This paper investigates the denotational invariants of non-wellfounded and circular proofs of linear logic with least and greatest fixed points, μLL, by providing a categorical semantics. More precisely the paper successively introduces semantics for (i) non-wellfounded pre-proofs, be they valid or not, (ii) valid pre-proofs exploiting their validity condition by considering an orthogonality construction on the given categorical model and finally (iii) circular strongly valid pre-proofs, exploiting both validity and regularity in order to define inductively the interpretation. Then the paper investigates the semantical content of the translation from finitary proofs to non-wellfounded proofs and, conversely, from (strongly valid) circular proofs to finitary proofs, showing that both translations preserve the interpretation.

The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (μMALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively. In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to μMALL with explicit (co)induction (i.e. μMALL^{ind}). We introduce a Tait-style system for μMALL called μMALL_ω where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property.

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.