The Category of Node-and-Choice Preforms for Extensive-Form Games

It would be useful to have a category of extensive-form games whose isomorphisms specify equivalences between games. Since working with entire games is too large a project for a single paper, I begin here with preforms, where a “preform” is a rooted tree together with choices and information sets. In particular, this paper first defines the category Tree\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Tree}$$\end{document}, whose objects are “functioned trees”, which are specially designed to be incorporated into preforms. I show that Tree\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Tree}$$\end{document} is isomorphic to the full subcategory of Grph\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Grph}$$\end{document} whose objects are converging arborescences. Then the paper defines the category NCP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {NCP}$$\end{document}, whose objects are “node-and-choice preforms”, each of which consists of a node set, a choice set, and an operator mapping node-choice pairs to nodes. I characterize the NCP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {NCP}$$\end{document} isomorphisms, define a forgetful functor from NCP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {NCP}$$\end{document} to Tree\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Tree}$$\end{document}, and show that Tree\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {Tree}$$\end{document} is equivalent to the full subcategory of NCP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {NCP}$$\end{document} whose objects are perfect-information preforms. The paper also shows that many game-theoretic entities can be derived from preforms, and that these entities are well-behaved with respect to NCP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {NCP}$$\end{document} morphisms and isomorphisms.


Introduction
Category theory has been used to systematize many subjects in mathematics and elsewhere. For example, Grph is the category of directed graphs. Grph morphisms can be used to state that one directed graph is embedded within another. Further, Grph isomorphisms can be used to state that two directed graphs are equivalent.
Similarly, it would be useful to have a category of extensive-form games whose morphisms would allow one to systematically compare extensive-form games. As yet, little has been done. 1,2 Lapitsky [13] and Jiménez [9] define categories of normal-form games. Machover and Terrington [15] define a category of simple voting games. Finally, Vannucci [26] defines categories of various kinds of games, but in its category of extensive-form games, every morphism merely maps a game to itself.
Building a category of extensive-form games with nontrivial morphisms is a large project because each extensive-form game has so many components: each is a rooted tree with choices, information sets, players, strategies, chance probabilities, and preferences. Accordingly, this paper takes a small necessary step: it builds a category of preforms, where a "preform" is a rooted tree with choices and information sets.
Since a preform incorporates a tree, this paper's first step is yet smaller: it develops a category of trees which are specially designed to be incorporated into preforms. These trees are called "functioned trees" because they consist of a set T of nodes together with an immediate-predecessor function p which specifies the immediate predecessor p(t) of each nonroot node t. Section 2 defines not only functioned trees but also morphisms between them. Theorem 2.5 shows that the resulting category Tree is well-defined. Theorem 2.6 characterizes the Tree isomorphisms by the bijectivity of their node transformations. Finally, Theorem 2.8 shows that Tree is isomorphic to the full subcategory of Grph whose objects are those directed graphs that are converging arborescences.
A functioned tree determines several derivative entities. These include the tree's root node, its stage function, its (strict) precedence relation, and its weak precedence relation (which partially orders its set of nodes). Further, a functioned tree determines its set of decision nodes and its collection of plays (which are the maximal chains in its partially ordered set of nodes). The propositions of Section 2 develop these entities and show that they are well-behaved with respect to the morphisms and isomorphisms of Tree.
Section 3 then defines a "node-and-choice preform" to consist of a set T of nodes, a set C of choices, and an operator ⊗. The operator ⊗ is a new concept. It maps node-and-choice pairs to nodes. In particular, each node-and-choice pair in the operator's domain is mapped to the node that follows the pair's node by way of the pair's choice. Importantly, this operator determines both a functioned tree (T, p) and a collection H of information sets. Preform morphisms are then defined, and Theorem 3.6 shows that the resulting category NCP is well-defined. Theorem 3.7 characterizes the NCP isomorphisms by the bijectivity of their node and choice transformations. Theorem 3.9 establishes a forgetful functor from NCP to Tree, which serves to make Section 2's Tree results readily accessible. Finally, Theorem 3.13 shows that Tree is equivalent to the full subcategory of NCP whose objects are those preforms that have perfect information.
As already mentioned, a preform determines a collection H of information sets. More precisely, each choice determines an information set as the set of nodes from which the choice is feasible (several choices may determine the same information set). Section 3 explores this construction with examples and propositions. In addition, a preform determines a previous-choice function q which specifies the choice q(t) that is previous to any nonroot node t. The propositions of Section 3 show that p and q together constitute the inverse of the operator ⊗, and that this inverse leads to useful characterizations of the NCP morphisms. Finally, the propositions of Section 3 show that ⊗, p, q, and H are well-behaved with respect to NCP morphisms and isomorphisms. Section 4 sketches out future applications and extensions. In particular, Section 4.1 uses a collection of examples to suggest that this paper's abstract NCP framework nests as special cases the particular formulations of [4,11,18,22,27] (each of these particular formulations continues to have its own advantages and disadvantages). 3 Finally, Section 4.2 briefly discusses how we plan to build players, strategies, chance probabilities, and preferences on top of the preforms defined here.

