From: <puite@math.uu.nl>
To: categories@mta.ca
Subject: Re: Category Theory from RFC Walters' book
Date: Wed, 1 Nov 2000 12:05:43 +0100 (MET) [thread overview]
Message-ID: <200011011105.MAA22504@schartriller.math.uu.nl> (raw)
> Hello Category Theory community,
>
> I am rereading RFC Walters' "Categories and Computer Science".
> In chapter 1 Section 3 (starting on pg . 10), Walters discusses the
> the notion of generators and relations to generate categories. He
> gives several examples of this concept.
On "generators":
Given a graph (set of objects and set of (formal) arrows between objects),
the original objects together with the finite *paths* (f,g,h,...,k) in
this graph (as morphisms) constitute a category (called
the "free category on this graph"):
- identities are the empty paths;
- composition is concatenation of paths.
On "relations":
Given a category and a relation R on its arrows (i.e. a subset of A x A)
relating only arrows with the same domain and codomain, let R' be the
smallest congruence relation containing R.
[A congruence relation S is an equivalence relation (i.e. S is
reflexive, symmetric and transitive) that moreover
is closed under composition: aSb and cSd imply (a.c)S(b.d).]
[Check that the " *smallest* congruence relation containing R "
is a well-defined notion, as the class of congruence relations containing R
is closed under arbitrary intersections. So the intersection of those
congruence relations containing R is also a congruence relation
containing R, indeed the smallest one.]
Now the original objects together with the congruence classes [f] of R'
(as morphisms) constitute a category (called the quotient category):
- identities are the classes of the identities [1];
- the composite [f].[g] is by definition [f.g], which is well-defined.
There is a more explicit/constructive description of R' in terms of R.
Consider the following relation:
aR"b iff there exist n \geq 0 and an n-chain:
(n+1) morphisms a_0, a_1, ..., a_n such that
a = a_0;
a_n = b;
for all 0 \leq i < n there are morphisms u,v,c,d such that
a_i = u.c.v and a_{i+1} = u.d.v and either cRd or dRc
This clearly is a congruence relation containing R, whence R' \subseteq R".
The other way around, you show by induction on n that every n-chain is
between R'-congruent formulas. Hence R' = R".
There is much more to say about presentations in general, but I hope the
above details for categories answer your questions.
Let me briefly return to your specific example:
> E.g. "Example 15. Consider
> the category presented by one object A, one arrow e:A->A, and the
> one relation e.e.e.e = 1subA. From this, he deduces by hand the
> category has additional arrows e .e, e.e.e. I have two questions:
Observe that the free category on this graph (one object, one arrow)
has infinitely many arrows: (); (e); (e,e); (e,e,e); (e,e,e,e); ...
Let us denote them by e^0, e^1, e^2, e^3, e^4, ...
Recall that these arrows are actually the paths in the graph, and
that they are all different!
According to your example R = { (e^4,e^0) }.
Check that R' = R" = { (e^n,e^m) | n-m is a 4-fold }.
Check that there are only four congruence classes in the quotient
category: [e^0], [e^1], [e^2], [e^3].
An example of a composite: [e^2].[e^3] = [e^2.e^3] = [e^5] = [e^1].
> 1) Walters shows that e.e.e = e is not "deducible". What kind of
> formal system/inference rule system is "at work" here that
> allows us to deduce
> formally any additional arrows and allows us to deduce
> arrow relations from the "axiom" relation, i.e. e.e.e.e = 1subA??
> Is this some kind of equational logic? Please specify in detail.
You do not deduce additional arrows; the arrows are the original paths.
R' can be described by the following deductive system:
axioms: the instances of R and reflexivity;
rules: symmetry, transitivity, closure.
> 2) Given generators and relations are we guaranteed to get a category?
Yes, since we agreed upon considering the smallest congruence relation
containing your relations.
Regards,
Quintijn Puite
next reply other threads:[~2000-11-01 11:05 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-11-01 11:05 puite [this message]
[not found] <200011011244.eA1CiKt07396@nmh.informatik.uni-bremen.de>
2000-11-02 10:30 ` Ronnie Brown
[not found] <F112R3XgbPhRaUMNGCX000043d1@hotmail.com>
2000-11-01 10:04 ` Steve Vickers
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