2 Mar 19:36 2008

### Minimal abelian subcategory

```My colleague Walter Mazorchuk has the following question.

Being abelian is a *property* of a category, not extra structure.  Given
an abelian category A, it therefore makes sense to define a subcategory of
A to be an ABELIAN SUBCATEGORY if, considered as a category in its own
right, it is abelian.  Note that a priori, the inclusion need not preserve
sums, kernels etc.

Now let R be a ring and M an R-module.  Is there a minimal abelian
subcategory of Mod-R containing M?  If so, is there a canonical way to
describe it?

Any thoughts or pointers to the literature would be welcome.  Feel free to
assume hypotheses on R (it might be a finite-dimensional algebra etc), or
to answer the question for full subcategories only.

Thanks,
Tom

```
3 Mar 01:21 2008

### Re: Minimal abelian subcategory

```Hi Tom,

Silly observation, but wouldn't the contractible category consisting only
of M and its identity morphism constitute an abelian subcategory by this
definition, albeit one that is trivial?  It would seem that the question
for full subcategories is more interesting (and harder).

Best,
Josh

On Sun, 2 Mar 2008, Tom Leinster wrote:

> My colleague Walter Mazorchuk has the following question.
>
> Being abelian is a *property* of a category, not extra structure.  Given
> an abelian category A, it therefore makes sense to define a subcategory of
> A to be an ABELIAN SUBCATEGORY if, considered as a category in its own
> right, it is abelian.  Note that a priori, the inclusion need not preserve
> sums, kernels etc.
>
> Now let R be a ring and M an R-module.  Is there a minimal abelian
> subcategory of Mod-R containing M?  If so, is there a canonical way to
> describe it?
>
> Any thoughts or pointers to the literature would be welcome.  Feel free to
> assume hypotheses on R (it might be a finite-dimensional algebra etc), or
> to answer the question for full subcategories only.
>
> Thanks,
> Tom
```

3 Mar 01:27 2008

### Re: Minimal abelian subcategory

```A couple of people have pointed out to me - in private, I think - that the
question has a trivial answer (namely, the subcategory consisting of just
M and its identity map).  Sorry.  I probably misinterpreted what Walter
said to me.

Tom

>> -----Original Message-----
>> From: cat-dist <at> mta.ca [mailto:cat-dist <at> mta.ca] On Behalf Of
>> Tom Leinster
>> Sent: Monday, March 03, 2008 5:37 AM
>> To: categories <at> mta.ca
>> Subject: categories: Minimal abelian subcategory
>>
>> My colleague Walter Mazorchuk has the following question.
>>
>> Being abelian is a *property* of a category, not extra
>> structure.  Given an abelian category A, it therefore makes
>> sense to define a subcategory of A to be an ABELIAN
>> SUBCATEGORY if, considered as a category in its own right, it
>> is abelian.  Note that a priori, the inclusion need not
>> preserve sums, kernels etc.
>>
>> Now let R be a ring and M an R-module.  Is there a minimal
>> abelian subcategory of Mod-R containing M?  If so, is there a
>> canonical way to describe it?
>>
>> Any thoughts or pointers to the literature would be welcome.
>> Feel free to assume hypotheses on R (it might be a
>> finite-dimensional algebra etc), or to answer the question
```

3 Mar 01:04 2008

### Re: Minimal abelian subcategory

```> Now let R be a ring and M an R-module.  Is there a minimal abelian
> subcategory of Mod-R containing M?  If so, is there a canonical way to
> describe it?

This question, as posed, is too easy: Just take M and its identity
arrow.  It will be a zero-object in that subcategory.  There may be a
better question here guiding  Walter Mazorchuk's intuition, but it will
have to require something more than just containing the one object.

Colin

```
3 Mar 15:37 2008

### Re: A small cartesian closed concrete category

```Folk,

At Thu, 14 Feb 2008 15:06:49 -0500 I wrote,
"Is there a cartesian closed concrete category which
is small enough to write out explicitly?"

At Fri, 15 Feb 2008 08:47:57 +0000 Philip Wadler srote,
"... please summarize the replies ... and send ... to the ... list?

I've counted 16 respondents!  The question is
answered well.  With my limited knowledge, the
summary probably fails to credit some of the
responses adequately but this is not intentional.
Thanks to everyone who replied!

5 messages mentioned Hyting-algebras.
Never heard of them.  Lawvere & Schanuel
do not mention them in the 1997 book.
Will store the terms for future reference.

Fred Linton wrote,
"... skeletal version of the full category
... having as only objects the ordinal numbers 0 and 1.

Here 0 x A = 0, 1 x A = A, 0^1 = 0, 0^0 = 1, 1^A = 1.
In other words, B x A = min(A, B), B^A = max(1-A, B)."

My product diagrams are at
http://carnot.yi.org/category01.jpg
```

3 Mar 22:30 2008

### Re: A small cartesian closed concrete category

```
Peter Easthope  points out that in
Lawvere & Schanuel there is no
mention of Arend Heyting. That is
unfortunate, especially since
pp 348-352 are devoted to
introducing Heyting's Algebras
and one of their possible
objective origins. The 2nd edition
should correct this omission.

Summarizing the 16 responses,
a common thought of many must
have been
"If small implies finite
then any example must be a poset
(category in which any two parallel
maps are equal) because of Freyd's
theorem.  A CC poset is almost
by definition a Heying Algebra.
There are linearly ordered ones of
any size, but if the size is four or more,
there are also examples that are not
linearly ordered....

On the other hand if infinite examples
are allowed, and posetal ones are not,
it is hard to think of a  CCC smaller than
a skeletal category of all finite sets."