Definition of Functioned Trees
By definition, a pair (T, p) is a functioned tree iff there exist t o ∈ T and X ⊆ T such that p is a nonempty function from T {t o } onto X, and (1a) Call T the set of nodes t and call t o the root node. Further, call p the immediate-predecessor function. (1a) states that every nonroot node t is assigned an immediate-predecessor p(t). (1b) states that every nonroot node t is eventually preceded by the root node. Theorem 2.8 (Section 2.5) will show the formal sense in which functioned trees are equivalent to nontrivial and possibly infinite converging arborescences.
Here are some further remarks about definition (1).
[i] Since (1a) implies that t o is the only node outside the function's domain, a functioned tree determines its t o .
[ii] (1a) implies p(t o ) is undefined, and (1b) precludes the existence of a t = t o such that p(t) = t. Hence (/ ∃t) p(t) = t.
[iii] The existence of X is not restrictive. Rather, (1a) defines X to be both the range and the codomain of p. Call X the set of decision nodes.
[iv] Since p is nonempty by (1a), there exists t = t o . Thus (1b) implies t o ∈ X. In other words, the root node must be a decision node.

Entities Derived from Functioned Trees
Throughout this subsection, let (T, p) be a functioned tree. By (1b), there exists a function k:T →N 0 such that k(t o ) = 0, and (3a)  (2) function, and call k(t) the stage of node t. (Graph theorists might call k(t) the "height" of node t.) Define the (strict) precedence relation ≺ on T by Say that t 1 (strictly) precedes t 2 iff t 1 ≺ t 2 . Equivalently, say that t 2 (strictly) succeeds t 1 . Note that the range of p is the set of nodes that precede at least one node. Thus, since the range of p is X by definition (1a), X is the set of nodes that precede at least one node. Equivalently, X is the set of nodes that have at least one successor. Although this may suggest other names for X, the previous section called X the decision-node set, and I will continue to favour that term. 5 Define the weak precedence relation on T by Notice that the term "precedence" without the modifier "weak" refers to strict precedence. The following proposition shows that (T, ) is a partially ordered set whenever (T, p) is an functioned tree. There is no converse because there are partially ordered sets that cannot be constructed from functioned trees. In particular, Alós-Ferrer and Ritzberger [2,3] define games over more general partially ordered sets. Apparently, their "discrete" partially ordered sets ([4, Section 3]) correspond to those partially ordered sets that can be derived from functioned trees. 5 I avoid the term "nonterminal node" because I avoid the term "terminal node". I avoid the latter because it is natural to expect that the set of "terminal nodes" would be in one-to-one correspondence with the set of plays. This does not happen because there can be infinite plays that do not correspond to individual nodes. In general, Proposition 2. 2(b) shows that infinite plays correspond to sequences of nodes rather than individual nodes. To illustrate this, the paragraph after the proposition discusses the centipede example (2), which has an infinite play. Proposition 2.1. Suppose (T, p) is a functioned tree with its ≺ and . Then (T, ) is a partially ordered set, and ≺ is the asymmetric part of . (Proof: Lemma A. 1

(b,d).)
Finally, let Z be the collection of maximal chains in (T, ), and call Z ∈ Z a play. In general, plays can be either finite or infinite. Accordingly, Z = Z ft ∪Z inft , where Part (a) of the following proposition shows that each finite play can be uniquely associated with a nondecision node. It does so by means of the maximization operator for . Meanwhile, part (b) shows that each infinite play can be uniquely associated with an infinite sequence of nodes (there is no single node associated with an infinite play). For this result, define the function E from Z inft into T N 1 by where each t v is the unique element t of Z for which k(t) = v. 6 Call E the enumeration operator.
Proposition 2.2. Suppose (T, p) is a functioned tree with its t o , X, k, ≺, , Z ft , Z inft , and E. Then the following hold. (a) Z ft Z → max Z is a bijection onto T X. Its inverse is

(Proof A.2.)
For example, consider the centipede example (2) of Figure 1. Here the stage function is defined by (∀n) k(n) = n and k(n) = n+1, and the 6 The sequence E(Z) = (t v ) v≥1 is defined to start with a stage-1 node rather than the stage-0 node t o . This is notationally convenient because the index v = 0 would lead to the redundant and awkward equation t 0 = t o . Incidentally, the sequence E(Z) = (t v ) v≥v * could have been defined to start with any v * ≥ 0. I believe that a variant of Proposition 2.2(b) would still hold because the result is fundamentally concerned with the tails of the sequences.
(strict) precedence relation ≺ is {(m, n)|m < n} ∪ {(m,n)|m ≤ n}. Proposition 2.2(a) implies that the maximization operator is a bijection from the finite-play collection implies that the enumeration operator E is a bijection from the (singleton) infinite-play collection In accord with footnote 6, the node sequence in Y begins with the stage-1 node 1 rather than the stage-0 node 0.

Functioned-Tree Morphisms
Let a functioned-tree morphism be a triple γ = [(T, p), (T , p ), τ] such that (T, p) and (T , p ) are functioned trees, (8a) The following proposition characterizes functioned-tree morphisms in terms of the category Set. The functions τ | T {t o } :T {t o }→T {t o } and τ | X :X→ X are well-defined by (9a) and the codomain definitions after (9b). These two functions as well as p and p are displayed in the Set diagram of Figure 2.
where the codomain of τ | T {t o } is defined to be T {t o } and where the codomain of τ | X is defined to be X . (Proof A.3.) (9) can be interpreted. (9a) states that the image of every nonroot node is a nonroot node, and that the image of every decision node is a decision node. (9b) states that the image of the predecessor of a nonroot node is the predecessor of the image of the nonroot node. By Proposition 2.3, every morphism satisfies (9a) and (9b). Proposition 2.4 shows that a morphism has many other properties as well. Many of these properties are proved via property (b), which concerns iterations of the predecessor functions p and p . Iterations of a predecessor function are not compositions within the category Set because the domain and codomain of a predecessor function are different. In particular, the root node is in the codomain but not the domain. Property (b) avoids this complication by assuming that t 1 = p m (t 2 ), which implicitly entails that (∀i < m)

The Category Tree
This paragraph and the following theorem define the category Tree, which is called the category of functioned trees. Let an object be a functioned tree (T, p). Let an arrow be a functioned-tree morphism [(T, p), (T , p ), τ]. Let source, target, identity, and composition be where id T is an identity in Set.
Theorem 2.5. Tree is a category. (Proof A.5.) The following theorem characterizes the isomorphisms in Tree. This characterization is then used by the subsequent proposition to establish the relationships between the entities derived from isomorphic trees.

