Before Groups from Categories: a Category Refresher

Jan 20 2008 Published by under Category Theory, Group Theory

So far, I've spent some time talking about groups and what they mean. I've also given a
brief look at the structures that can be built by adding properties and operations to groups -
specifically rings and fields.

Now, I'm going to start over, looking at things using category theory. Today, I'll start
with a very quick refresher on category theory, and then I'll give you a category theoretic
presentation of group theory. I did a whole series of articles about category theory right after I moved GM/BM to ScienceBlogs; if you want to read more about category theory than this brief introduction, you can look at the category theory archives.

Like set theory, category theory is another one of those attempts to form a fundamental
abstraction with which you can build essentially any mathematical abstraction. But where sets
treat the idea of grouping things together as the fundamental abstraction, category
theory makes the idea of mappings between things as the fundamental abstraction.

A category is thus a simple structure consisting of a collection of objects,
connecting by arrows (arrows are also called morphism). For intuition, you
can think of objects as sets, and arrows as functions: an arrow is a mapping from one object
to another, much like a function is a mapping from one set to another. (But that's only a very
vague intuition: in category theory, we never consider the internal structure of the objects
or the arrows: the objects and arrows are the primitive atoms of category theory - and there
are some categories where that intuition will lead you astray.) If there's an arrow f from
object a to object b, we'll write that as "f:a→b".

In some sense, the objects of a category are almost unnecessary; what we care about are
the arrows. The objects are only there to give us something to connect with arrows. It's like
a connect-the-dots puzzle: the dots aren't important parts of the picture: they're just there
to show you where to draw the lines.

In addition to the arrows, there's a single fundamental operation on arrows, called
composition. Given two arrows f:a→b and g:b→c, we can compose f and g,
getting a new arrow from a to c. We write the composition of f and g "gºf". Arrow
composition has to meet two properties, which are going to look pretty familiar from group
theory: associativity, and identity. Associativity says that
aº(bºc)=(aºb)ºc; and identity says that that for any arrow f:a→b,
there are arrows 1a and 1b such that 1bºf = f =
fº1a. (In other words, for any arrow, there's something you can compose it
with that won't change it.)

Category theorists have a lot of jargon for describing arrows with additional properties.
There's a rundown of the basic ones here.
For an example that's important for looking at group theory related topics, we say than an
arrow is iso (i.e., it is a iso-morphism) if it's reversable - which in category
theoretic terms means that if f is an iso-arrow, then there is an arrow f-1 such
that fºf-1=1.

In category theory, one useful technique is diagrams. You can draw a set of objects and arrows as a diagram. The diagram is said to commute if any two paths in the diagram that have the same start and end-points compose to the same arrow. It's important to realize that a diagram is an explanatory tool - not a proof. (That's a common error - I can't count how many CS papers I've seen that show a diagram without proving that it actually commutes!) Just because you can draw it doesn't mean that it commutes. You've got to show that.

principle-arrow.png

For an example of a diagram, there's a type of arrow between two objects A and B called a principle morphism. The idea of a principle morphism is that a morphism M from A to B is principle if and only if every self-arrow of A (or endomorphism, an arrow from A to A), composed with an arrow from A to B yields M. As a very visual thinker, I find that paragraph hard to follow. But in category theory, we can say that m is principle if and only if for every arrow x from A to A, and for every arrow y from A to B, the diagram to the right commutes.

The key to reading diagrams is to realize that you really need to look carefully at the labels. The same object can (and often will) appear multiple times in the same diagram. But any two paths between the same objects has to be equivalent. So in the example above, the object A appears two different times. That helps make clear that we're saying that composing an arrow from A to A with an arrow from A to B always yields m; if we only drew A once, we'd have a looped arrow on A, and two arrows from A to B. It wouldn't be clear what we were asserting about which arrows compose with which. By drawing A twice, we've done something which is equivalent, but easier for a human to read.

