Re: Internal diagrams

Re: Internal diagrams

[Venkata Rayudu Posina]

Hi Eduardo Ochs,

Just in case you missed, internal diagrams of maps are dual to graphs
(of maps) and hence called cographs (Conceptual Mathematics, pp.
293-4).  As noted in the announcement of Sets for Mathematics: "The
standard tools for analyzing an arbitrary map are the induced
equivalence relation, co-equivalence relation, graph and cograph
(cographs have been very frequently pictured in practice but only
rarely recognized explicitly); all four of these are shown to arise
inevitably as Kan quantifications, along the two possible
interpretations of the generic map as half of the splitting of a
generic idempotent" (http://www.mta.ca/~rrosebru/setsformath/; Sets
for Mathematics, pp. 29-30, 80, 190).

Thank you,
posina

On Feb 12, 2018 06:05, "Eduardo Ochs" <eduardoochs@gmail.com> wrote:

Hi list,

Lawvere and Schanuel use, in "Conceptual Mathematics" - from p.13
onwards - an idea that they call the "internal diagram" of a function:
the internal diagram for f:A->B "looks inside" A and B and shows how
each element of a A is taken to an element of B. The extension of this
idea to functors is trivial: the internal diagram of a functor F:C->D
shows how objects and morphisms of C are taken to objects and
morphisms of D.

The idea of internal diagrams for functors, natural transformations,
etc, feels (to me!) as something that is not only folklore, but also
something that I guess that is kept to the oral culture of courses on
Category Theory - a tool that is taught informally to students as a
way to help them visualize things and do the calculations, but that is
practically never published in details, even in course notes...

Since I started looking for these internal diagrams in CT a few months
ago I stumbled on only one other place where they are mentioned - in
Emily Riehl's "Category Theory in Context", in pages 17, 19, 60, and a
handful of other places there.

Anyone knows where else I can look for these things? All pointers will
be greatly appreciated - I just wrote a section about internal
diagrams for CT in these notes,

    http://angg.twu.net/LATEX/2017yoneda.pdf

but it would be great if I could connect that to other work.

    Cheers, and thanks in advance,
      Eduardo Ochs
      http://angg.twu.net/math-b.html
      http://angg.twu.net/logic-for-children-2018.html

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Internal diagrams

[Eduardo Ochs]

Hi list,

Lawvere and Schanuel use, in "Conceptual Mathematics" - from p.13
onwards - an idea that they call the "internal diagram" of a function:
the internal diagram for f:A->B "looks inside" A and B and shows how
each element of a A is taken to an element of B. The extension of this
idea to functors is trivial: the internal diagram of a functor F:C->D
shows how objects and morphisms of C are taken to objects and
morphisms of D.

The idea of internal diagrams for functors, natural transformations,
etc, feels (to me!) as something that is not only folklore, but also
something that I guess that is kept to the oral culture of courses on
Category Theory - a tool that is taught informally to students as a
way to help them visualize things and do the calculations, but that is
practically never published in details, even in course notes...

Since I started looking for these internal diagrams in CT a few months
ago I stumbled on only one other place where they are mentioned - in
Emily Riehl's "Category Theory in Context", in pages 17, 19, 60, and a
handful of other places there.

Anyone knows where else I can look for these things? All pointers will
be greatly appreciated - I just wrote a section about internal
diagrams for CT in these notes,

   http://angg.twu.net/LATEX/2017yoneda.pdf

but it would be great if I could connect that to other work.

   Cheers, and thanks in advance,
     Eduardo Ochs
     http://angg.twu.net/math-b.html
     http://angg.twu.net/logic-for-children-2018.html


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]