Reference search: new categories by replacing morphisms with diagrams

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

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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