Products and Coproducts (category theory)

I’m working on the next installment of cobordism theory and we’re going to need a bit of category theory.  Fortunately I’ve come across some excellent videos on YouTube put up years ago (2007-2008) by Eugenia Cheng (regularly at the University of Sheffield, currently visiting UChicago) and  Simon Willerton.  They start with initial and terminal objects:

Initial and Terminal Objects -1

The names are apt.  An initial object is an object $0\in\mathcal{C}$ such that for any other object $A\in\mathcal{C}$ there is a unique morphism $0\to A$.  Terminal objects are dual to initial objects meaning that the  arrows get reversed.  So a terminal object is an object $1\in\mathcal{C}$ such that for any other object $A\in\mathcal{C}$ there is a unique morphism $A\to 1$.  See?  Same words, reversed arrows.  The videos above are under ten minutes and the presenter, Cheng, is lively and very clear.  I came across these videos partly from a recommendation from Simon and also because I wanted to see their take on coproducts.  I’m a convert, so much so that I’ll just refer you to their videos.

Products and Coproducts 1

Products and Coproducts 2

Products and Coproducts 3

Products and Coproducts 4

Products we all have experience with, the common example being the cartesian product. It should be realized that for these constructions it’s not just the object that’s a product, but it’s the object plus a pair of morphisms that play the role of projections that is the full construction.  While products are probably inspired by such things as cartesian products and direct sums of groups there’s many more examples (posets pretty much always have a neat interpretation on account of the inequalities becoming the morphisms, i.e  $x becomes $x\to y$.

I was gratified recently while reading Kock (whose text has a certain helical pedagogy  in that it revisits topics over and over again with deeper and deeper significance).  He was defining cobordisms again, but this time with a diagram.  Given (n-1)-manifolds M and N a cobordism is an n-manifold W along with two orientation preserving embeddings $p_M, p_N$,

$M\overset{p_M}{\longrightarrow} W\overset{p_N}{\longleftarrow} N$

such that if V is another cobordism (so, n-manifold with embeddings) there exisgts a unique morphism from W to V causing the following diagram commutes,

“Ah ha!” I said…., “a coproduct.”  Little victories are better than no victories.  I still haven’t told you what a coproduct is yet (though Catsters will do a better job).  Though you might guess with names like product and coproduct that these definitions are dual and they are. In nCob, the product is disjoint union and the coproduct is cobordism.  However, I’m still working on that one,  turns out there needs to be some equivalence relations before properly defining the category.  It seems there is an issue with composition of cobordisms and also decomposition.  How do to you ‘stitch’ together two manifolds (compose)?  There’s no unique way, but it’s unique up to diffeomorphism.

If you’re new to category theory, I really recommend playing with any novel examples you come across in a book or video.  Examples like Set and Top are a bit boring.  Examples like nCob or posets are better because they can help you break your set-minded perspectives.  Category theory is much more interested in how things are related (via morphisms and functors) than what’s inside of them (set membership).  I’ve been noticing this trend the more I look at advanced mathematics, the focus is on functions (groups or algebras of…) telling us about the structure of the space.  This ends up being really important in the development of Noncommutative Geometry.

One more word on category theory, and the above ideas.  Equivalence, as I’ve alluded to before, is up to an isomorphism. In category theory we won’t be worried with whether two things are equal, but isomorphic (in whatever sense that takes on meaning in a particular category).  Another theme you’ll see is that of a universal property.  The product/coproduct notion above is one.  If there is another cobordism…..then there exists a unique…. The italicized phrase captures the idea as best as I can currently.

Really, watch the videos.  They’re good, they’re funny, you’ll learn something cool (like why the category of fields is tricky and the category of groups is convenient).

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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4 Responses to Products and Coproducts (category theory)

1. sheabrowne says:

I’ll be sure to check these videos out, thanks!

2. Simon and I were chatting today and we came up with a tongue-in-cheek coproduct that nicely illustrates the generality of the above construction. Suppose I have a poset with three items: apples, bananas and peaches. I like apples better than bananas and apples better than peaches but I don’t have any preference between bananas and peaches. My preference partially orders the set of these three fruits. We can treat a poset as a category, if $x then $x\to y$ so if bananas<apples then $latex\mbox{bananas}\to\mbox{apples}$ for example. Thus we have a coproduct,

$\mbox{bananas}\rightarrow \mbox{apples}\leftarrow \mbox{peaches}$

Funny as it sounds, this is completely rigorous. Three cheers for generality.

• And furthermore, ‘apples’ is the terminal object of that category. No initial object though.

3. Further comment: I think I’m being a bit careless here. How are cobordisms coproducts? What category? It cannot be nCob simply because the arrows in that category are not smooth functions. This construction must be in the category of smooth manifolds. So in that category you can construct coproducts and in that sense if the coproduct exists it’s a cobordism.