Connection to Grph
This subsection shows that Tree, the category of functioned trees, is isomorphic to a full subcategory of Grph, the category of directed graphs.
By definition, let a nontrivial converging arborescence be a quadruple (T, E, init, ter) such that (T, E, init, ter) is a nontrivial oriented tree, and In this context, nontriviality is equivalent to E being nonempty, which implies that T has at least two elements. In (10b), the (unique) path linking init(e) = t o and t o is well-defined by [ As noted in the previous paragraph, a nontrivial converging arborescence is a special kind of directed graph. Accordingly, let Grph ca be the full subcategory of Grph whose objects are those directed graphs that are nontrivial converging arborescences.
The following theorem shows that Tree and Grph ca are isomorphic ([14, p. 14]). A related result is [25, Theorem VI.1, p. 126], which implies that in a converging arborescence, every nonroot node is the initial node of exactly one edge. In terms of the following theorem, that result concerns the welldefinition of H 0 .
Theorem 2.8. Tree and Grph ca are isomorphic. In particular, define G from Tree to Grph ca by Conversely, define H from Grph ca to Tree by , ter(e)) | e∈E }, and

Definition of Node-and-Choice Preforms
As in Section 2, let T be a set and call t ∈ T a node. Further, let C be a set and call c ∈ C a choice. A (node-and-choice) preform Π is a triple (T, C, ⊗) such that Call ⊗ the node-and-choice operator. Note that equation ( Streufert (2015bStreufert ( , 2015c.) (13a) states that the operator ⊗ is a function from a subset of T ×C into a subset of T . Thus it maps node-choice pairs to nodes. Let (t, c, t ) ∈ ⊗ be equivalent to ⊗(t, c) = t , and let this also be equivalent to t⊗c = t . Call t⊗c the result of the node-choice pair (t, c).
Further, (13a) states that the range of ⊗ is T {t o }. This determines the root node t o as the only node t that is not in the range of ⊗. Hence T has no superfluous elements: every node t other than the root node t o is the result of some node-choice pair.
Further, (13a) states that the domain of ⊗ is F ⊆ T ×C. Thus Since F is a subset of T ×C, F can be regarded as a (nonempty-valued) correspondence whose domain is some subset of T and whose range is some subset of C. In accord with this perspective, let the statement (t, c) ∈ F be equivalent to the statement c ∈ F (t). Thus by (14), c ∈ F (t) iff t⊗c exists. Accordantly, F (t) is called the set of choices that are feasible from the node t. Now consider the range of F . This set consists of those choices c that are feasible from some node. By (13c) and the fact that a partition consists of nonempty sets, each inverse image F −1 (c) is nonempty. Thus the range of F is all of C. Hence C has no superfluous elements: each choice c is feasible from at least one node.
Finally, note that the domain of F is F −1 (C). This set consists of those nodes with at least one feasible choice. Accordantly, the elements of F −1 (C) are called the decision nodes.
(13b) defines the function p:T {t o }→F −1 (C). Lemma C.1(a) shows that (13a) implies that p is well-defined and surjective. This function maps any t in the nonroot-node set T {t o } to its immediate predecessor p(t ) in the decision-node set F −1 (C).
Substantively, (13b) assumes that (T, p) is a functioned tree, as defined by (1a)-(1b). Given (13a) and Lemma C.1(a), (1a) adds only that that the function p is nonempty. More significantly, (1b) adds that every nonroot node is eventually preceded by the root node. Lemma C.1(b) shows that the root node t o of the preform (T, C, ⊗) equals the root node of the derived tree (T, p) (the root node of a tree is also denoted by t o in Section 2). Further, Lemma C.1(c) shows that the decision-node set F −1 (C) of the preform (T, C, ⊗) equals decision-node set of the derived tree (T, p) (the decision-node set of a tree is denoted by X in Section 2).
Related to (13b), and for future reference, define q:T {t o }→C by By Lemma C.2, q is well-defined and surjective. Call q the previous-choice function, and call q(t ) the choice previous to t . The function q resembles the function α defined by [16, p. 227]. Further, note that the definition of q in (15) closely resembles the definition of p in (13b). This resemblance is not coincidental: Proposition 3.1(b) shows that p is the first component of ⊗ −1 , and that q is the second component of ⊗ −1 .
(13c) defines the collection H of information sets H. This important construction will be discussed at length in Section 3.2.
In summary, many entities can be derived from a preform Π = (T, C, ⊗). In particular, (13) and (15) define F , t o , p, q, and H. Further, T and p define a tree (T, p) from which more entities can be derived. In particular, (3)-(7) define k, ≺, , Z, Z ft , Z inft , and E. Finally, as noted four paragraphs ago, Π's decision-node set F −1 (C) equals (T, p)'s decision-node set X.

