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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]