Props in Network Theory
Props in Network Theory
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网络理论中的道具
DOI：
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发表时间：
2017
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通讯作者：
Franciscus Rebro
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作者：
J. Baez;Brandon Coya;Franciscus Rebro
早在费曼图发明之前，工程师们就使用类似的图来推理电路和包含机械、液压、热力学和化学成分的更一般的网络。我们可以使用props来形式化这个推理：也就是说，严格对称monoidal范畴，其中对象是自然数，对象的张量积由加法给出。在这种方法中，每种网络对应一个道具，每种网络都是该道具中的一个态射。一个有m个输入和n个输出的网络是从m到n的一个态射，把网络串联在一起是合成，把它们并排放在一起是张量。在这里，我们将详细介绍这种方法在各种电路中的应用，从仅由理想的完美导电导线组成的电路开始，然后是具有无源线性元件的电路，然后是具有电压和电流源的电路。每一种电路都对应一个数学上自然的道具，我们用道具之间的态射来描述这些电路的“行为”。特别是，我们给出了方和第一作者证明的黑盒定理的一个新的证明，与原来的证明不同，这个新的证明很容易推广到非线性元件的电路。我们也使用一个态射的道具，以澄清之间的关系，电路图和控制理论中的信号流图。从技术上讲，关键的工具是Rosebrugh-Sabadini-Walters的结果，将电路与特殊的交换Frobenius幺半群联系起来，在props和签名之间的monadic adjunction，以及一个结果，说明哪些对称幺半群范畴等价于props。
Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using props: that is, strict symmetric monoidal categories where the objects are natural numbers, with the tensor product of objects given by addition. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with $m$ inputs and $n$ outputs is a morphism from $m$ to $n$, putting networks together in series is composition, and setting them side by side is tensoring. Here we work out the details of this approach for various kinds of electrical circuits, starting with circuits made solely of ideal perfectly conductive wires, then circuits with passive linear components, and then circuits that also have voltage and current sources. Each kind of circuit corresponds to a mathematically natural prop. We describe the "behavior" of these circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem proved by Fong and the first author; unlike the original proof, this new one easily generalizes to circuits with nonlinear components. We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory. Technically, the key tools are the Rosebrugh-Sabadini-Walters result relating circuits to special commutative Frobenius monoids, the monadic adjunction between props and signatures, and a result saying which symmetric monoidal categories are equivalent to props.