In the present paper, we develop the theory of log homotopy exact sequences associated to proper log smooth morphisms and morphisms whose characteristic sheaves are locally constant with stalks isomorphic to the monoid of natural numbers. In the process of developing this theory, we also show the existence of a logarithmic version of the Stein factorization and develop the theory of algebraization of log formal schemes.
"The exactness of the log homotopy sequence." Hiroshima Math. J. 39 (1) 61 - 122, March 2009. https://doi.org/10.32917/hmj/1237392380