As I said, in category theory, we're studying structures in terms of mappings. One of the
major tools of the field is to use higher-order mappings to study the structures of
categories. If you've got a category with arrows, you can define a higher-order mapping called
a functor, which is a mapping from arrows to arrows. (And it's also an arrow in the
category of categories.) Above that, there's something called a natural
transformation
which, very loosely described, is a mapping from functors to functors. The
idea of all of these mappings is to provide a tool for talking about structures, how they
work, and what they mean. In every case, you can view a mapping as something that preserves
some kind of structure. An arrow in a category is a mapping between things that have some
structural similarity, and the mapping preserves that. A functor maps between arrows in a
category, preserving the structure described those arrows. A natural transformation maps
between functors, preserving the structure described by those functors.

Since from the viewpoint of abstract algebra, we're building sets with a bit of structure
to them (provided by the operations), category theory is a great tool for studying groups,
rings, fields, and the other things in abstract algebra.

No responses yet

  • Magus says:

    I'm slightly confused about the way you explain identity morphisms. You said that f:a->b, and I assume that 1a:a->a and 1b:b->b. If that's the case, wouldn't(to borrow some Haskell syntax) 1b . f = f = f . 1a not be correctly typed? I'd think that it would instead be 1a . f = f = f . 1b
    Was that a typo, or am I missing something fundamental?

  • Nathan says:

    Composition is performed from right to left, so composing f with 1_a means perform 1_a:a->a first, and then f:a->b, so he's correct.

  • Blaise Pascal says:

    I found it very frustrating when I took Category Theory (a nice 400-level course that filled an upper math requirement but didn't have any prerequisites listed) to have groups simply described as "a group is a monad (i.e., category with one object) where every arrow is an isomorphism". Perfectly true, but not particularly helpful, especially as I had yet to take Abstract Algebra.
    I trust you'll be a bit more expressive than that?

  • Alex Besogonov says:

    Actually, the problem with composition (or functions or arrows) is that there are _different_ notations!
    I've read books that treat AoB as right-to-left and left-to-right compositions.

  • Graeme says:

    Mark, I think you've typed 'principle' in some places when you mean 'principal'.

  • Andrew says:

    The description of an isomorphism is a little lacking:
    an arrow f:A->B is an isomorphism if there is a g:B->A
    such that g o f = idA and f o g = idB

  • ray burchard says:

    In accordance with reduction mathematics, where the defined magnitude is 9, doesn't the zero '0' serve as the collective arrow indicating sequential flow, as left to right and/or right to left.
    Example;
    (01+02+03+04 thru 09) = 045 = (00+04+05) = 09 = (0+9) = 9
    (90+80+70 back thru 10) = 450 = (40+50+00) = 90 = (9+0) = 9
    Therefore, (9+9) = 18 = (1+8) = 9.
    Where (09+09+09+09+09) = 045 = (0+4+5) = 9
    minus (01+03+05+07+09) = 025 = (0+2+5) = 7
    Equals (08+06+04+02+00) = 020 = (0+2+0) = 2
    While (90+90+90+90+90) = 450 = (4+5+0) = 9
    Minus (90+70+50+30+10) = 250 = (2+5+0) = 7
    Equals (00+20+40+60+80) =200 = (2+0+0) = 2
    Therefore 045 is equivalent to 01,one nine, while 450 is then equivalent to 10, ten nines and (01+10) = 11 nines or 99, where (9+9) = 18 =(1+8) = 9.
    Divide 99 by 11 = 9, now divide 09 by 11 = 0.8181... Plus 90 divided by 11 = 8.1818... Therefore (0.8181...+8.1818...) = 8.9999 as the fractional equivalent of 9.
    This is the method of conjugating the dichotomy of (dynamic/static) as 0 thru 9, (ten digits while nine values) with 1 thru 9, (nine digits with nine values)
    [(09+09+09+09+09) + (90+90+90+90+90)] = (99+99+99+99+99) = 495 where 495 is built from a concentric center out (045+450) = 495.
    Therefore:
    [(08+06+04+02+00) + (00+20+40+60+80)] = (08+26+44+62+80) = 220
    [(01+03+05+07+09) + (90+70+50+30+10)] = (91+73+55+37+19) = 275
    __________________________________________________________
    [(09+09+09+09+09) + (90+90+90+90+90)] = (99+99+99+99+99) = 495
    Where (2+2+0) = 4 and (2+7+5) = 14 = (1+4) = 5 then (4+5) = 9.
    Divide 495 by 11 = 45, then divide 045 by 11 = 4.0909... Plus 450 divided by 11 = 40.9090... Equals (4.0909... + 40.9090...) = 44.9999...
    Now try the principles of 495, as 220 plus 275.

