Re: Quantum computation and categories

[Mark Weber <mark.weber.math@googlemail.com>]

Greetings all and thanks to John for the friendly provocation ...

I was very interested at Mark Weber's reaction to this problem.  He said,
> roughly, "So dagger-categories aren't really categories with extra
> structure.  Okay: they're something else!  And that's fine."  (I'd be happy
> for him to correct my rough summary and make his point more precisely.)
>

OK. First of all when thinking about cobordisms, or paths, or homotopies,
the act of reversing orientation is fundamentally strictly involutive. So at
first glance, it's not at all clear what would be gained by weakening in
this case, despite the light that replacing equations by isomorphisms sheds
on so many other mathematical situations.

Algebraically also, it appears natural to think of the strict involutions as
more fundamental. Just as categories are algebras of a very nice and easily
described monad on the presheaf topos of graphs, "dagger categories" (I
think the terminology "involutive category" would be better) are algebras of
an analogous monad on the presheaf topos of involutive graphs. An involutive
graph is a graph together with an involution on the set of edges which
switches sources and targets. Formally the dagger category monad is obtained
by a canonical lifting of the category monad through the forgetful functor
from involutive graphs to graphs, so we have a canonical monad distributive
law describing this situation.

The observations of the previous paragraph generalise a lot and rather
easily, and this encourages me to resist any urge to weaken the involutory
aspects of the notion of dagger category. So the "something else" of John's
post is that dagger categories are *involutive graphs* with structure, and
their higher dimensional analogues are *involutive n-globular sets* with
structure. So all the way up the dimensional ladder, regardless of how weak
your higher compositions happen to be, if involutions (to model orientation
reversals) are to be part of the picture, then I think they should be strict
and strictly compatible with all the compositions and coherence data.

For those still interested, I'll now give a few more details. First some
background. In

http://arxiv.org/abs/0909.4715

Michael Batanin, Denis-Charles Cisinski and I reformulated much of the
"Batanin approach" to defining higher categories as the study of monads on
categories of enriched graphs, particularly those that arise from
multitensors. Briefly, given a category V, one can associate to any
"distributive multitensor on V" (which is a lax monoidal structure on V such
that the n-ary tensor product functor V^n-->V preserves coproducts in each
variable), a monad on the category GV of graphs enriched in V. So for
example this process takes the cartesian product for Set to the category
monad on Graph. The operads used by Batanin to define weak higher
categories, seen as certain monads on the presheaf topos G^n(Set) of
n-globular sets, also arise in this way.

One can also consider the category G_i(V) of involutive graphs enriched in
V, and so begin to consider structures defined by monads on the presheaf
topos (G_i)^n(Set) of what would be sensible to call "involutive n-globular
sets". An involutive graph enriched in V is a V-graph X together with, for
each pair of objects a,b from X, maps
i_(a,b) : X(a,b) --> X(b,a)
in V such that for all a,b, i_(b,a)i_(a,b) = identity. It is easy to verify
that both processes V |-> GV and V |-> G_i(V) preserve presheaf toposes, so
(G_i)^n(Set) really is a presheaf topos.

To spell out the generalisation alluded to above, let E be a distributive
multitensor on V, and write (as in the above paper) Gamma(E) for its
associated monad on GV -- corollary(4.5) of our paper indicates an explicit
formula. This formula is easily adapted to the involutive case to describe
the monad Gamma_i(E) on G_i(V) and this is by definition a canonical lifting
of Gamma(E) though the forgetful G_i(V)-->GV.

In summary, for any higher categorical structure of interest, there is an
involutive version (eg one can define involutive Gray categories), and from
the above remarks we understand as much about the monads which describe them
as we do their non-involutive counterparts, and moreover there's a canonical
distributive law relating them.

For me the interesting question now is how to adapt this to give an explicit
description of monads which describe weak higher groupoids (with strictly
involutive "inverse operations"). Steve Lack and I observed recently that
ordinary groupoids are algebras for a monad on the category of involutive
graphs, which arises via a *weak* distributive law in the sense of

http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html

between the category monad on Graph and the involutive graph monad on Graph,
but I really don't see yet how this generalises.

Best new years wishes to all,

Mark Weber


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