Re: Tensor of monads

[Richard Garner <r.h.g.garner@gmail.com>]

Further to my earlier question:

   -- Given idempotent monads S, T on a category C for which we can speak of
> the tensor of S and T, is it always the case that S * T is isomorphic to S +
> T?
>

I think I'm now happy that the answer is "no". Consider, as in my previous
message, a presheaf category [D^op, Set]. Let S and T be the idempotent
monads corresponding to two
Grothendieck topologies J and J' on [D^op, Set]. Then S + T is the monad
whose algebras are presheaves which are simultaneously J-sheaves and
J'-sheaves. On the other hand,
S * T has as algebras those presheaves X which are both J-sheaves and
J'-sheaves, but which satisfy an additional axiom (*). This axiom may be
expressed most expediently when D has finite products; so let us assume that
now. The condition says:

(*) Let f_i : U_i --> U be J-covering, and let g_k : V_k --> V be
J'-covering. Let ( x_ik \in X(U_i x V_k) ) be a compatible family for ( f_i
x g_k : U_i x V_j --> U x V ). Then the two natural ways of patching to an
element of X(U x V) agree.

These two ways of patching are as follows. For the first, note that since (
f_i x V_k | i \in I ) is J-covering for each k in K, we may patch to obtain
elements ( y_k \in X(U x V_k) | k \in K ). Then since ( U x g_k | k \in K )
is J'-covering, we may patch these to obtain an element z \in X(U x V). For
the second way of patching, we proceed entirely analogously, but this time
going via a family ( y'_i \in X(U_i x V) | i \in I).

Now (*) is a genuine extra condition which as far as I can see is not a
consequence of being both a J-sheaf and a J'-sheaf, so that S + T algebras
are not the same as S * T algebras. Note, however, that (*) _is_ a
consequence of being a (J n J')-sheaf, since ( f_i x g_j ) is covering in J
n J'. On the other hand, I'm not sure if (*) implies being a (J n J')-sheaf,
as I conjectured in my previous message; I don't have an Elephant to hand to
check.

Richard


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