  • Jesse Farmer says:

    A good, basic overview of category theory, but you're missing the "why." Most mathematicians I know don't study category theory in itself. Rather, category theory is a tool used to study other branches of mathematics, usually when there's a strong connection between two branches.
    I'd love to see an article titled "Why We Care About Category Theory."
    Historically it came about as generalizations from the field of Algebra Topology, which tries to connect groups to topological spaces. That's when you start getting into fun stuff: fundamental groups, covering spaces, cohomology, etc.

  • Pseudonym says:

    It's "principal morphism", not "principle morphism".

  • ray burchard says:

    Jesse Farmer,
    To answer your query, "Most mathematicians I know don't study category theory in itself. Rather, category theory is a tool used to study other branches of mathematics, usually when there's a strong connection between two branches"
    It is the "whole" versus fractionalization (specialization), (dynamic/static), antithetical duality. The answer can be seen in why Academia's impetus has been to promote Einstein's 'Relativity' while basically ignoring E.A.Milne's 'Kinematic Relativity'.
    This is when Academia's leadership is then guided by a collective psychological predilection to redefine mathematics to facilitate commerce and thereby creating a mathematical foundation for a cultural logic of exclusion (solipsism), predicated on fractionalization (specialization) where the "individuals" dominate the "whole" instead of being subordinate to the "whole".
    Then as to, "Why We Care About Category Theory.", you have already answered your own question in your blog, "Memo to OpenSocial: It's about distribution, stupid!", where magnitude (the "whole" and/or co-homology) rules. "That's when you start getting into fun stuff:"

  • Stephen Wells says:

    Ray Burchard: strongly suggest that you open a window and allow the crack fumes to disperse before commenting again.

  • Anonymous says:

    Ok, I'm a bit confused about principal morphisms.
    Suppose that m: a -> b is a principal morphism. Then let y: a -> b also be a morphism connecting a to b, and 1_a is a morphism connecting a to a, so:
    y = y . 1_a = m
    ??
    This would seem to say that every morphism y connecting a to b is the same as m, or in other words, if there is a principal morphism, then it is the only morphism that connects the two. That seems kind of odd to me, so I guess I'm missing something.

  • Doug Spoonwood says:

    [In accordance with reduction mathematics, where the defined magnitude is 9, doesn't the zero '0' serve as the collective arrow indicating sequential flow, as left to right and/or right to left.]
    No, it doesn't serve as an arrow in your examples. In your examples 0 MERELY serves as a place holder. 1+2+...+9=45, as does 9+8+...+1=45, as does 01+02+...+09=45, and 90+80+...+10=450. Concerning dynamic/static, fractional/whole... have you ever heard of the fallacy of reification? If not, look it up FAST.
    Mark,
    [Arrow composition has to meet two properties, which are going to look pretty familiar from group theory: associativity, and identity. Associativity says that aº(bºc)=(aºb)ºc; and identity says that that for any arrow f:a→b, there are arrows 1a and 1b such that 1bºf = f = fº1a.]
    So, if we have f:a->b and f(-1):b->a, then
    f*f(-1)=b->b, and f(-1)*f=a->a. So, 1a=a->a, and 1b=b->b. In such a case, we have inverses for categories. Or did I do something wrong and we only have a monoid structure?

