On Wed, Jun 22, 2011 at 5:03 PM, Gregg Reynolds <span dir="ltr"><<a href="mailto:dev@mobileink.com">dev@mobileink.com</a>></span> wrote:<div><br><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div class="gmail_quote"><div>Well, you're way ahead of me. I don't even "get" adjunctions, to tell you the truth. By which I mean that I have no intuition about them; it's not so hard to understand the formal definition, but it's another thing altogether to grasp the deep significance.</div>
</div></blockquote><div><br></div><div>In short, an adjunction is the relationship that for every "limit" of type A, there is a corresponding limit of type "B", and vice-versa, realized by a functor F that maps A to B and a functor F* that maps B to A.</div>
<div><br></div><div>Since F and F* map limits to limits, they preserve more algebraic structure than just any old functors F :: A -> B and G :: B -> A. In particular, adjoint functors are "continuous". (There are very strong parallels with topology, owing to Category theory's history as a language for describing topological constructs without reference to point sets.)</div>
<div><br></div><div>Every adjunction gives rise to a monad -- a generalized closure operator (topology and lattice theory are very intimately related). In particular, if a category is complete (contains all its limits), then it will have the same structure as the monad generated by an adjunction.</div>
<div><br></div><div>There are some good videos on youtube for gaining intuition about some of these issues:</div><div><a href="http://www.youtube.com/watch?v=loOJxIOmShE&feature=related">http://www.youtube.com/watch?v=loOJxIOmShE&feature=related</a></div>
<div>is a good place to start the series.</div></div></div>