The Construction of Information Sets
(13c) defines the information-set collection H as {F −1 (c)|c}. This generalizes a similar construction by [20, p. 97].
For example, consider Figure 3a, which depicts a preform corresponding to Selten's horse game. 8 To be specific, the horse-like diagram depicts (T, C, ⊗), where T = {0, 1, 2, 3, 4, 5, 6, 7, 8}, C = {r S , d S , r G , d G , e, f}, and Better, one can read T , C, and ⊗ directly from the diagram: Nodes and choices are as usual, and the eight triples in ⊗ are the eight node-choicenode segments in the diagram. Note that F consists of the eight node-choice 8 To tell a story, suppose a student (S) must decide between the right choice (r S ) of doing her homework and the dumb choice (d S ) of not doing her homework. Knowing that the homework has been finished (node 1), a goat (G) must decide between the right choice (r G ) of taking a nap and the dumb choice (d G ) of eating the homework. Knowing that either the student played dumb (node 3) or that the student played right and the goat played dumb (node 4), the teacher must choose between excusing the student (e) and failing the student (f). The preform's three information sets correspond to the student, the goat, and the teacher.    Figure 3b depicts the first information set by the curved dashed line above the nodes 0 and 1. This new information set is absentminded in the sense of [19].
This way to construct information sets imposes a mild notational restriction. To see this restriction, recall again that (13c) specifies each information set H ∈ H as the set F −1 (c) of decision nodes from which a choice c is feasible. This implies that each choice determines exactly one information set (though several choices may determine the same information set). Thus each choice is associated with exactly one information set (though several choices can be associated with the same information set). Thus information sets cannot share choices. In other words, each information set must have its own choices. This notational restriction is conclusion (16b) of Proposition 3.2.
To illustrate this notational restriction, suppose that one wants to use a node-and-choice preform to express the rooted tree, choices, and information sets of Selten's horse game. That well-known game has three information sets with two choices each. So, since each information set must have its own choices, one must specify 3×2 = 6 choices. This is what Figure 3a does.
As a whole, Proposition 3.2 collects some general observations about the information sets constructed from a preform. It requires some introduction.
Further, Proposition 3.2(16b) shows that the information sets constructed here have an additional property. As with any correspondence, the value F (H) of the correspondence F at the set H is defined to be { c | (∃t∈H) c∈F (t) }. Proposition 3.2(16a) implies that (∀H∈H) t ∈ H ⇒ F (t) = F (H). Hence each F (H) is readily interpreted as the feasible-choice set of the information set H. Thus, Proposition 3.2(16b) states that each information set has its own choices. This is the notational restriction introduced four paragraphs ago. It is satisfied by both of Figure 3's preforms.
Section 4.1 will provide four more examples of node-and-choice preforms. Among other things, these examples illustrate that information sets constructed via (13c) satisfy (16a) and (16b). All four examples express the rooted tree, choices, and information sets of Selten's horse game. Figure 3a does the same.

Preform Morphisms
A (preform) morphism is a quadruple α = [Π, Π , τ, δ] such that Π = (T, C, ⊗) and Π = (T , C , ⊗ ) are preforms, (17a) Lemma C.6 shows that a quadruple [Π, Π , τ, δ] is a morphism iff its satisfies (17a)-(17b), where Π determines F and Π determines F .   (19) can be interpreted like (18) can be interpreted. (19a) states that the image of every feasible node-choice pair is a feasible node-choice pair, and that the image of every nonroot node is a nonroot node. (19b) states that Alternatively, recall ⊗ −1 = (p, q) by Proposition 3.1(b). Lemma C.9 uses this identity to show that a quadruple [Π, Π , τ, δ] is a morphism iff it satisfies (17a)-(17b), where Π determines p and q, and Π determines p and q . Then Lemma C.10 shows that a quadruple is a morphism iff it satisfies (17a)-(17b), where Π determines t o , p, and q, and Π determines t o , p , and q . Below     (22) can be interpreted like (21) can be interpreted. (22a) states that the image of every nonroot node is a nonroot node, and that the image of every decision node is a decision node. (22b) states that the image of the predecessor of a nonroot node is the predecessor of the image of the nonroot node.
(22c) states that the image of the previous choice of a nonroot node is the previous choice of the image of the nonroot node. A morphism implies relationships between [a] the entities derived from the source preform and [b] the entities derived from the target preform. For example, a result about the two feasibility correspondences is in the first half of (19a), a result about the two nonroot-node sets in in the second half of (19a), a result about the two predecessor functions is in (20a), a result about the two previous-choice functions is in (20b), and a result about the two decision-node sets is in the second half of (22a). A result about the two information-set collections is in Proposition 3.5 below. Finally, results about the two trees are established categorically by Corollary 3.10 in the next subsection.

The Category NCP
This subsection defines the category NCP, which is called the category of node-and-choice preforms. Let an object be a node-and-choice preform Π = (T, C, ⊗). Let an arrow be a preform morphism α = [Π, Π , τ, δ]. Let source, target, identity, and composition be where id T and id C are identities in Set. 10 The symbol τ is overloaded. See footnote 7.
Theorem 3.6. NCP is a category. (Proof C.13.) Theorem 2.6 characterized the isomorphisms in Tree by the bijectivity of τ . The following theorem provides an analogous characterization for the isomorphisms in NCP. 16.) The following proposition uses Theorem 3.7 to provide some properties of NCP isomorphisms. Corollary 3.11 will categorically derive many more properties.
and H, and where Π = (T , C , ⊗ ) determines F , q , and H . Then the following hold.
This paragraph makes two observations. First, by (13b), a preform incorporates a functioned tree. Second, since (20a) and (8c) are identical, a preform morphism incorporates a functioned-tree morphism. In accord with this second observation, the left-hand side of Figure 5 for preforms is identical to Figure 2 for functioned trees (recall that the preform's decision-node set F −1 (C) equals the functioned tree's decision-node set X by Lemma C.1(c)).
Together, these two observations suggest that there is a forgetful functor from NCP to Tree. Theorem 3.9 establishes this result. Three corollaries follow immediately.

