categorical "varieties of algebras" (fwd)

categorical "varieties of algebras" (fwd)

[Michael Barr]

I am forwarding this to the categories list, where I am sure there will be
many answers, but I have never myself delved into these interesting
questions.  Please be sure to copy your answers to him (although he should
probably subscribe to the list).

--M

---------- Forwarded message ----------
Date: Sun, 13 Dec 2009 23:15:11 +0200
From: Alexandru Chirvasitu <chirvasitua@gmail.com>
To: barr@math.mcgill.ca
Subject: categorical "varieties of algebras"

Dear Prof. Barr,


My name is Alexandru Chirvasitu, and I am a first-year mathematics graduate
student at UC Berkeley. I apologize for bothering you wih this, especially
since you don't know me, but I was kind of at a loss: I don't really know
any people working in the areas I am interested in personally, so I thought
I'd give this a go :). I'm quite sure you'll be able to clear this out
straight away.

Before coming to Berkeley, I was interested in applying category-theoretic
methods to study coalgebras, Hopf algebras, and other such creatures:

http://arxiv.org/abs/0907.2881

It became clear later on that to get some further insight into the universal
constructions useful for these problems (Hopf envelopes of co (or bi)
algebras, free Hopf algebra with bijective antipode on a Hopf algebra,
etc.), it would be useful to apply some Tannaka reconstruction techniques
and move the free constructions "up the categorical ladder": free monoidal
category on a category, (left) rigid envelope of a monoidal category, etc.
Unfortunately, I couldn't find any results stating clearly (clearly for
someone who is perhaps not *too* familiar with the higher categorical
machinery) that such free categories always exist. Of course, the few
constructions I needed can easily be done by hand, but what I had in mind
was some kind of higher categorical analogue of the fact that the forgetful
functor from a variety of algebras to another variety of algebras with
"fewer operations" has a left adjoint.

There's also an issue of how strict things should be. For what I was doing,
the following setting is typical: consider the category whose objects are
(not necessarily strict, but that's not very important here) monoidal
categories with a specified left dual and specified (co)evaluation maps for
every object, and whose morphisms are the functors which preserve all of
this structure *strictly*. Then I wanted to conclude that the forgetful
functor from this to *Cat* has a left adjoint, which should be easy enough.

To state my question properly, I'm thinking about a category whose objects
are categories *C* endowed with certain "operations", consisting of functors
from *C^n x (C opposite)^m* to *C* (example: the specified left dual in the
above example is a contravariant functor), appropriately natural
transformations between such functors (example: the evaluation maps in a
rigid tensor category as above form, together, a dinatural transformation),
and equations involving these natural transformations; the morphisms are
functors which preserve all the structure *strictly*. Consider the forgetful
functor from this to a similar category, but with "fewer operations" (this
could easily be made precise). Then, is there a  result stating that such a
functor always has a left adjoint?

I expect it should be easy enough to employ a form of the Adjoint Functor
Theorem (working with universes say, to have everything set-theoretically
sound) to prove something like this, and I was thinking about writing it up
for further reference. My problem was that I can't seem to be able to tell
exactly what is well-known and has been written up, what is folklore and
trivial, etc. Also, even though a statement as outlined above and to which I
could refer would be completely satisfactory, I realize that after
destrictification things become much more interesting, and you probably get
some neat bicategorical results.

I must once again apologize for intruding on your time like this, for the
potential silliness of a newbie's question, and for the ramble factor and
length of this message :). I hope you do get to reply.


Thank you,


Alexandru


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Re: categorical "varieties of algebras"

[Steve Vickers]

Dear Alexandru,

I would recommend the paper "Partial Horn logic and cartesian
categories" that I wrote with Eric Palmgren (Annals of Pure and Applied
Logic 145 (3) (2007), pp. 314 - 353; doi:10.1016/j.apal.2006.10.001).

Free categories of various kinds usually do exist, an idea that comes
out well in Higgins's 1971 "Notes on categories and groupoids" (now
reissued as a TAC reprint).