  • ray burchard says:

    Doug Spoonwood,
    Your having to much trouble getting your thinking around the (antithetical duality) concept. The issue at hand is as Jesse Farmer states in post #8, "category theory is a tool used to study other branches of mathematics". What we have is two conjugated, (using the same numerical system), systems of 'Applied' mathematics with a (dynamic/static) relationship. The same relationship that couples two distinctly opposites while maintaining separation as in, animal and vegetation, muscle and bone.
    One the calculative system, that as you are conversant with, which is designed to mathematically fragment and/or fractionalize the static principles of an inanimate 'whole' for management and/or manipulation.
    Then conjoined with another system of 'Applied' mathematics, as Life's Owners Manual, designed through inverse numerical symmetries to example the multi-dimensional, animated systems matrix of living entities and organisms.
    As to your statement, "In your examples 0 MERELY serves as a place holder", you are partially correct, the zero '0' does serve as a spacer, but also it serves as a directional indicator. While individually people can walk forward or backward, but they have a predilection to travel in the direction their sensory receptors are pointed. Also the '0' implies a ten to one, one to ten ratio (01 mirrored image 10).
    Example:
    (The Meaning of Division: Quotient Groups post #30)
    the "whole"-----directional bifurcation-----------sub-units (90 +09) = 099
    ----------------------A
    A) 4554-------= (4500 + 0054) =--------------------(4050 + 0504)
    ----4653-------= (4600 + 0053) =--------------------(4050 + 0603)
    ----4752-------= (4700 + 0052) =--------------------(4050 + 0702)
    ----4851-------= (4800 + 0051) =--------------------(4050 + 0801)
    ----4950 b)----= (4900 + 0050) =--------------------(4050 + 0900)
    a) 5049-------= (5000 + 0049) =--------------------(5040 + 0009)
    ----5148-------= (5100 + 0048) =--------------------(5040 + 0108)
    ----5247-------= (5200 + 0047) =--------------------(5040 + 0207)
    ----5346-------= (5300 + 0046) =--------------------(5040 + 0306)
    ----5445 B)---= (5400 + 0045) =--------------------(5040 + 0405)
    -------------------------------- B
    _+_________+__________________________+_____________
    ---49995--------49500 + 00495----------------------45450 + 04545
    MIRROR IMAGE (01 to 10) RATIO WHERE;
    (4653-4554) = 099 while (4554-3564) = 990
    ----4554-------= (4500 + 0054) =---------------(4050 + 0504)
    ----3564-------= (3500 + 0064) =---------------(3060 + 0504)
    ----2574-------= (2500 + 0074) =---------------(2070 + 0504)
    ----1584-------= (1500 + 0084) =---------------(1080 + 0504)
    ----0594-------= (0500 + 0094) =---------------(0090 + 0504)
    _+_________+______________________+___________
    --12870 =------(12500+00370)----------------(10350 + 2520)
    ----9405-------= (9400 + 0005) =---------------(9000 + 0405)
    ----8415-------= (8400 + 0015) =---------------(8010 + 0405)
    ----7425-------= (7400 + 0025) =---------------(7020 + 0405)
    ----6435-------= (6400 + 0035) =---------------(6030 + 0405)
    ----5445-------= (5400 + 0045) =---------------(5040 + 0405)
    _+_________+______________________+___________
    --37125 =------(37000+00125)----------------(35100 + 2025)
    --12870 =------(12500+00370) =--------------(10350 + 2520)
    --37125 =------(37000+00125) =--------------(35100 + 2025)
    +_________+_______________________+____________
    =49995 =------(49500+00495) =--------------(45450 + 45450)
    Doug I can prove this system is One, the Mobius Band continuum and Two, it's axiomatic which then splits with magnitude.

  • jeff says:

    I'm confused. Shouldn't the diagram be
    y
    A -> B
    ^
    |x
    A
    with m on the undrawn diagonal up from A to B? (The ascii diagram is defeating me.) The difference is in the direction of the arrow x from A to A. It says that y composed with x is m but the diagram shown doesn't show that y composes with x, or does it?