Hence it obeys properties (a)-(j) of Proposition 2.7.
Corollary 3.12. Recall the functor G from Theorem 2.8. Then G•F is a well-defined functor from NCP to Grph ca .

Perfect Information
A preform is said to have perfect information iff Thus, in light of (13c), a preform has perfect information iff each of its information sets is a singleton. For example, Figure 6 depicts Π = (T, C, ⊗), where T is defined in (2), In contrast, the horse-like preform of Figure 3a does not have perfect information because it has a non-singleton information set. Several general observations can be made. [1] Perfect information implies that {F −1 (c)|c} partitions F −1 (C). Hence perfect information implies (13c). Therefore, a triple (T, C, ⊗) is a perfect-information preform iff it satisfies (13a)-(13b) and (23). [2] Proposition 3.8(e) implies that isomorphisms preserve perfect information (nothing similar can be said for morphisms). [3] Lemma D.2 shows that a preform has perfect information iff its previouschoice function q is bijective.
Let NCP p be the full subcategory of NCP whose objects are the preforms with perfect information. The following theorem shows that NCP p and Tree are equivalent ( [14, p. 18]). Incidentally, the natural isomorphisms ([14, p. 16]) used to establish Theorem 3.13(a) appear in the statement of Lemma D.6.
Theorem 3.13. NCP p and Tree are equivalent. In particular, let F p be the restriction of Theorem 3.9's functor F to NCP p . Conversely, define E from Tree to NCP p by

t)∈p }, and by
For example, consider again Figure 6's preform Π. Closely related is the "spartan" preform Π s = (T, C s , ⊗ s ), where T is again defined defined by (2), but These two preforms share the same tree. Formally, where (T, p) is defined by (2). The "expansion" of that tree is the spartan preform. Formally, agrees with Theorem 3.13(b). Since Π s and Π are isomorphic, [2] implies that E 0 •F 0 (Π) and Π are isomorphic but unequal. Their being isomorphic agrees with Theorem 3.13(a). Their being unequal shows that F p and E do not form a pair of isomorphisms between NCP p and Tree. Finally, Corollary 3.14 follows from the two theorems shown in Figure 7. In addition, the figure almost shows Theorem 3.9's forgetful functor F, which goes from all of NCP to Tree.

Applications
The intention behind developing categories for extensive-form games is to systematically compare the results which are obtained within different strands of the game-theory literature. Each of these strands has its own way of formulating games, and each these formulations has its own advantages and disadvantages.
For example, Figure 3a depicted a horse-like NCP preform. Figures 8 and 9 depict four more horse-like NCP preforms, all of which are isomorphic to the first. Each of these four additional preforms is formulated according to a particular strand of the game-theory literature. From the abstract perspective of NCP, each of these specialized formulations has its own particularly convenient means of specifying the node-and-choice operator ⊗. Figure 8a uses a "choice" formulation in the sense that it expresses each node as a choice-sequence. In particular, Figure 8a depicts the NCP preform Better, one can read T , C, and ⊗ from the diagram directly: Nodes and choices are as usual, and the eight triples in ⊗ are the eight node-choice-node segments in the diagram. Notice that the node-and-choice operator ⊗ takes the form t⊗c = t⊕(c), where ⊕ is the concatenation operator for sequences.
This formulation is popular. It appears in the textbook of Osborne and Rubinstein [18], 12 and there, choicesequences are called "histories". Figure 8b uses a "node" formulation in the sense that it expresses each choice as a node-set. In particular, Figure 8b depicts the NCP preform (T, C, ⊗) where T consists of nine nodes such as t = 3 and t = 6, where C consists of six node-sets such as c = {6, 8}, and where ⊗ consists of the diagram's eight node-choice-node triples such as (t, c, t ) = (3, {6, 8}, 6).
. This formulation appears in Alós-Ferrer and Ritzberger [4, p. 94], and is related to the classic formulation of Kuhn [11].
The two formulations of Figure 8 are "dual" in the sense that [a] choice preforms express each node in terms of choices and [b] node preforms express 12 Figure 8a is meant to be emblematic of some, but not all, Osborne-Rubinstein structures. To be precise, let a "structure" be a rooted tree together with choices and information sets. Then Figure 8a is meant to be emblematic of those Osborne-Rubinstein structures that can be expressed as NCP preforms. In accord with Proposition 3.2(16b), this rules out Osborne-Rubinstein structures in which different information sets share the same choices.
[Such a qualification is not needed for Figures 8b, 9a, and 9b because the node, choice-set, and outcome formulations all require that each information set has its own choices.] Figure 9. a A "choice-set" preform where t⊗c = t∪{c}. b An "outcome" preform where t⊗c = t∩c each choice in terms of nodes. We plan to show categorically that either of these special formulations is effectively as general as all of NCP. The result for choice forms would be related to [10]'s non-categorical link between "OR trees" and "KS trees". Figure 9a uses a "choice-set" formulation. This is the same as the "choice" formulation of Figure 8a, except that it expresses each node as a choice-set rather than a choice-sequence. In particular, Figure 9a  This formulation was introduced by Streufert [22], and was used to advantage by [21]. Figure 9b uses an "outcome" formulation in the sense that it expresses each node and each choice as an outcome-set. Routinely, outcomes are in one-to-one correspondence with plays, and in this example, the outcomes are 2, 5, 6, 7, and 8 (these labels correspond with the five nondecision nodes in Figure 8b). Accordantly, Figure 9b [27], and was extended to allow infinite plays by Alós-Ferrer and Ritzberger [4].
The two formulations of Figure 9 are "dual" in the sense that [a] the ⊗ of choice-set preforms uses a union, and [b] the ⊗ of outcome preforms uses an intersection. This dual pair is slightly less general than Figure 8's dual pair because only the formulations of the former pair can accommodate absentmindedness. Given this caveat, [4] essentially develops a noncategorical equivalence that links the node and the outcome formulations. Meanwhile, [23] develops a non-categorical equivalence that links the choice, the choice-set, and the outcome formulations. We plan to synthesize and strengthen these equivalences categorically.
In summary, the operator ⊗ provides a unified way of comparing apparently dissimilar formulations.

