We employ the lens provided by formal truth theory to study axiomatizations of Peano Arithmetic ${\textsf {(PA)}}$. More specifically, let Elementary Arithmetic ${\textsf {(EA)}}$ be the fragment $\mathsf {I}\Delta _0 + \mathsf {Exp}$ of ${\textsf {PA}}$, and let ${\textsf {CT}}^-[{\textsf {EA}}]$ be the extension of ${\textsf {EA}}$ by the commonly studied axioms of compositional truth ${\textsf {CT}}^-$. We investigate both local and global properties of the family of first order theories of the form ${\textsf {CT}}^-[{\textsf {EA}}] +\alpha $, where $\alpha $ is a particular way of expressing “${\textsf {PA}}$ is true” (using the truth predicate). Our focus is dominantly on two types of axiomatizations, namely: (1) schematic axiomatizations that are deductively equivalent to ${\textsf {PA}}$ and (2) axiomatizations that are proof-theoretically equivalent to the canonical axiomatization of ${\textsf {PA}}$.

The benefits of a ‘holistic’ approach to transitional justice are enhanced by considering how synergies between different transitional mechanisms may be optimized. Drawing upon multiple examples, this article explores the potential contribution of truth seeking to reparation efforts at a normative, institutional and operational level. The article emphasizes the importance of an awareness of the reparative potential of truth seeking on the part of those implicated in its design and implementation, as well as an appreciation of the influence of contextual factors on a delicate process. It cannot be conceived of simply as a technocratic exercise, but as an inherent part of empowering victims.

We analyze flat ${{S}_{3}}$ -covers of schemes, attempting to create structures parallel to those found in the abelian and triple cover theories. We use an initial local analysis as a guide in finding a global description.

In this article we use the theories of relative algebraic geometry and of homotopical algebraic geometry (cf. [HAGII]) to construct some categories of schemes defined under Specℤ. We define the categories of ℕ-schemes, 1-schemes, -schemes, +-schemes and 1-schemes, where (from an intuitive point of view) ℕ is the semi-ring of natural numbers, 1 is the field with one element, is the ring spectra of integers, + is the semi-ring spectra of natural numbers and 1 is the ring spectra with one element. These categories of schemes are linked together by base change functors, and all of them have a base change functor to the category of ℤ-schemes. We show that the linear group Gln and the toric varieties can be defined as objects in these categories.

Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$ . There is a complex ${{\hat{C}}^{\cdot }}\left( X \right)$ of topological ${{\mathcal{O}}_{X}}$ -modules, called the complete Hochschild chain complex of $X$ . To any ${{\mathcal{O}}_{X}}$ -module $\mathcal{M}$ —not necessarily quasi-coherent—we assign the complex $Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,\mathcal{M} \right)$ of continuous Hochschild cochains with values in $\mathcal{M}$ . Our first main result is that when $X$ is smooth over $\mathbb{K}$ there is a functorial isomorphism

$$Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{C}^{\cdot }}\left( X \right),\,M \right)\,\cong \,\text{R}\,Hom_{{{\mathcal{O}}_{X}}^{2}}^{{}}\,\left( {{\mathcal{O}}_{X}},\,M \right)$$

in the derived category $\text{D}\left( \text{Mod}\,{{\mathcal{O}}_{{{X}^{2}}}} \right)$ , where ${{X}^{2}}\,:=\,X\,{{\times }_{\mathbb{K}}}\,X$ .

The second main result is that if $X$ is smooth of relative dimension $n$ and $n!$ is invertible in $\mathbb{K}$ , then the standard maps $\text{ }\!\!\pi\!\!\text{ }\,\text{:}\,{{\hat{C}}^{-q}}\left( X \right)\,\to \,\Omega _{X/\mathbb{K}}^{q}$ induce a quasi-isomorphism

$$Hom_{{{\mathcal{O}}_{X}}}^{{}}\,\left( \underset{q}{\mathop \oplus }\,\,\Omega _{X/\mathbb{K}}^{q}\,\left[ q \right],\,M \right)\,\,\to \,Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,M \right).$$

When $M\,=\,{{\mathcal{O}}_{X}}$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem.

Combining the two results above we deduce a decomposition of the global Hochschild cohomology

$$\text{Ext}_{{{\mathcal{O}}_{{{X}^{2}}}}}^{i}\,\left( {{\mathcal{O}}_{X}}\,,\,M \right)\,\cong \,\underset{q}{\mathop \oplus }\,\,\,{{\text{H}}^{i-q}}\,\left( X,\,\left( \underset{{{\mathcal{O}}_{X}}}{\overset{q}{\mathop \Lambda }}\,\,{{T}_{X/\mathbb{K}}} \right)\,{{\otimes }_{{{\mathcal{O}}_{X}}}}\,M \right),$$

where ${{T}_{X/\mathbb{K}}}$ is the relative tangent sheaf.