Re: "Databases are Categories"

[David Spivak <dspivak@math.mit.edu>]

Hi all,

The basic idea of my talk was this: one can think of any category C as a
"database schema": the objects of C are called "tables" and an arrow f:A-->B
is called a "column of table A with values in table B".  Now a functor
C-->Sets is a "state" of that database: it fills every table with a set of
rows.  Leaf objects of C (objects with no outgoing arrows) correspond to
"pure data."  One can thus visualize a category as a system of tables;
commutative diagrams correspond to "rules" such as "the secretary of a
department must be in that department."

Using sketches instead of categories allows a little more flexibility, but
the basic idea is as above.  The model is nice for a variety of reasons,
most of all its simplicity.  In polite contradiction to one of Diskin's
claims, two database administrators (at two large multi-national
corporations) that I know think it is a viable model.  They do not balk at
the idea of data columns being considered as foreign keys.  It puts
everything on the same playing field.

The database=category idea furthermore allows us to "migrate data" from one
schema to another.  Given a morphism of schemas (functor F:C-->D) one pushes
and pulls data between the two categories using F^*, F_*, and F_!.  One can
thus achieve unions, joins, etc.  Use of F^*, F_*, and F_! may have a
variety of applications including a solution to the "view update problem."
  The Grothendieck construction applied to a database state C-->Set yields
its "RDF graph," which is an important component of the semantic web, and of
which conversions to and from databases is of high interest.  The point is
that its all very natural in this model.

Finally, in reference to Diskin's objection below (regarding people owning
cars, and both having addresses) we can consider the category [1]x[1]:
(excuse the typography -- V's are the tips of downward arrows)

O----->P
  |         |
  |         |
V        V
C----->A

A functor from this category to sets is a set of ownerships, each of which
has an associated person and an associated car, each of which has an
associated address (and these addresses must agree).  The same idea can be
used to model situations where an employee could work for any number of
departments (as in Diskin's other objection).

If anyone wishes to engage me more personally either in support or denial of
these ideas, I suggest we talk by phone; email me and I'll send you my
number.  Of course I also don't mind continuing the conversation by email,
either privately or publicly.  Also, let me extend my appreciation to Vasili
for bringing this whole thing up!

Best regards,
David

PS. I've informed the people at Galois of your comments on the camera-work;
they said they'd take it into consideration in the future.

PPS. A simplicial set (or any presheaf) can be regarded as a database state
(on a schema with countably many tables).  Can anyone imagine a database
management system like Oracle being of use in answering mathematical
questions about such things?  In other words, could modern, incredibly fast
database technology be useful to a mathematician studying the dynamics of a
presheaf category such as simplicial sets?


2010/8/16 Zinovy Diskin <zdiskin@gsd.uwaterloo.ca>

> 2010/8/14 Mattias Wikström <mattias.wikstrom@gmail.com>:
>> I think it makes sense to regard database schemas as theories and
>> databases as models of such theories. Given a theory T that models
>
> yes, this is the basic underlying assumption in a series of papers of
> Michael  Johnson and the moderator of this list.
>
>> some database schema S, a term in the language of T may be thought of
>> as a "database query" (to obtain the result of the query, simplify the
>> term),
>
> a query is a formula (defining a  derived table) rather than a term
>
> while a statement that two terms in the language of T are equal
>> may be thought of as a "database constraint" that one may want to add
>> to S.
>
> there are many types of non-equational constraints, for example,
> inclusion P=>Q with P,Q formulas
>
>>
>> What sort of theory should a database schema be? This surely depends
>> on what exactly one is trying to model: A schema in Company A's DBMS
>> (database management system) is rarely the same thing as a schema in
>> Company B's DBMS, and in any case one probably wants to work with some
>> idealised mathematical model.
>>
>
> a good question is what the place of SQL on the scale of logical
> doctrines is. Since models of SQL-theories include predefined infinite
> domains with operations (think of Integer and String), and  finite
> domains for tables, model theory for SQL is not classical. I'm not
> sure but it seems model-theorists call it "metafinite model theory"
> (Gradel and Gurevich).
>
>> David Spivak seems to offer two different answers. On the one hand, a
>> database schema may be the same thing as a category. On the other
>> hand, a database schema may be a labeled simplicial set. Both answers
>> may be found at http://www.uoregon.edu/~dspivak/cs/ .
>>
>
> I took a look at the slides of  "databases are categories" talk. It
> seems to be an overly simplified model.  Since an employee may work
> for several or none departments, data should be modeled by a functor
> into Rel rather than Set. 2-category structure may be important, and
> we often have a lax functor F(a;b)<F(a);F(b). For example, a Person
> _owns_ a Car, which is _parked_ at an Address, where the person
> _lives_, but a Person may not have a Car but still _live_ at an
> Address (arrow names are underlined, _lives=owns;parked_).
> Considering all columns as foreign keys, that is, disregarding the
> difference between value-valued and object-valued attributes, would
> look very strange for a practitioner.
>
> Z.
>

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