The underlying machinery, again recognized for a long time, relies on
the notion of left exact theories, alias finite limit theories,
cartesian theories or essentially algebraic theories. In the formulation
as an essentially algebraic theory, the operators may be partial but
their domains of definition can be expressed in terms of equations using
operators defined "earlier" (i.e. there is an ordering on the
operators). For categories, the core example is composition, which is
partial and defined when an equation holds between the domain of one
morphism and the codomain of the other. But the same phenomenon arises,
for example, when adding morphisms in an Abelian enriched category or
composing 2-cells in a 2-category.

The fundamental result for left exact theories is that a forgetful
functor, from a category of algebras for one theory to that for another
with fewer operators (or fewer equations, for that matter), has a left
adjoint.

I believe the result is covered in chapter 4 Barr and Wells' "Toposes,
Triples and Theories", or at least is inferrable from Kennison's Theorem
 as stated there.

Palmgren and I proved the fundamental freeness result (our Theorem 29,
Free Partial Model Theorem) in a way that makes much clearer the
connection with the well known result for algebraic theories. (For
algebraic theories, with all operators total, you just take a term model
and factor out a congruence.) We used a logic, minimally adapted from
the standard account of categorical logic as in the Elephant, in which
terms are only partially defined. We also described a simple
"quasi-equational" mode of theory presentation equivalent to left exact
theories. The proof is then very similar to the algebraic case, taking a
partial term model and factoring out a partial congruence. It is also
more elementary than that in TTT.

I hope this will answer your question in most cases.

Strictness is an issue, as you mention. Our treatment is very syntactic
in nature, and expects strictness of homomorphisms. However, we do have
techniques that allow this to be relaxed - see our Theorem 56.

We also give various examples from category theory, as well as
discussing some of the history of the result.

Regards,

Steve Vickers.

Michael Barr wrote:
> From: Alexandru Chirvasitu <chirvasitua@gmail.com>
>
> Dear Prof. Barr,
>
> ...
> Unfortunately, I couldn't find any results stating clearly (clearly for
> someone who is perhaps not *too* familiar with the higher categorical
> machinery) that such free categories always exist. Of course, the few
> constructions I needed can easily be done by hand, but what I had in mind
> was some kind of higher categorical analogue of the fact that the forgetful
> functor from a variety of algebras to another variety of algebras with
> "fewer operations" has a left adjoint.



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Re: A well kept secret

[Dusko Pavlovic]

i am wondering why is the public image of category theory so important for
us.

i mean, if category theory is a powerful and useful tool, as it is, then
it should be able to take care for itself. bread does not need
advertising.

i have been with categories for many years. i think in categories, and i
used them in each and every one of my research projects, in every piece of
software that i designed, in every paper that i wrote. but sometimes it is
easier to get to the point without spelling out all definitions in full
generality. and without tackling the opprobium.

e.g., i worked on networks, and have papers about trust networks, and
reputation networks, and recommender systems. a network is a weighted
graph, and it composes to some extent, because a friend of a friend is
almost like a friend, but a friend of a friend of a friend etc, six hops
removed --- is probably not a friend. but you can do with networks a lot
of what you can do with categories: make arrow networks, adjoin
colimits... anyway, i defined all that, and mentioned categories, but did
not really advertise them. was that a mistake? maybe it wasn't such a good
work, and i would have done a disservice to category theory by advertising
it in a bad paper.

i have been using categories in my little crypto modules, and in serious
reduction proofs, for more than 5 years (cryptography is a theory of
functions after all!), but i first gave a crypto talk using categories
last week. and this was not a talk to hard-core cryptographers.

i think category theory sometimes suffers from our advertising. even in a
good paper, advertising is advertising. there are places for that, and
there are places where it is better not to do.

i do understand that we need to take care for the public image of our
work. funding depends on that, hiring depends on that. but maybe we should
clearly state that this is a matter of advocacy and of influence, and not
mix it up with Promoting the Truth. i somehow think that the truth can
take care for itself.

but as always, maybe i am wrong.

-- dusko



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