Reference search: new categories by replacing morphisms with diagrams

Reference search: new categories by replacing morphisms with diagrams

[Jason Erbele]

Dear all,

I built a category from another category by keeping "the same" objects
and taking the morphisms to be diagrams from the old category that
satisfy certain properties.  The closest thing to what I'm doing that
I have been able to find is factorization systems, but there are some
major differences.

To be more specific, I am starting with an Abelian category.  If there
are morphisms f: A --> B, g: X --> B, and h: A --> X, it makes sense
to talk about the morphism f+gh: A --> B, which can be represented
with a non-commutative triangle.  I don't know how to draw that in
plain text, so I will depict it as the ordered triple (f,g,h).  The
category I built takes this type of non-commutative triangle as a
morphism (f,g,h): A --> B.

That is, the new category is storing extra information in the
morphisms by distinguishing between the part that goes directly from A
to B and the part that takes a detour through an intermediate object,
X.  So while it may be possible for f+gh = f'+g'h' in the original
category, (f,g,h) and (f',g',h') would be different morphisms in the
new category unless f=f', g=g', and h=h'.  One nice feature of this
construction is the original category can be embedded in the new
category by taking X to be the zero object.

The people I have shown this to have told me they have never seen
anything like my construction.  I am at a loss for search terms --
everything I have tried either turns up nothing or thousands of
unrelated articles.  The closest I've found is factorization systems,
which involve a commutative triangle, f=gh, for some g and h with
certain properties.

If any of you know a reference or keyword associated with expanding a
category by replacing the morphisms with diagrams (with a specified
property/shape), I would greatly appreciate the assistance.

Sincerely,
Jason Erbele


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Re: Reference search: new categories by replacing morphisms with diagrams

[Jason Erbele]

Ah, it figures I would leave something out in my first post here.  My
apologies to all of you that were scratching your heads over the
missing rules for composition.  At least five of you have responded
with essentially the same question, so this will be sort of a blanket
response.

Composition of (f,g,h): A --> B through X and (f',g',h'): B --> C
through Y will go through the biproduct X (+) Y with (f'f, [f'g, g'],
[h, h'(f+gh)]^T): A --> C.  That is, the first component composes in
the usual way, the second component is a row matrix, and the third
component is a column matrix.  In the original category, the various
morphisms in the composed triple correspond to A --> B --> C,
directly; X --> B --> C and Y --> C; and A --> X and A --> B --> Y,
where this A --> B is f+gh rather than f.

Identity morphisms in the new category are those with f=id and the
zero object for X, which uniquely determines g and h.  The embedding I
mentioned of the original category into the new category is a functor,
after all.

Best,
Jason

On Wednesday, September 24, 2014, Tom Hirschowitz
<tom.hirschowitz@univ-savoie.fr> wrote:
>
>
> Dear Jason,
>
> How do your triples compose?
>
> Best,
> Tom
>
> Jason Erbele <erbele@math.ucr.edu> writes:
>
>> Dear all,
>>
>> I built a category from another category by keeping "the same" objects
>> and taking the morphisms to be diagrams from the old category that
>> satisfy certain properties.  The closest thing to what I'm doing that
>> I have been able to find is factorization systems, but there are some
>> major differences.
>>
>> To be more specific, I am starting with an Abelian category.  If there
>> are morphisms f: A --> B, g: X --> B, and h: A --> X, it makes sense
>> to talk about the morphism f+gh: A --> B, which can be represented
>> with a non-commutative triangle.  I don't know how to draw that in
>> plain text, so I will depict it as the ordered triple (f,g,h).  The
>> category I built takes this type of non-commutative triangle as a
>> morphism (f,g,h): A --> B.
>>
>> That is, the new category is storing extra information in the
>> morphisms by distinguishing between the part that goes directly from A
>> to B and the part that takes a detour through an intermediate object,
>> X.  So while it may be possible for f+gh = f'+g'h' in the original
>> category, (f,g,h) and (f',g',h') would be different morphisms in the
>> new category unless f=f', g=g', and h=h'.  One nice feature of this
>> construction is the original category can be embedded in the new
>> category by taking X to be the zero object.
>>
>> The people I have shown this to have told me they have never seen
>> anything like my construction.  I am at a loss for search terms --
>> everything I have tried either turns up nothing or thousands of
>> unrelated articles.  The closest I've found is factorization systems,
>> which involve a commutative triangle, f=gh, for some g and h with
>> certain properties.
>>
>> If any of you know a reference or keyword associated with expanding a
>> category by replacing the morphisms with diagrams (with a specified
>> property/shape), I would greatly appreciate the assistance.
>>
>> Sincerely,
>> Jason Erbele
>>

