I'd want also to add the others in giving John a big Thank You for all the effort and guidance, the helping hand of a field expert has done charms and made a difference, and prepared us for better self-learning. I'm going to miss this advice. Can't …
In page 24 there is helpful insight on the hints of the end paragraph of previous lecture. Efforts like this aimed at a wider categorical outreach out of closed circles shine. Thanks to the author!
We've seen pairs of adjoint functors, and we've seen 'duals' in compact closed categories. In fact they're closely related! There is a general concept of adjunction that has both of these as special cases!
Interest stirred up!
What a motivational booster! One question, assuming legit that monad language, would \(\mathbf{Cocomp}\) be equivalent to the Eilemberg-Moore category of that presheaf "monad"?
Hey, there's a very recent (July!) paper from Fiore, Gambino, Hyland and Winskel addressing precisely the size issues of the profunctor part of the analogy I wrote about earlier, how good!
Part 2/2
Now, in the profunctor side of things, John starts with a teaser:
Sums, existential quantifiers, colimits and coends are all special cases of 'weighted colimits'. Basically they're all ways to think about addition.
And here drops the…
Part 1/2
Sorry for elaborating the excursus, but wanted to give another stab to the profunctors vs linear algebra analogy trying to tie together what John has said. It's fun/profitable to pursue this, juicy, for me is work in progress. The gist of …
Hi Matthew,
Your question here is quite intriguing, I haven't an answer but wanted to share what I've found up to now.
Let's fix a field \(\mathbb{K}\) for the rest. One can speak of the category FdVectWB of finite dimensional vector spaces over \…
Some minor stuff, when
[ \Phi : X^{\text{op}} \times Y \to \mathbf{Bool} ]
we write \( \Phi : X\nrightarrow Y \) but nLab, Wikipedia and Borceaux write \( \Phi : Y\nrightarrow X \) (from the covariant to the contravariant argument).
For me the course has already been greatly beneficial and I'm finding excruciating that the obligations of a 'day job' don't let me engage and commit fully to this opportunity to squeeze out all the goodness. In my case I cannot afford to fight the …
Hi Cheuk, don't think that analogy is going to help. You can look at this math.SE quetion, this paper, and in ch. 3 of this book of Chris Heunen there is a nice diagram of the wider situation. So you can complicate the problem in terms of enrichin…
I'm lagging a bit, but a question: I'm working to see the general idea that the German-Italian business exemplify, and seeking for a more systematic grip.
Would it be correct to say that in our example \(Lan_G\) sends sets to identities? (or better…
Hi Bruno, sorry for the delay. That table of binary relations exemplifies exactly what I had in mind. I didn't knew your cibernetic theorem but had shared your inklings about the role of control theory in this area. You mention several excelent moti…
Hi Pete, I agree with your view on quantifiers and that one can view \(\forall x \in X: \phi(x)\) as \(\bigwedge_{x \in X} \phi(x)\) (similarly for "exists"). In a posetal category meets are products and the empty product (a limit of an empty diagra…
One concern also after a wiggly road ahead warning.
Thinking about the example, and about what naturality means with the actual data, I started to think laterally and came to the suspicion that what was sought was Puzzle 151. I was thinking, the ca…
Matthew, I'm an imperative languages guy (did some lisp ages ago...) but find itchy that in your enumeration you seem to put Haskellite monads in other shelf than the category-theory ones. I had the vague impression that in Haskell there was more or…
I agree with Matthew that pushing the parallelism is a bad idea. In Aristotle I recall that categories were supreme genera, maximal elements in an order with resemblance to proper classes in set theory but not much fruitful analogy with modern ones …
As another example:
This is a lattice of lattices from section 6.3 in the book on Formal Concept Analysis from Wille and Ganter (ISBN 978-3-642-59830-2).
That is an example of a concept lattice. Another example can be this classification in wiki…
Keith, nice inquire. There's a very didactic book called Picturing Quantum Processes by Bob Coecke and Aleks Kissinger also in e-book form that treats systematically the commonalities you consider, attending to the string diagrammatic view of both s…
Hi Julio,
Let \(\mathcal{I}\) be the category generated by the graph $$1 \overset{i}{\rightarrow} 2$$ with only one nontrivial morphim.
Then two functors \(F,G:\mathcal{I} \to \mathbf{Set}\) correspond to two old functions: \(F(i):F(1) \to F(2)\),…
Thanks John and Matthew, I was thinking more in what John describes, but will give more thought to what you Matthew wrote.
Do you mean if we treat monoids as categories with one object, and ask what adjunctions between categories reduce to in th…
Perhaps an ill posed question, but here it goes anyway. Does the notion of adjunction in the case of monoids reduce to something formerly known? I mean: When the categories are preorders, an adjoint situation is a Galois connection. In the case when…
Hi, I think one can argue that rotation by 350º and by -10º are the same morphism in Vect. They are equal element-wise ("extensionally"), so that would leave us only with one inverse. Another option without this problem is the complex exponential. T…
Another for Puzzle 141, in Set for a surjection defining a labeled partition, the function in the converse direction sending partition labels to any of the elements in its block is a left inverse.