Brègman运算范畴与认知（共）单子
Categories of Brègman operations and epistemic (co)monads
摘要：
我们构造了一个非线性后量子推理的范畴框架，以适当的自反Banach空间的凸闭集的嵌入为对象，以Brègman拟非扩张映射的拉回（特别是Brègman相对熵的约束最大化）为态射。它提供了一个非线性凸解析模拟的Chencov的程序的几何研究类别的线性正映射之间的空间的状态，一个工作模型的Mielnik的非线性发射机，并设置为非线性资源理论（幺半群的Brègman准非扩张映射作为自由操作，其渐近不动点集作为自由集，和Brègman相对熵作为资源单调）。我们构造了一系列半有限JBW-代数和任意W*-代数的具体例子。由于相对熵的不对称性，所有的构造都有左和右版本，勒让德对偶在它们定义明确的限制之间诱导范畴等价。这些范畴的内广群实现了统计等价的概念。由熵投影给出的态射子范畴的hom-set具有偏序交换幺半群的结构（因此，它们是Fritz意义下的资源理论）。进一步将对象限制为仿射集，将布莱格曼的相对熵变成了一个函子。最后，Lawvere的伴随范式演绎逻辑，但语义扭曲代表Jaynes和Chencov的统计推断的意见，我们引入了一个类别理论的多（合作）代理设置归纳推理理论，实现家庭的单子和comonads。我们表明，brègmanian方法提供了一些特殊的情况下，这种设置。
英文摘要：
We construct a categorical framework for nonlinear postquantum inference, with embeddings of convex closed sets of suitable reflexive Banach spaces as objects and pullbacks of Brègman quasi-nonexpansive mappings (in particular, constrained maximisations of Brègman relative entropies) as morphisms. It provides a nonlinear convex analytic analogue of Chencov's programme of geometric study of categories of linear positive maps between spaces of states, a working model of Mielnik's nonlinear transmitters, and a setting for nonlinear resource theories (with monoids of Brègman quasi-nonexpansive maps as free operations, their asymptotic fixed point sets as free sets, and Brègman relative entropies as resource monotones). We construct a range of concrete examples for semi-finite JBW-algebras and any W*-algebras. Due to relative entropy's asymmetry, all constructions have left and right versions, with Legendre duality inducing categorical equivalence between their well-defined restrictions. Inner groupoids of these categories implement the notion of statistical equivalence. The hom-sets of a subcategory of morphisms given by entropic projections have the structure of partially ordered commutative monoids (so, they are resource theories in Fritz's sense). Further restriction of objects to affine sets turns Brègman relative entropy into a functor. Finally, following Lawvere's adjointness paradigm for deductive logic, but with a semantic twist representing Jaynes' and Chencov's views on statistical inference, we introduce a category-theoretic multi-(co)agent setting for inductive inference theories, implemented by families of monads and comonads. We show that the brègmanian approach provides some special cases of this setting.
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