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Re: Reference search: new categories by replacing morphisms with diagrams

[Steve Lack]

Dear Jason,

I’m travelling at the moment, and can’t look up details, but similar things have been
done in the past, in particular by Bob Walters. If you start with a monoidal category, 
then you can define a bicategory with the same objects, and in which a morphism 
from A to B consists of an object X and a morphism A—> X @ B, where @ denotes 
the tensor product.

There is a dual version in which the original category in which morphisms have the form A@X->B.

Your version is a combination of both of these. (Once again, you get a bicategory rather 
than a category,  unless for some reason + is strictly associative.)

Another closely related notion is that of Elgot automaton, studied by Walters and various collaborators.

Regards,

Steve Lack.


On 25 Sep 2014, at 4:49 am, Jason Erbele <erbele@math.ucr.edu> wrote:

> Ah, it figures I would leave something out in my first post here.  My
> apologies to all of you that were scratching your heads over the
> missing rules for composition.  At least five of you have responded
> with essentially the same question, so this will be sort of a blanket
> response.
> 
> Composition of (f,g,h): A --> B through X and (f',g',h'): B --> C
> through Y will go through the biproduct X (+) Y with (f'f, [f'g, g'],
> [h, h'(f+gh)]^T): A --> C.  That is, the first component composes in
> the usual way, the second component is a row matrix, and the third
> component is a column matrix.  In the original category, the various
> morphisms in the composed triple correspond to A --> B --> C,
> directly; X --> B --> C and Y --> C; and A --> X and A --> B --> Y,
> where this A --> B is f+gh rather than f.
> 
> Identity morphisms in the new category are those with f=id and the
> zero object for X, which uniquely determines g and h.  The embedding I
> mentioned of the original category into the new category is a functor,
> after all.
> 
> Best,
> Jason



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Re: Reference search: new categories by replacing morphisms with diagrams

[Robin Cockett]

An addition to Steve's comment:

Going even further back in history these ideas were used in Richard Wood's
thesis: there he started with a bicategory and created a new bicategory by
setting the new 1-cells  (f,X): A --> B := f: A -> X @ B as Steve
suggested.  Composition then uses the associativity isomorphism  ... and
the resulting 1-cell composition is certainly bicategorical.  One can, of
course, also do the I-cell dual construction and, indeed your construction
seems to amalgamate these two constructions.

One can also always extract a category from a bicategory by identifying
isomorphic 1-cells ...

-robin
(Robin Cockett)

On Thu, Sep 25, 2014 at 1:52 PM, Steve Lack <steve.lack@mq.edu.au> wrote:

> Dear Jason,
>
> I’m travelling at the moment, and can’t look up details, but similar
> things have been
> done in the past, in particular by Bob Walters. If you start with a
> monoidal category,
> then you can define a bicategory with the same objects, and in which a
> morphism
> from A to B consists of an object X and a morphism A—> X @ B, where @
> denotes
> the tensor product.
>
> There is a dual version in which the original category in which morphisms
> have the form A@X->B.
>
> Your version is a combination of both of these. (Once again, you get a
> bicategory rather
> than a category,  unless for some reason + is strictly associative.)
>
> Another closely related notion is that of Elgot automaton, studied by
> Walters and various collaborators.
>
> Regards,
>
> Steve Lack.
>

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