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{\bf Downcasing Types}

Proto-abstract, 2009nov04


When we represent a category $\catC$ in Type Theory it becomes a
7-uple: $(\catC_0, \Hom_\catC, \id_\catC, _\catC; {assoc}_\catC,
{idL}_\catC, {idR}_\catC)$, where the first four components are
``structure'' and the last three are ``properties''.

We can define a ``protocategory'', $\catC^-$, as a 4-uple $(\catC_0,
\Hom_\catC, \id_\catC, _\catC)$ --- just the ``syntactical part'' of
$\catC$, leaving out the ``logical part'' --- and we can do the same
for functors, natural transformations, adjunctions, isos, limits, and
so on.

We can take many constructions and proofs from the ``real world'' ---
where entities have both their ``syntactical'' and their ``logical''
parts --- and project them onto the ``syntactical world'' --- by
keeping only their syntactical parts.

The projections can be seen as the ``syntactical skeletons'' of the
real entities, and they are especially amenable to diagrammatic
representation. There is a simple way to attribute a precise meaning
to each entity --- nodes, arrows, etc --- in each such diagrammatic
representations, and we will show how to formalize two such diagrams
--- the proof of the Yoneda lemma and the definition of monadic
functor --- in Coq.

For most applications in Categorical Semantics one further trick is
needed: ``downcasing types'' --- for example, a morphism $f: A \to B$
is ``downcased'' to $a \mto b$.

In a hyperdoctrine, if $P$ is an object over $B×C$, then $P$ ``is'' a
subset $\ssst{(b,c)}{P(b,c)}$ of $B×C$. The downcasing of $P$ is
$\psst{b,c}{P(b,c)}$, and the Beck-Chevalley Condition for $\forall$
says that the natural morphism from $f^* \Pi_{\pi_{BC}} P$ to
$\Pi_{\pi_{AC}} (f×C)^* P$ should be an iso... in the downcasing, this
becomes $\psst{a}{c.P(fa,c)} \mto \psst{a}{c.P(fa,c)}$... however,
if we give a unique tag to each node and arrow and define the BCC
morphism by a diagram showing its construction, then the BCC morphism
becomes an iso between the two objects that ``are'' the same {\sl in
  the archetypal hyperdoctrine}, $\Sub(\Set)$.

Roughly, what is the happening is the following: the formal definition
of hyperdoctrine generalizes {\sl some} of the structure of
$\Sub(\Set)$; with our way of interpreting diagrams we can define all
this structure diagrammatically, in a notation that suggests that we
are in $\Sub(\Set)$ (we say: ``in the archetypical case''), and then
we can ``lift'' these definitions to diagrams with the same
two-dimensional structure, but in any of the standard notations.

Several categorical theorems become quite clear when we find an
archetypical diagram for their (proto-)proofs, and then we lift that
to standard notation; one example is $\_\_\_$.

% (find-books "__cats/__cats.el" "jacobs")


[Note for the referees: this paper is not about proving new theorems
  --- it is just about formalizing a language that might make old
  theorems clearer, and that might help constructing dictionaries
  between standard notations, and with formalizing categorical proofs
  in proof assistants.

I would like to be able to characterize, for example, when a proof
done in the ``syntactical world'' can be lifted to the ``real world''
--- but this looks as a distant goal at the moment. My guess is that a
good first step towards that would be to formalize the realtion
between the ``real world'' and the ``syntactical world'' in terms of
the language of institutions --- but for that I would have try to have
first a notion of proto-institution and (maybe) an archetype for
institutions, and I haven't had the time to learn enough about
institutions yet...]



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