On Fri, Feb 16, 2007 at 05:08:06PM -0400, Peter Selinger wrote:
I do not know whether Robin Houston's construction, when applied to a dagger compact closed category, yields dagger-biproducts.
The answer is no (unless I have overlooked something below, which is all too easy to do when cooking up pathological counterexamples). We're looking for a category that: - has a dagger compact-closed structure, - has biproducts - does *not* have dagger-biproducts. I've tried to keep the description quite concrete, perhaps too much so for the taste of some readers of this list. 1. Let A be the category with: - the set of objects is the set of integers; - an arrow is a rational number: A(n,n) = Q, and A(m,n) = {0}, for m != n; - composition is multiplication; - the following compact closed structure: [ The symbol @ denotes tensor below] m @ n := m + n for objects m, n q @ r := q x r for arrows q, r, The tensor unit is the object 0, n* := -n; - the obvious preadditive structure: addition within a hom set is just addition of rational numbers; - the trivial dagger: q^\dagger := q for each arrow q. Note that A is a (preadditive) dagger compact closed category. 2. Let B be the category of matrices over A, i.e. the result of freely adding finite biproducts to A. Concretely, an object of B is a finite tuple of integers (n_i | i<k) for some natural number k. An arrow f: (m_i | i<k) --> (n_j | j<k') is a k x k' matrix of rational numbers, with the property that the (i, j)th entry is 0 unless m_i = n_j. Composition is matrix multiplication. The biproduct of two objects is their concatenation as tuples. The dagger compact closed structure distributes over the formal biproducts, in the standard way. 3. Let C be the full subcategory of B determined by those objects (n_i | i<k) with the property that: for all i<k there exists j<k s.t. n_j = -n_i. Note that C is closed under tensor, * and biproduct. Of course there is a 'standard dagger' (transposition), and a standard compact closed structure that's compatible with it. I'll define a new dagger on C, and exhibit a compact closed structure that's compatible with this new dagger. The purpose of the restriction in (3) is to make it possible -- or at any rate easier -- to define this compact closed structure. Finally I'll show that C does not have dagger-biproducts, with respect to this new dagger. Given a matrix f: (m_i | i<k) --> (n_j | j<k') let f^\dagger be the transpose of f, multiplied by the scalar 2 ^ (max {n_j | j<k'} - max {m_i | i<k}) If either k or k' is zero, this expression is ill-defined, but the matrix is also empty so that doesn't matter. Below I shall write max(n_i) as an abbreviation for max {n_i | j<k}; the intended k will be clear from the context. This dagger is clearly involutive. Also max( (m_i | i<k) @ (n_j | j<k') ) = max {m_i + n_j | i<k, j<k'} = max(m_i) + max(n_j), from which it follows that this dagger is compatible with tensor. It's thus easy to check that C is dagger symmetric monoidal w.r.t this dagger. For the compact closed structure, consider an object A and denote the standard unit eta_A: (0) --> A* @ A. Define eta'_A to be the product of eta_A with the scalar 2 ^ -max(A). By the restriction of (3), we know that max (A* @ A) = 2 max(A), hence (eta'_A)^\dagger = (2 ^ max(A)) x epsilon_A where epsilon_A is the standard counit. Thus the additional scalar factor in (eta'_A)^\dagger is the reciprocal of that in eta'_A, so they cancel out as required. Finally, to see that C does not have dagger biproducts, consider the objects (0) and (-1,1). The injection i: (0) --> (0,-1,1) is a matrix [ q ] for some rational q. Then i^\dagger = [ 2q 0 0 ], [ 0 ] [ 0 ] so i^\dagger . i : (0) -> (0) is the 1x1 matrix [ 2q^2 ]. For this to be the identity, we would need to have 2q^2 = 1, but that equation famously has no solution in the rationals! Can anyone see any errors in the above, or think of a simpler example? Robin