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Given a category fibered in groupoids over schemes with a log structure, one produces a category fibered in groupoids over log schemes. We classify the groupoid fibrations over log schemes that arise in this manner in terms of a categorical notion of "minimal" objects. The classification is actually a purely category-theoretic result about groupoid fibrations over fibered categories, though most of the known applications occur in the setting of log geometry, where our categorical framework encompasses many notions of "minimality" previously extant in the literature.

We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let $\mathrm{\Omega }$ be a regular uncountable cardinal. Let $m<\omega$ and $M$ be an $m$-sound premouse and $\mathrm{\Sigma }$ be an $(m,\mathrm{\Omega }+1)$-iteration strategy for $M$ (roughly, a normal $(\mathrm{\Omega }+1)$-strategy). We define a natural condensation property for iteration strategies, inflation condensation. We show that if $\mathrm{\Sigma }$ has inflation condensation then $M$ is ${(m,\mathrm{\Omega },\mathrm{\Omega }+1)}^{\ast }$-iterable (roughly, $M$ is iterable for length $\le \mathrm{\Omega }$ stacks of normal trees each of length $<\mathrm{\Omega }$), and moreover, we define a specific such strategy ${\mathrm{\Sigma }}^{\mathrm{\text{st}}}$ and a reduction of stacks via ${\mathrm{\Sigma }}^{\mathrm{\text{st}}}$ to normal trees via $\mathrm{\Sigma }$. If $\mathrm{\Sigma }$ has the Dodd-Jensen property and $\mathrm{\text{card}}(M)<\mathrm{\Omega }$ then $\mathrm{\Sigma }$ has inflation condensation. We also apply some of the techniques developed to prove that if $\mathrm{\Sigma }$ has strong hull condensation (introduced independently by John Steel), and $G$ is $V$-generic for an $\mathrm{\Omega }$-cc forcing, then $\mathrm{\Sigma }$ extends to an $(m,\mathrm{\Omega }+1)$-strategy ${\mathrm{\Sigma }}^{+}$ for $M$ with strong hull condensation, in the sense of $V[G]$. Moreover, this extension is unique. We deduce that if $G$ is $V$-generic for a ccc forcing then $V$ and $V[G]$ have the same $\omega$-sound, $(\omega ,\mathrm{\Omega }+1)$-iterable premice which project to $\omega$.

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation $\pi$ to be $k$-pass sortable if $\pi$ is sortable using $k$ passes through the stack. Permutations that are $1$-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of $2$-pass sortable permutations in terms of their basis. We also show all $k$-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer $k$. Finally, we define the notion of tier of a permutation $\pi$ to be the minimum number of passes after the first pass required to sort $\pi$. We then give a bijection between the class of permutations of tier $t$ and a collection of integer sequences studied by Parker [The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)]. This gives an exact enumeration of tier $t$ permutations of a given length and thus an exact enumeration for the class of $(t+1)$-pass sortable permutations. Finally, we give a new derivation for the generating function in [S. Parker, The combinatorics of functional composition and inversion, PhD thesis, Brandeis University (1993)] and an explicit formula for the coefficients.