Kan Extensions

Let’s set some notation up. Let \mathscr{A,B,C} be categories, and suppose that F:\mathscr{A}\to \mathscr{B} and G:\mathscr{A}\to \mathscr{C} are functors. Our goal is to define a functor H:\mathscr{B}\to\mathscr{C} that in some reasonable sense extends F (like, maybe we could make a commuting triangle?).

In particular, I’d like my H to also come with a natural transformation \eta:HF\to G (this is what justifies the term “extension”). And I’d like H to be the “best” such functor. That is, if K:\mathscr{B}\to\mathscr{C} comes with a natural transformation KF\to G, then I want to have a natural transformation K\to H (presumably making appropriate diagrams commute). If I can find an H with this property, I will call it the (right) Kan extension of G along F.

Calling this the right extension has always confused me. I always, in fact, had it backwards – I would have called this the left extension. Since HF\to G, I think of H as being on the left of G, so I thought it was the left extension. I guess instead I should be thinking about how it shows up on the right of any other extension (there’s a natural transformation to it).

Since I had my extension on the wrong side, I’ve always been confused by the pointwise construction of the extension. Let’s have H be the right extension, in the “correct” sense above, so that HF\to G. In several places (references at the bottom) I’ve seen that one can compute H(b) as a limit over an arrow (slice) category:

H(b)=\lim\limits_{\substack{Fa\to b\\ \in F\downarrow b}}G(a).

For convenience, let me suppose F:\mathscr{A}\to\mathscr{B} is the inclusion of a sub-category. Then I can suppress it from the notation, and the above becomes

H(b)=\lim\limits_{\substack{a\to b\\ \in \mathscr{A}\downarrow b}}G(a).

But here’s where I always got confused – I’m supposed to get a natural transformation H\to G. But this means I’m looking for a family of maps, each of which is a map out of a limit. I don’t like mapping out of limits. That’s not what they are for. Limits are for mapping to.

Today, I finally realized that you do, actually, map “out” of this limit, in a sense. Taking a\in \mathscr{A}, we’re supposed to get a map H(a)\to G(a) – that is, a map

\left(\lim\limits_{\substack{a'\to a\\ \in \mathscr{A}\to a}}G(a')\right)\to G(a).

The map we want actually comes from part of the definition of a limit. Since id:a\to a\in \mathscr{A}\to a, this arrow is an object that the limit is taken over. And the value of the functor whose limit we are taking, evaluated at this object, is just G(a). And so, by definition of limit, we always have a “projection” from the limit to the functor evaluated at any of the objects we are taking the limit over, and so we’ve got our map.

Of course, you’re supposed to check this is “works” – that’s its natural and universal and probably other things as well. But I’m happy now, because I finally have things on the standard side, and see where the maps come from.

While I’m on the subject, I should point out that if \mathscr{A} is a full subcategory of \mathscr{C}, then H(a)=G(a) for all a\in\mathscr{A}, because id:a\to a is an initial object in the arrow category you take the limit over. If the inclusion isn’t full, though, this need not happen.

As another (final) note, I should mention why I’m looking at Kan extensions. Possibly after changing limits to homotopy limits in the above, Kan extensions are useful because they preserve homotopy limits. With the notation above, \text{holim }H\simeq \text{holim }G. So if you’ve got a functor out of some big category, but show that it’s equivalent to the extension of a functor on a subcategory, you can work with the smaller category to think about homotopy limits.

References: When Wikipedia isn’t enough (Kan extension), I look at MacLane’s “Categories for the Working Mathematician”. I’m also a big fan of Borceux’s 3 volume “Handbook of Categorical Algebra”. Perhaps with my new understanding, I should go see what I can make of MacLane’s statement that “All Concepts Are Kan Extensions”…

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3 Responses to “Kan Extensions”

  1. sumidiot Says:

    “Understand” may be the wrong word above. I met with my advisor yesterday, and I’m not sure any of the above is correct. My impression from him is that _right_ kan extensions are _right_ adjoints. Since that’s helpful.

  2. Nick Says:

    So my limits above should be taken in _under_ categories, instead of _over_ categories. Or, the functor I’m trying to extend should be contravariant, in which case the above is, I think, legit.

  3. Topos Says:

    When I first had to figure this out in ‘prehistoric times, 1959’ in Dan Kan’s course I reverted back to my Category Understanding Methods for Idiots, drawing the pictures at each category using Mac Lane’s metaphor that Functors map commutative diagrams to commutative diagrams. I drew & followed a typical triangle diagram as it went ‘down into’ C, and how it travelled to B and then to C. [ I cheat with diagrams and color each diagram]. I then saw the triangular prisms and facets as they ‘flowed’ – natural transformations – in C which, with the color melding made the definition of RANk and LANk clear and unforgettable. Where was PIXAR animations when I needed them? 🙂

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