Categorical models of Linear Logic with fixed points of formulas

We develop a categorical semantics of μLL, a version of propositional Linear Logic with least and greatest fixed points extending David Baelde's propositional μMALL with exponentials. Our general categorical setting is based on Seely categories and on strong functors acting on them. We exhibit two simple instances of this setting. In the first one, which is based on the category of sets and relations, least and greatest fixed points are interpreted in the same way. In the second one, based on a category of sets equipped with a notion of totality (non-uniform totality spaces) and relations preserving it, least and greatest fixed points have distinct interpretations. This latter model shows that μLL enjoys a denotational form of normalization of proofs.