L. Alberucci and A. Facchini, Abstract, The Journal of Symbolic Logic, vol.5, issue.04, pp.1367-1400, 2009.
DOI : 10.1007/s11225-006-8301-9

L. Alberucci and A. Facchini, On Modal????-Calculus and G??del-L??b Logic, Studia Logica, vol.8, issue.2, pp.145-169, 2009.
DOI : 10.1007/s11225-009-9170-9

A. Arnold and D. Niwi´nskiniwi´nski, Rudiments of µ-calculus, 2001.

A. Bauer and P. L. Lumsdaine, On the Bourbaki???Witt principle in toposes, Mathematical Proceedings of the Cambridge Philosophical Society, vol.155, issue.01, pp.87-99, 2013.
DOI : 10.1002/mana.3210040138

N. Bertrand and P. Schnoebelen, Computable fixpoints in well-structured symbolic model checking, Formal Methods in System Design, vol.21, issue.3, pp.233-267, 2013.
DOI : 10.1007/s10703-012-0168-y
URL : https://hal.archives-ouvertes.fr/hal-00906826

N. Bezhanishvili and D. De-jongh, Intuitionistic Logic. Tech. rep., Institute for Logic, Language and Computation, pp.2006-2031, 2006.

P. Clairambault, Strong functors and interleaving fixpoints in game semantics. RAIRO - Theor, Inf. and Applic, vol.47, issue.1, pp.25-68, 2013.

J. R. Cockett and D. Spencer, Strong categorical datatypes II: A term logic for categorical programming, Theoretical Computer Science, vol.139, issue.1-2, pp.69-113, 1995.
DOI : 10.1016/0304-3975(94)00099-5

D. Agostino, G. Hollenberg, and M. , Logical questions concerning the ??-calculus: Interpolation, Lyndon and ??o??-Tarski, The Journal of Symbolic Logic, vol.16, issue.01, pp.310-332, 2000.
DOI : 10.1016/0890-5401(89)90031-X

D. Agostino, G. Lenzi, and G. , On the <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>??</mml:mi></mml:math>-calculus over transitive and finite transitive frames, Theoretical Computer Science, vol.411, issue.50, pp.4273-4290, 2010.
DOI : 10.1016/j.tcs.2010.09.002

F. Dau, Some Notes on Proofs with Alpha Graphs, 14th International Conference on Conceptual Structures, pp.172-188, 2006.
DOI : 10.1007/11787181_13

S. Frittella, L. Santocanale, P. Höfner, P. Jipsen, W. Kahl et al., Fixed-point theory in the varieties Dn, Relational and Algebraic Methods in Computer Science - 14th International Conference, pp.446-462, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01509386

G. Gentzen, Untersuchungen ???ber das logische Schlie???en. I, Mathematische Zeitschrift, vol.39, issue.1, pp.176-210, 1935.
DOI : 10.1007/BF01201353

S. Ghilardi and M. Zawadowski, Sheaves, Games, and Model Completions: A Categorical Approach to Nonclassical Propositional Logics, 2011.
DOI : 10.1007/978-94-015-9936-8

S. Ghilardi and M. W. Zawadowski, Model completions and r-Heyting categories, Annals of Pure and Applied Logic, vol.88, issue.1, pp.27-46, 1997.
DOI : 10.1016/S0168-0072(97)00012-2

G. Kelly, Basic Concepts of Enriched Category Theory. No. 64 in Lecture Notes in Mathematics, republished in: Reprints in Theory and Applications of Categories, pp.1-13, 1982.

A. Kock, Strong functors and monoidal monads, Archiv der Mathematik, vol.1, issue.1, pp.113-120, 1972.
DOI : 10.1007/BF01304852

D. Kozen, Results on the propositional ??-calculus, Theoretical Computer Science, vol.27, issue.3, pp.333-354, 1983.
DOI : 10.1016/0304-3975(82)90125-6

K. Lehtinen and S. Quickert, Deciding the first levels of the modal mu alternation hierarchy by formula construction, 24th EACSL Annual Conference on Computer Science Logic, CSL 2015 Schloss Dagstuhl -Leibniz-Zentrum fuer Informatik, pp.457-471, 2015.

S. I. Mardaev, Least fixed points in grzegorczyk's logic and in the intuitionistic propositional logic, Algebra and Logic, vol.49, issue.3, pp.279-288, 1993.
DOI : 10.1007/BF02261708

S. I. Mardaev, Convergence of positive schemes in S4 and int, Algebra and Logic, vol.49, issue.3, pp.95-101, 1994.
DOI : 10.1007/BF00739995

A. M. Pitts, Abstract, The Journal of Symbolic Logic, vol.52, issue.01, pp.33-52, 1992.
DOI : 10.1016/0022-4049(83)90104-4

W. Ruitenburg, On the period of sequences (An(p)) in intuitionistic propositional calculus, The Journal of Symbolic Logic, vol.344, issue.03, pp.892-899, 1984.
DOI : 10.2307/2274142

A. Visser, A. Middeldorp, V. Van-oostrom, and F. Van-raamsdonk, L??b???s Logic Meets the ??-Calculus, Processes, Terms and Cycles: Steps on the Road to Infinity, pp.14-25, 2005.
DOI : 10.1007/11601548_3

S. Ghilardi and D. Di-matematica, Università degli Studi di Milano E-mail address: silvio.ghilardi@unimi

M. Gouveia, C. Lif, and C. Umr, Portugal E-mail address: mjgouveia@fc.ul, pp.1749-1765