Borel函子与无穷解释
Borel Functors and Infinitary Interpretations
摘要：
我们引入结构的无限解释的概念。一般来说，结构之间的解释会导致它们的自同构群之间的连续同态，而且，它会导致每个结构的副本范畴之间的函子。我们证明了，对于无限解释的情况下，反转也是真的：每个Baire-measurable同态之间的自同构群的两个可数结构是由一个无限解释，和每个Baire-measurable函子之间的集合的副本的两个可数结构是由一个无限解释诱导。此外，我们证明了复杂性保持在这个意义上，如果函子是$\mathbfΔ^0_α$，那么导致它的解释是$Δ^{\mathtt{in}}_α$到$\mathbfΔ^0_α$等价。
英文摘要：
We introduce the notion of infinitary interpretation of structures. In general, an interpretation between structures induces a continuous homomorphism between their automorphism groups, and furthermore, it induces a functor between the categories of copies of each structure. We show that for the case of infinitary interpretation the reversals are also true: Every Baire-measurable homomorphism between the automorphism groups of two countable structures is induced by an infinitary interpretation, and every Baire-measurable functor between the set of copies of two countable structures is induced by an infinitary interpretation. Furthermore, we show the complexities are maintained in the sense that if the functor is $\mathbfΔ^0_α$, then the interpretation that induces it is $Δ^{\mathtt{in}}_α$ up to $\mathbfΔ^0_α$ equivalence.
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