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

[Jason Erbele <erbele@math.ucr.edu>]

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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