```

3 Mar 18:15 2008

### Re: Minimal abelian subcategory

```Hmm.  I suppose that restricting to subcategories which respect the group
structure on the Hom-sets would be enough to render the problem harder
(the group structure of course can be recovered canonically from the
underlying category, so this merely refines the class of subcategories we
are considering).  I would imagine this restriction would also have more
repercussions for algebra, anyway...

Josh

On Sun, 2 Mar 2008, Colin McLarty wrote:

>> Now let R be a ring and M an R-module.  Is there a minimal abelian
>> subcategory of Mod-R containing M?  If so, is there a canonical way to
>> describe it?
>
> This question, as posed, is too easy: Just take M and its identity
> arrow.  It will be a zero-object in that subcategory.  There may be a
> better question here guiding  Walter Mazorchuk's intuition, but it will
> have to require something more than just containing the one object.
>
> Colin
>
>
>

```
4 Mar 02:59 2008

### Re: Re: A small cartesian closed concrete category

```> 5 messages mentioned Hyting-algebras.
> Never heard of them.Lawvere & Schanuel
> do not mention them in the 1997 book.
> Will store the terms for future reference.

Nowadays when I hear "Never heard of x" my subconscious seems to turn it
into "never heard of Wikipedia."   When five people tell you x is the
the point of the answer.  (As one of the five, my examples consisted of
the finite nonempty chains and the finite Boolean algebras, which I
pointed out to Peter gave an example of every finite positive
cardinality, and two for the powers of two.  My mistake was to lump
these examples together under the common rubric of "Heyting algebra,"
which appears to have made what was meant to be a simple answer
incomprehensible.)

As Bill points out, a Heyting algebra is almost the same thing as a CCC
in the case of categories that are posets.  This is exactly the case
when there are finitely many objects (a case where Heyting algebras and
distributive lattices are "the same thing" in the sense that they have
the same underlying posets), and is close to true modulo existence of
joins in the infinite case.  In particular a Heyting algebra needs the
empty join 0 in order to define negation as x->0, whence the negative
integers made a category with its standard ordering is cartesian closed
but is not a Heyting algebra for want of a least negative integer.  More
generally Heyting algebras are required to have all finite joins, not a
requirement for posetal cartesian closed categories.

Vaughan Pratt

```

4 Mar 01:53 2008

### Minimal abelian subcategory (corrected)

```Apologies for the previous trivial question.  Here is the correct version.

(The mistake was omitting to say that the subcategory must contain all
endomorphisms of M.)

*

My colleague Walter Mazorchuk has the following question.

Being abelian is a *property* of a category, not extra structure.  Given
an abelian category A, it therefore makes sense to define a subcategory of
A to be an ABELIAN SUBCATEGORY if, considered as a category in its own
right, it is abelian.  Note that a priori, the inclusion need not preserve
sums, kernels etc.

Now let R be a ring and M an R-module.  Is there a minimal abelian
subcategory of Mod-R containing M and all its endomorphisms?  If so, is
there a canonical way to describe it?

Any thoughts or pointers to the literature would be welcome.  Feel free to
assume hypotheses on R (it might be a finite-dimensional algebra etc), or
to answer the question for full subcategories only.

Thanks,
Tom

```
4 Mar 16:20 2008

### How to motivate a student of functional analysis

```This semester I am teaching rudimentary category theory at graduate
level. It is somewhat scary that I should be doing this, but other
faculty members do not seem to do much general category theory.

I have only few students (and they are very bright) but their areas of
research are quite diverse: discrete math/computer science, algebra,
algebraic topology, and functional analysis.

I can plenty motivate categories for discrete math and computer science,
with things like "initial algebras are inductive datatypes, final
coalgebras are coinductive (lazy) datatypes".

I also know enough general algebra to motivate algebraists with
tquestions like "What is an additive category with a single object?".
And we will study algebraic theories as well.

Algebraic topologists are self-motivated. Nevertheless, we'll do some
sheaves towards the end of the course.

But how do I show the fun in categories to a student of functional
analysis? I would like to give him a class project that he will find
close to his interests. The course is covering (roughly) the following
material: basic category theory (limits, colimits, adjoints, we
mentioned additive and enriched categories), Lawvere's algebraic
categories, monads (up to stating Beck's theorem and working out some
examples), basics of presheaves and sheaves with a slant toward
topology. There must be some functional analysis in there.

I would very much appreciate some suggestions.

```