Extensions
As discussed in the introduction, this paper is the first step in a larger agenda to develop a category of extensive-form games. [24] takes the second step. It develops a category of node-and-choice forms, where a "form" is a preform from this paper, augmented with players. There, [i] a form assigns each information set (and concurrently each decision node and each choice) to a player, and [ii] a form morphism allows players to be renamed and merged. In addition, we are developing a third paper which augments forms with [a] pure player strategies and [b] player preferences over non-probabilistic outcomes (i.e. plays).
This paper, and the two papers just discussed, admit continuum feasible sets and infinite-horizon trees. They do so at relatively little cost because they rely on set theory and category theory alone. In contrast, measuretheoretic issues will arise when there are mixed player strategies with continuum supports, or chance moves with continuum supports, or player preferences over outcome lotteries with infinite supports. Accordingly, our next step will avoid these issues by considering only [a] mixed player strategies with finite support, [b] chance moves with finite support, and [c] player preferences over outcome lotteries with finite support. This intermediate step will likely entail a restriction to finite-horizon trees. The measure-theoretic issues can then be left for last.

A.1. Objects
Then the following hold.
Proof. (a). The reverse direction follows immediately from the definition of ≺. To see the forward direction, suppose t 1 ≺ t 2 . Then by the definition of ≺, there exists an m ≥ 1 such that This and the definition of m imply both k(t 2 ) > k(t 1 ) and . By the definition of , it suffices to prove that ≺ is asymmetric. This relation is asymmetric because if both t 1 ≺ t 2 and t 2 ≺ t 1 held, part (a) would imply both k(t 1 ) < k(t 2 ) and k(t 2 ) < k(t 1 ).
(c). By using the definition of for the first equivalence, and by using part (a) for the second equivalence, Reflexivity holds by the definition of . Transitivity holds by [1] the definition of and [2] the transitivity of ≺, which follows immediately from its definition. To show antisymmetry, suppose t 1 t 2 and t 2 t 1 . Then by two applications of part (c), k(t 1 ) = k(t 2 ). Thus by t 1 t 2 and part (c) again, (e). Let p 0 be the identity function, so that Suppose S is an infinite chain. Since S is a chain and since min S exists, I may number the elements of S so that min S = t 1 ≺ t 2 ≺ t 3 . . . . Thus by part (a), (∀n ≥ 1) k(t n ) < k(t n+1 ). Hence (∀n ≥ 1) k(t n ) ≥ n−1.
By part (e),S is a chain. Further, it is infinite because S is infinite. Thus it remains to be shown thatS is maximal. Accordingly, suppose that it were not maximal. Then there would be some t / ∈S such thatS∪{t } is a chain. This paragraph shows that (∀n ≥ 1) k(t ) ≥ n. Take any n ≥ 1. Since t / ∈S, and since t n and all its predecessors are inS, it must be that t t n . Thus by part (a), k(t ) > k(t n ). Thus, since k(t n ) ≥ n−1 by the second-previous paragraph, By the previous paragraph, k(t ) / ∈ N 0 . This contradicts the definition of k.
(g). Suppose S is a chain. On the one hand, suppose S is infinite.
Then part (f) shows that it is a subset of a member of Z inft . On the other hand, suppose S is finite. Then max S exists, and two cases arise. These cases are defined in the first sentences of the next two paragraphs. [ SinceS contains all the predecessors of t * , it must be that t t * . But this contradicts the assumption that t * does not have a successor.
[2] Suppose that max S has a successor and that every successor of max S has a successor. Then define S 1 by S 1 = S∪{t 1 } where t 1 is some successor of max S. Then, for every n ≥ 2, define S n = S n−1 ∪{t 2 } where t n is some successor of t n−1 . Then ∪ n≥1 S n is an infinite chain. Thus part (f) shows that it is a subset of a member The set membership holds because t ∈ Z and k(t) ≥ m ≥ 1. The set inclusion holds because [1] Z∪{p m (t )|t ∈Z, k(t ) ≥ m ≥ 1} is a chain by part (e) and [2] Z is maximal by the assumption Z ∈ Z ft ∪Z inft .
is a bijection onto T X, and that its inverse is These results follow from the next two paragraphs. This paragraph argues that the function (24) followed by the function (25) is the identity function on Z ft . Accordingly, take any Z ∈ Z ft . The remainder of this paragraph argues where the two arrows apply the functions (24) and (25), respectively. By inspection, the first arrow applies the function (24). Before applying the function (25), it must be shown that max Z exists and is an element of T X. First, max Z exists and is an element of T because Z ∈ Z ft by definition. Second, max Z is not an element of X, for if it were an element of X, [1] it would have a successor, thus [2] Z would not be a maximal chain, and thus [3] Z / ∈ Z ft in contradiction to the definition of Z. Accordingly, the second arrow in (26) applies the function (25) at t = max Z. To continue, the ⊆ direction of the equality in (26) holds by Lemma A.1(h) applied at t = max Z. To see the ⊇ direction, take any t ∈ Z. Because Z is a chain that contains max Z, either t max Z or max Z ≺ t. The former implies that t is in the left-hand side. The latter contradicts the definition of the maximum operator.
This paragraph argues that the function (25) followed by the function (24) is the identity function on T X. Accordingly, take any t ∈ T X. The remainder of this paragraph argues where the two arrows apply the function (25) and (24), respectively. By inspection, the first arrow applies the function (25). Before applying the function (24), it must be shown that S := {p m (t)|k(t) ≥ m ≥ 1}∪{t} is an element of Z ft . Since S is a finite chain by inspection, I only need to show that S is maximal. Accordingly, suppose there were a t / ∈ S such that S∪{t } was a chain. Because t ∈ S and S∪{t } is a chain, either t t or t ≺ t . The first case is impossible for it would imply that t ∈ S, in contradiction to the definition of t . The second case would imply [1] that t has a successor, and thus [2] that t ∈ X. This would contradict the definition of t. Accordingly, the second arrow in (27) applies the function (24) at Z = S. The equality is immediate.
