An automaton is called reachable if every state is reachable from the initial state. This notion has been generalized coalgebraically in two ways: first, via a universal property on pointed coalgebras, namely, that a reachable coalgebra has no proper subcoalgebras; and second, a coalgebra is reachable if it arises as the union of an iterative computation of successor states, starting from the initial state. In the current paper, we present corresponding universal properties and iterative constructions for trees. The universal property captures when a coalgebra is a tree, namely, when it has no proper tree unravellings. The iterative construction unravels an arbitrary coalgebra to a tree. We show that this yields the expected notion of tree for a variety of standard examples. We obtain our characterization of trees by first generalizing the previous theory of reachable coalgebras and of a minimal object in a category, related to projectivity. Surprisingly, both the universal property and the iterative construction for trees arise as instances of this generalized notion of reachability. Our iterative construction works for all analytic set functors.

翻译：若自动机的每个状态均可从初始状态到达，则称其为可达的。这一概念已在余代数框架下以两种方式得到推广：首先，通过带点余代数的泛性质，即一个可达余代数没有真子余代数；其次，若一个余代数是通过从初始状态开始迭代计算后继状态而得到的并集，则称其为可达的。在本文中，我们针对树结构提出了相应的泛性质与迭代构造方法。该泛性质刻画了余代数成为树的条件，即它不存在真的树展开。迭代构造则将任意余代数展开为一棵树。我们证明，对于一系列标准实例，该方法能得到预期的树概念。我们首先通过推广先前关于可达余代数以及范畴中与投射性相关的最小对象理论，从而得到树的特征刻画。值得注意的是，树的泛性质与迭代构造均可视为这种广义可达性概念的实例。我们的迭代构造适用于所有解析集合函子。