphism) total order. In fact, we’ve been implicitly thinking about an order all along. When we draw our diagrams, the objects consist of a set of marked points along the upper or lower edge of the diagram, which we can read in order from left to right.
Let’s pick a specific representation of each ordinal to be concrete about this. The ordinal will be represented by the set of natural numbers from to with the usual order relation. The monoidal structure will just be addition — .
The morphisms between ordinals are functions which preserve the order. A function between ordinals satisfies this property if whenever in then in . Note that we can send two different elements of to the same element of , just as long as we don’t pull them past each other.
So what sorts of functions do we have to play with? Well, we have a bunch of functions from to that skip some element of the image. For instance, we could send to by sending to , skipping , sending to , and sending to . We’ll say for the function that skips in its image. The above function is then . For a fixed , the index can run from to .
We also have a bunch of functions from to that repeat one element of the image. For example, we could send to by sending to , and both to , and to . We’ll say for the function that repeats in its image. The above function is then . Again, for a fixed , the index can run from to .
Notice in particular that “skipping” and “repeating” are purely local properties of the function. For instance, is the unique function from (the empty set) to , which clearly skips . Then can be written as , since it leaves the numbers from to alone, sticks in a new , and then just nudges over everything from (the old) to . Similarly, is the unique function from to that sends both elements in its domain to . Then all the other can be written as .
Now every order-preserving function is determined by the set of elements of the range that are actually in the image of the function along with the set of elements of its domain where it does not increase. That is, if we know where it skips and where it repeats, we know the whole function. This tells us that we can write any function as a composition of and functions. These basic functions satisfy a few identities:
- If then .
- If then .
- If then .
- If or then .
- If then .
We could check all these by hand, and if you like that sort of thing you’re welcome to it. Instead, I’ll just assume we’ve checked the second one for and the fourth one for .
What’s so special about those conditions? Well, notice that takes two copies of to one copy, and that the second relation becomes the associativity condition for this morphism. Then also takes zero copies to one copy, and the fourth relation becomes the left and right identity conditions. That is, with these two morphisms is a monoid object in this category! Now we can verify all the other relations by using our diagrams rather than a lot of messy calculations!
We can also go back the other way, breaking any of our diagrams into basic pieces and translating each piece into one of the or functions. The category of ordinal numbers not only contains a monoid object, it is actually isomorphic to the “theory of monoids” functor — it contains the “universal” monoid object.
So why bother with this new formulation at all? Well, for one thing it’s always nice to see the same structure instantiated in many different ways. Now we have it built from the ground up as , we have it implemented as a subcategory of , we have it as the category of ordinal numbers, and thus we also have it as a full subcategory of — the category of all small categories (why?).
There’s another reason, though, which won’t really concern us for a while yet. The morphisms and turn out to be very well-known to topologists as “face” and “degeneracy” maps when working with shapes they call “simplicial complexes”. Not only is this a wonderful oxymoron, it’s the source of the term “simplicial category”. If you know something about topology or homology, you can probably see how these different views start to tie together. If not, don’t worry — I’ll get back to this stuff.