(b). This paragraph shows that E is a well-defined function from Z inft into T N1 . Accordingly, take any Z ∈ Z inft . It must be shown that Take any v ≥ 1. First, consider uniqueness. It must be shown that there are not two nodes in Z at stage v. This holds because distinct nodes in a chain have different stages by Lemma A.1(a). Second, consider existence. Let S := {t ∈Z|k(t )≤v}. Since distinct nodes in a chain have different stages by Lemma A.1(a), S is finite. Thus since Z is infinite, Z S = {t ∈Z|k(t )>v} is nonempty. Take t * ∈ Z S and let t = p k(t * )−v (t * ). Note t ∈ Z by Lemma A.1(h) at its t equal to t * and its m equal to k(t * )−v. Further note that where the first equality holds by the definition of k(t * ), the second is a rearrangement, and the third holds by the definition of t. Thus k(t) = v by the definition of k(t).
This paragraph shows that E maps from Z inft into Y ⊆ T N1 . Accordingly, take any Z ∈ Z inft . By the previous paragraph, I may let E(Z) = (t v ) v≥1 . It must be shown that t o = p(t 1 ) and that (∀v ≥ 1) t v = p(t v+1 ). Since k(t 1 ) = 1 by the definition of E, p(t 1 ) = t o by the definition of k. Next take any v ≥ 1. By the definition of E, [1] Thus t v ≺ t v+1 because the alternative is impossible by [2], [3], and Lemma A.1(c). Finally, t v ≺ t v+1 implies t v = p(t v+1 ) by [2], [3], and Lemma A.1(a).
The next two paragraphs prove that E is a bijection from Z inft onto Y, and that its inverse is This paragraph argues that E followed by the function (28) is the identity function on Z inft . Accordingly, take any Z ∈ Z inft . I argue where the arrows apply the functions E and (28), respectively. The first arrow applies E by inspection. The second arrow applies (28) because E(Z) ∈ Y by the second-previous paragraph. To see the ⊆ direction of the equality, take any This paragraph argues that the function (28) followed by E is the identity function on Y. Accordingly, take any (t v ) v≥1 ∈ Y. I argue where the arrows apply the functions (28) and E, respectively. The first arrow applies (28) by inspection. Before applying E, it must be shown that S := {t o }∪{t v |v ≥ 1} belongs to Z inft . In other words, it must be shown that S is an infinite maximal chain. The definitions of (t v ) v≥1 and Y assure that S is a chain and that S contains a node of every stage. This easily implies that S is infinite. It also implies that S is maximal because distinct nodes in a chain have different stages by Lemma A.1(a). Hence S belongs to Z inft and the second arrow applies E. The equality follows from the fact that (∀v ≥ 1) k(t v ) = v by the definitions of (t v ) v≥1 and Y.
The initial step (i = 1) holds by (9b) of Proposition 2.3, applied at t = t 2 (note t 2 = t o because p m (t 2 ) exists and m ≥ 1). To show the inductive step (m ≥ i > 1), I argue The first equality is a rearrangement. The second equation holds by (9b) of Proposition 2.3, applied at t = p i−1 (t 2 ) (note p i−1 (t 2 ) = t o because p m (t 2 ) exists and m ≥ i). The third equation holds by the inductive hypothesis, and the fourth is a rearrangement. Finally, I argue τ (t 1 ) = τ (p m (t 2 )) = (p ) m (τ (t 2 )).
The first equality holds by the assumption t 1 = p m (t 2 ), the second holds by (29) at i = m.
(c). By the definition of k (τ (t)), it suffices to show The first equality follows from the definition of k (τ (t o )). The second equality holds because [a] t o = p k(t) (t) by the definition of k(t), and hence Then by the definition of ≺, there exists m ≥ 1 such that t 1 = p m (t 2 ). Thus by part (b), τ (t 1 ) = (p ) m (τ (t 2 )). Thus by the definition of ≺ , .
(e). Suppose t 1 t 2 . Then by the definition of , either t 1 = t 2 or t 1 ≺ t 2 . In the case of equality, τ (t 1 ) = τ (t 2 ). In the case of precedence, part (d) implies To show that τ | S is injective, suppose t 1 and t 2 are distinct members of S. Since S is a chain, t 1 ≺ t 2 without loss of generality. Hence τ (t 1 ) ≺ τ (t 2 ) by part (d).
To show that τ (S) is a chain, take any distinct t 1 and t 2 in τ (S). Since both are in τ (S), there exist distinct t 1 and t 2 in S such that τ (t 1 ) = t 1 and τ (t 2 ) = t 2 . Thus since S is a chain, t 1 ≺ t 2 without loss of generality. Hence τ (t 1 ) ≺ τ (t 2 ) by part (d). Hence t 1 ≺ t 2 by the definition of t 1 and t 2 .
(g). Take any Z ∈ Z inft . Since Z is an infinite chain in T , part (f) implies that τ (Z) is an infinite chain in T . Thus by Lemma A.1(f) applied to (T , p ) at S = τ (Z), there exists Z ∈ Z inft such that τ (Z) ⊆ Z .
(h). Take any Z ∈ Z ft . Since Z is a chain in T , part (f) implies that τ (Z) is a chain in T . Thus by Lemma A.1(g) applied to (T , p ) at S = τ (Z), there exists Z ∈ Z ft ∪Z inft such that τ (Z) ⊆ Z .
The equality is a rearrangement. The first inclusion holds by (8c) for γ, and the second inclusion holds by (8c) for γ .
The first paragraph of this proof shows that the identity arrow id (T,p) is welldefined for any functioned tree (T, p). The second paragraph shows that the composition γ •γ is well-defined for any morphisms γ and γ . The unit and associative laws are immediate. Thus Tree is a category (e.g. [5, Section 1.3]).
To show (31c), take any (t , t ) ∈ p . For notational ease, define t = τ −1 (t ) and t = τ −1 (t ). Thus it suffices to show that (t , t) ∈ p, or equivalently, that First, I argue If t = t o were true, [a] t t since t o precedes every element of T , thus [b] τ (t ) τ (t) by γ being a morphism and Proposition 2.4(e), and thus [c] t t by the definitions of t and t. This would contradict t t which follows from the assumption that p (t ) = t . Second, I argue The first equality follows from γ being a morphism, from (9b) of Proposition 2.3, and from (33). The second equality follows from the definition of t . The third holds by assumption. Finally, I argue that (32) holds: The first equality follows from applying τ −1 to both sides of (34). The second equality is the definition of t. Finally, Thus γ is an isomorphism (and γ −1 = γ * ). Proof. By Theorem 2.6, τ is bijective and γ −1 = [(T , p ), (T, p), τ −1 ]. These facts will be used implicitly. Take any Z ∈ Z. Then by Proposition 2.4(f) applied to γ at S = Z, τ (Z) is a chain. Hence it remains to be shown that τ (Z) is maximal. Suppose not. Then there is t / ∈ τ (Z) such that τ (Z)∪{t } is a chain. By Proposition 2.4(f) applied to γ −1 These facts will sometimes be used implicitly.
(g). This proof is similar to that of the previous part. Replace ≺ with , and replace Proposition 2.4(d) with Proposition 2.4(e).
(h)-(i). Let Z = Z ft ∪Z inft and Z = Z ft ∪Z inft . Since τ is a bijection, the cardinality of S equals the cardinality of τ (S) for any set S ⊆ T . Thus it suffices for both parts (h) and (i) to show that τ | Z is a bijection from Z onto Z . Lemma A.9 implies that τ | Z is a well-defined function from Z into Z . It is injective because τ is injective. To show that it is surjective, take any Z ∈ Z . By Lemma A.9 applied to α −1 , τ −1 (Z ) ∈ Z. Thus τ (τ −1 (Z )) = Z is in the range of τ | Z . (

