
Category Theory for Dummies [pdf] - karlzt
http://homepages.inf.ed.ac.uk/jcheney/presentations/ct4d1.pdf?
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xyzzyz
It seems that category theory is the new hype -- almost nobody actually
understands what it is about, or, more importantly, what it is for. Let me
tell you what it is for -- it's an important technical tool in mathematical
research, which gives you new, coherent language, sometimes provides you with
an additional insight in the structure of the stuff you are researching and
makes it easier to notice and classify similarities between different kind of
structures. Unfortunately, it is almost completely useless and uninteresting
by itself -- because, well, what's interesting in objects and arrows anyway?

What make category theory interesting are its connections with various field
and math and computer science. That's why introducing "category theory for
dummies" makes completely no sense -- it's like following Erlangen program to
teach kids about points, lines and circles on a plane. The need and the
significance of Erlangen program arise when you learn about many different
geometries, notice what they have in common and what they do not, and try to
find out what the geometry is all about. Without it, the Erlangen program is
all about abstract bullshit, and the situation is completely the same with
category theory. But nobody writes or posts Erlangen program for dummies on
HN. Why? "General theory of everything" hype, that's why. Erlangen program is
"general theory of geometry", but geometry seems a bit pale when compared to
everything.

If you really want to understand the significance of category theory, then
learn set theory, then algebra, then topology, then algebraic topology and
algebraic geometry, or take abstract programming languages theory path. If you
don't care about all this stuff, because you're hyped on the category theory,
then you're missing the point -- it's like you wanted to learn about algebraic
topology, but did not care about algebra or topology.

Also anything that has "for dummies" in title should invariably remind you of
Norvig's essay (google Peter Norvig 21 days).

Also my old comments about category theory on HN:
<http://news.ycombinator.com/item?id=2713315>

<http://news.ycombinator.com/item?id=2713510>

~~~
more_original
Category theory more than just a language for classifying and organizing
existing results. It is the mathematical theory of universal properties
(<http://en.wikipedia.org/wiki/Universal_property>).

In category theory one studies what structure can be captured by universal
properties. That is one tries to characterizes structure uniquely by its
properties, rather than giving a concrete construction, as is common in set
theory. The emphasis on universal properties leads to a different way of
thinking and mathematical taste. The study of what one can do in general with
universal constructions is thus interesing in itself, though perhaps not for
dummies.

Anyway, what I want to say is that while category theory has been overhyped in
the context of Haskell, for example, it is not true that it is "completely
useless and uninteresting by itself".

~~~
xyzzyz
This exactly what I mean by:

 _sometimes provides you with an additional insight in the structure of the
stuff you are researching and makes it easier to notice and classify
similarities between different kind of structures_

I just tried to avoid the notions which average HNer is unlikely to know, and
understanding them takes too much time.

Anyway, I have yet to see a deep and nontrivial result in a pure category
theory, and mind you, I have the Mac Lane's book in front of me. Most of it is
easy symbol juggling which only result in something interesting when you think
what it means in context of concrete structures.

------
DanielRibeiro
For a not so short, and way scarier introduction: _Physics, Topology, Logic
and Computation: A Rosetta Stone_ [1]

However it doesn't shy away from the fact that the sets of all sets is not
set, and therefore we need a higher level of abstraction to talk about these
non-set things. The same, of course, applies to other set _like_ structures:

All sets do not form a set, but form a category

All monoids do not form a set, but form a category

All graphs do not form a set, but form a category

[1] <http://math.ucr.edu/home/baez/rosetta.pdf>

~~~
antimora
Thank you. It seems like a good resource - very interesting.

------
antimora
Out of curiosity, I did some search on Category Theory popularity on Google's
"books ngrams" and found it peaked around 1990 and started picking up again in
2000

Here is the source:

[http://ngrams.googlelabs.com/graph?content=category+theory&#...</a>

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antimora
Here is a great "An introduction to Category Theory for Software Engineers"

<http://www.cs.toronto.edu/~sme/presentations/cat101.pdf>

~~~
T_S_
Thanks for that link. It is much better than the one headlined.

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charlieflowers
Did you know that mathematicians refer to Category Theory as "General Abstract
Nonsense"? It's true. And they mean it as a _compliment_!

It is entirely possible that Category Theory is a practical joke that the
mathematicians are playing on the rest of us.

~~~
MaysonL
<http://en.wikipedia.org/wiki/Abstract_nonsense>

------
petrilli
This is a heretofore unknown definition of "dummies".

------
antimora
I am curious how many people know about the Category Theory?

I asked one of the Math professors and he had no idea, though we was teaching
Group Theory at the time.

~~~
zitterbewegung
Which college? Note that category theory is heavily used in topology but
homological algebra is used in abstract algebra (group theory). See
<http://en.wikipedia.org/wiki/Category_theory>

~~~
eldina
I think category theory is rarely essentially used in a first or second
typical course on some topic in math. Once in a while if students can be
assumed to have had a course in category theory, then it is more elegant and
efficient e.g. simply to show that some functor has an adjoint hence this or
that limit is preserved, but to me category theory, except if it is the main
topic of interest, is useful because it eases communication and allows you to
quickly get some understanding of a construction that might look very "local"
to some category that you are not familiar with. Unlike the person who created
the cited notes, I think it actually can help understanding the things under
study or the associated constructions, assuming sufficient command of category
theory. I remember when I first understood the definitions of things like
products, coproducts, push-outs and pull- backs etc. in my last year as an
undergraduate. Suddenly, for many of the constructions from topology and
algebra it became easier to remember them, how they were constructed and which
properties they had. To me it is kind of like with design patterns, mainly I
don't use them as tools picked up from my tool box when solving a problem,
rather they simply allow me to communicate more easily and gives me another
level of abstraction where I can reuse thinking I have done earlier.

