## The

LNaturality Gap^{ p}## F.E.J. Linton

Proposed for the International Category Theory Conference “CT04” at the

Wesleyan University

Middletown, CT 06459 USA

University of British Columbia (Vancouver, BC, Canada), 18-24 July 2004The spaces

l(^{ p}I) ofI-indexed sequences with absolutelyp^{th}-power summable entries (1 ≤p≤ ∞) are covariantly functorial inIforp= 1, contravariantly functorial inIforp= ∞, but nothing much for otherp.

By the same token, thel-style “products” of^{ p}I-tuples of Banach spaces serve as their coproducts forp= 1, their products forp= ∞, but nothing much (pacetheir Hilbert space structure whenp= 2) for otherp.

Similar observations pertain to the variousl-style “cross-norms” on tensor products of Banach spaces.^{ p}

Again, the various spacesl(^{ p}I) behave somewhat like the components of a gradedl^{ ∞}(I)-algebra, but not in an entirely satisfactory way.

The talk will fill in some of the details behind the remarks above, and, so far as they are available, behind theirLanalogues.^{ p}

Last modified: 29 May 2004.