Some miscellaneous notes about the concept of form, which I sketched (ahem) in a series of posts TDMO1, TDMO2, TDMO3, TDMO4,TDMO5, TDMO6, TDMO7, TDMO8, and TDMO9. This series builds up to an explanation of the concept of form in the paper Graph-Based Logic and Sketches by Atish Bagchi and me. I am now embarking on a series of posts with further explanations and comments.
More about a model of a form
For any constructor space (which will be the sketch for a kind of category, a special case of a doctrine), take any object
in the FL Cattheory of
and adjoin a new arrow
where 1 is the terminal object.
is what we call a C-form and the enhanced category is denoted by
. In TDMO9 I described the node
for reflexive function spaces in a cartesian closed category; it is an example of a CCC-form.
A model of a -form
“in a C-category
” means that
is a model of
. In particular, in
,
is nonempty.
The connection with sketches is this: If you have a sketch in some doctrine , the sketch consists of a graph with some diagrams, cones and cocones. There is a node
in the FL Cattheory of
each of whose elements in a model of
(in other words in a
-category) will be such a sketch.
Example: The FP sketch for magmas
A magma is a set with a binary operation defined on it (Note 1). It does not have to be associative or commutative or anything. In the FP doctrine its sketch consists of one diagram
(Diagram1)
and one cone
and nothing else. The FP-Cattheory for this sketch is (equivalent to) the Lawvere theory of magmas.
The FL-Cattheory for FP categories, described in some detail in TDMO8, contains a node whose inhabitants in any model of
(in other words in any FP-category) are all such sketches (the diagram and the cone). This means that the FP sketch for magmas corresponds to an FP-form. In this way you can see that all sketches in Ehresmann’s sense are forms in my sense.
This node can be constructed as the limit of a cone over a diagram in
as was done in previous posts. You have to make the diagram become a description of the diagram and cone above, using the arrows in the constructor space
, for example
,
,
and others, and including formally commutative diagrams that say for example that Cone1’s projections go to the same object (using
and
). Maybe someday I will produce this diagram in a post but right now I have a cold. (Excuses, excuses…)
Adjoining a global element to this limit node will result in an FL-sketch which contains the FL-Cattheory for
along with that global element.
So a model of the form for magmas in an FP category is a model of
for which the model of the underlying cattheory
is
; in other words it is the category
with a distinguished element f of the node
. That distinguished element is a particular diagram and cone like the ones shown above for a particular object
(because the projections onto
include a particular projection to
). That object
with the arrows corresponding to
and
is a particular magma, a model of the sketch for magmas given above.
Notes
Note 1 “Magma” was the term used by Bourbaki for this structure. As far as I know, very few people ever used the word until it was published in [1]. When I was a grad student in 1962-65 it was called a “groupoid”, which means something else now (something much more important than a magma in my opinion). Now the name occurs in examples all over Wikipedia.
References
[1] M. Hazewinkel (2001), “Magma“, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
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