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Structure and Structure-Preserving Morphisms

January 23, 2014

In one of my past lives (plz don’t ask me about reincarnation ;) I used to be good at measuring neurons in plastic dishes and in behaving brains.  Let’s see if I’m any good at measuring dynamical systems, processes, graphs and things of such nature.

You look at me and you see that my hair color is black.  You look at yourself and you see the black color of your hair.  What we have here is a measurement

color: A → B

with A = {you, me}, B = {black, blue} and

color (you) = black

color (me) = black

I’d have died of boredom a long, longtime ago had the things around me were as structure-less as abstract sets.  Thankfully there are graphs with little more structure which will keep us entertained for the rest of this post.

A functor

A: GS

from the category of graphs G to the category of sets S is an S-valued measurement of graphs in G.  But what about graphs is the functor A measuring?  When I look at a graph, I see arrows (somewhat like the way you saw, amongst innumerable attributes, my hair color).  So we say that the functor

A: GS

extracts from each graph G its sets of arrows AG

A(G) = AG

I could have, instead of seeing arrows, attended to dots.  So we have another measurement given as another functor

D: GS

extracting from each graph G its set of dots DG

D(G) = DG

If you have two measurements, say, height and weight, and not a whole lot happening in your life, then you start playing with your measurements and, god willing, come up with sky-piercing theories like height increases with weight ;)  So we play with our measurements of graphs

A: GS

D: GS

There is a natural transformation (Sets for Mathematics, page 241)

s: A → D

from the arrows functor A to the dots functor D assigning to each graph

G

(in the category of graphs G) an arrow

D(G)

^

|

sG

|

A(G)

i.e. the function

DG

^

|

sG

|

AG

(in the category of sets S) and to each graph map (morphing one graph into another without tearing; Conceptual Mathematics, page 210, 371)

g: G → G’

(in G) a commutative diagram

D(G) – D(g) –> D(G’)

^                      ^

|                       |

sG                     sG’

|                       |

A(G) – A(g) –> A(G’)

i.e. the square of functions

DG  – gD –>  DG’

^                      ^

|                       |

sG                     sG’

|                       |

AG  – gA –>  AG’

satisfying gD º sG = sG’ º gA, where ‘º’ denotes composition of functions in the category of sets S.

If upon looking at the component

DG

^

|

sG

|

AG

and the commutative diagram

DG  – gD –>  DG’

^                      ^

|                       |

sG                     sG’

|                       |

AG  – gA –>  AG’

of the natural transformation

s: A → D

you are experiencing déjà vu, then it’s because we have been modeling graphs G as a pair of sets

set of arrows AG

set of dots DG

and a pair of functions

s, t: AG → DG

assigning to each arrow in AG its source, target dots in DG and a graph map

g: G → G’

as a pair of functions

gA: AG → AG’

gD: DG → DG

satisfying

gD º sG = sG’ º gA

gD º tG = tG’ º gA

(see Conceptual Mathematics, pp. 141 – 2).  Isn’t it fascinating that morphisms of structures can be identified with natural transformations.

I’m in a rush, but I’ll be back with measurements of dynamical systems (cf. natural transformation from the functor extracting states to the [same] state functor) and processes.

7 Comments
  1. The notion of a map f: A –> B respecting specified endomaps of A and of B is defined on page 60
    (as the obvious commutativity condition), but just how commutativity corresponds to an intuitive notion of “respecting” is first mentioned in passing on page 155 (in the middle of a discussion of more complicated matters) and really explained on pages 172ff.
    Blass

    see also Types of structure (Conceptual Mathematics, pp. 149 – 51)

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