  • ray burchard says:

    Be mindful of the horizontal and vertical inverse sequencing and their coparcener (01 to 10) ratio.
    ..(00..+..90) = ....(090..+..900) = ....990
    ..(01..+..80) = ....(081..+..810) = ....891
    ..(02..+..70) = ....(072..+..720) = ....792
    ..(03..+..60) = ....(063..+..630) = ....693
    ..(04..+..50) = ....(054..+..540) = ....594
    +___ +______+______+_______+____
    (010 + 350) = ....(360 + 3600) = ...3960
    ..(05..+..40) = ....(045..+..450) = ....495
    ..(06..+..30) = ....(036..+..360) = ....396
    ..(07..+..20) = ....(027..+..270) = ....297
    ..(08..+..10) = ....(018..+..180) = ....198
    ..(09..+..00) = ....(009..+..090) = ....099
    +___ +______+_____ +_______+____
    (035 + 100) = ....(135 + 1350) = ...1485
    (010 + 350) = ....(360 + 3600) = ...3960
    (035 + 100) = ....(135 + 1350) = ...1485
    +_______________________________
    (045 + 450) = ....(495 + 4950) = ...5445, or (01+10) = 11 x 495.
    Then;
    (009..+..090) = ....099, and (018..+..081) = ....099, and 027 + 072 etc...
    (090..+..900) = ....990, and (180..+..810) = ....990, and 270 + 720 etc...
    +__________________ +_______________________
    (099..+..990) = ...1089, and (198..+..891) = ..1089, and 297 + 792 etc...
    Therefore the correlation you seek can be seen demonstrated through a mathematical juxtaposing of the principle and sub-principles of (The Meaning of Division: Quotient Groups post #30) and the other half represented in post # 14 of this thread.
    #30
    ...23760 = ....(23500..+..00260)....and....(20250..+..3510)
    ...12870 = ....(12500..+..00370)....and....(10350..+..2520)
    +________+________+_______....and..+______________
    = 36630 = ....(36000..+..00630)....and..= (30600..+..6030)
    #14
    ...26235 = ....(26000..+..00235)....and....(25200..+..1035)
    ...37125 = ....(37000..+..00125)....and....(35100..+..2025)
    +________+________+_______....and..+______________
    = 63360 = ....(63000..+..00360)....and....(60300..+..3060)
    Therefore #30 plus #14 totals are:
    # 30....36630 = (36000..+..00630)....and....(30600..+..6030)
    # 14....63360 = (63000..+..00360)....and....(60300..+..3060)
    _________________________________________________
    ......... 99990 = (99000..+..00990)....and....(90900..+..9090)
    Here is the functional Mobius twist in conjugating the dynamic with the static. While both # 30 and # 14 demonstrate a difference between their principles as, # 30 (23760 - 12870) = 10890 and # 14 (37125 - 26235) = 10890 look at the directional flow of the paired order of magnitudes and the twist required in # 14 in order to subtract the smaller amount from the larger. Demonstrating that while the both flow matrix are designed the same, their direction of flow are opposite. Extrapolate # 30 forward, (12870 + 10890) = (23760 + 10890) = (34650 + 10890) = 45540. Now reverse the process and extrapolate # 14 backward, (37125 - 10890) = (26235 - 10890) = (15345 - 10890) = 04455. Now add the two together (45540 + 04455) = 49995 and (12870 + 37125) = 49995 and (23760 + 26235) = 49995 etc...
    Take the # 30's 34650 and take the leading 3 from it's position and add it to the 0 position, 04653. Now take # 14's 15345 and take the leading 1 from it's position and add it to the rear 5 position, 05346. Now add the paired set reversals, (04653 + 05346) = 09999. Now reverse the process and 99990. Try # 30's, 23760 and # 14's, 26235.
    I have taken this quadratic concentric system much, much further, as 198, 165 and 132 also equal 495 and 1089 is 33 squared while it's sequence reverse 9801 is 99 squared.