[FOM] Question about Congruence

Vaughan Pratt pratt at cs.stanford.edu
Tue Jan 29 18:56:53 EST 2008

```hdeutsch at ilstu.edu wrote:
> I was asking two questions in my earlier posting.  First, is the simple
> fact about closer under equivalence well-known?

It's less a fact than a pair of (to my thinking) not terribly insightful
ways of looking at a very natural property: compatibility or
commutativity of two operations, respectively subobject and quotient.
Although I said as much in my original reply, true to form I was in
retrospect too cryptic.  A diagram will help set the stage.

B ----> A
|       |
|       |
V       V
Q ----> P

Interpreting this diagram in Set, B, A, and P are as per your
terminology.  B --> A is the given inclusion (a monic) and A --> P the
given quotient (an epi).  Your two criteria are indeed equivalent to
each other, but also to the criterion that the composite B ---> A ---> P
(exhibited as an epi-mono factorization) has a mono-epi factorization,
exhibited in the diagram as the two morphisms through Q.  When such
exists it is necessarily unique in Set, and in many other categories
admitting EM factorization systems.  That is, Q is the unique subset of
P that is also a quotient of B and that makes the diagram commute.

One place to read about EM systems from this perspective specifically
for Set is the vicinity of page 37 of Lawvere and Rosebrugh, "Sets for
Mathematics".

> Secondly, does anyone
> have anything to say about the philosophical import of the principle?

Well, first that it illustrates how easily even philosophers can find
themselves drawn to epi-mono factorizations without (presumably) ever
having had any exposure to them.  Very much like when I invented dynamic
logic as a formalization of Floyd-Hoare logic for my semantics class at
MIT in 1974 and a couple of my students had to point out to me that I
was essentially doing Kripke semantics.

Second, this square is simultaneously a pullback of the monic Q --> P
along A ---> P (hence making B --> A a monic, with no assumption as to
whether the vertical morphisms are epis), and completely dually a
pushout of the epi B --> Q along A --> P (so A --> P is an epi).  There
is probably a lot more that can be said about squares this well endowed
but I'm more of a set theorist than a category theorist (to the extent
that I could be considered either), maybe Colin McLarty can shed more
light on this.

Vaughan Pratt
```

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