Appendix B. Between Tree and Grph ca
This entire appendix is concerned with the proof of Theorem 2.8.
Lemma B.1. Suppose that (T, p) is a functioned tree. Then G 0 (T, p) is a nontrivial converging arborescence, where G 0 is defined in Theorem 2.8.
Proof. Derive t o from (T, p). Let (T, E, init, ter) = G 0 (T, p). I will show that (T, E, init, ter) is a nontrivial converging arborescence in four steps.
Step 1 This step shows that (T, E) is a nontrivial graph. By remark (ii) in the paragraph following the definition (1) of a functioned tree, (/ ∃t) p(t) = t. This, and the nonemptiness of p (1a), imply that E is a nonempty collection of two-element subsets of T . Thus by the definitions of [6, p. 2], (T, E) is a nontrivial graph.
Step 2 This step shows that (T, E) is a nontrivial tree. If neither t A nor t B is t o , two applications of (1b) imply the existence of m A ≥ 1 and where p 0 is the identity function. This m * exists because the set in its definition contains 0 by the definitions of m A and m B .
By the definition of E, there is a path 13 and also a path Suppose these two paths shared a node t = t * . Then t = t * being on (36a) would imply the existence of n ≥ 1 such that and p n (t ) = t * , where p 0 is the identity function. Further, p n (t ) = t * and t being on (36b) would imply Since n ≥ 1, (37a) and (37b) contradict the definition of m * . Hence the sets (36a) and (36b) do not share a node other than t * . Hence the union of the two paths (36a) and (36b) is a path. This path links t A and t B .
13 As in [6, p. 6, note 3], I denote a path by a natural sequence of its nodes. Pairs of nodes that are adjacent in the sequence are adjacent in the sense of [6, p. 3].

For [b]
, take any edge e ∈ E and suppose that (T, E {e}) is connected. The following four paragraphs will derive a contradiction with (1b).
First, by the definition of E, there is some (t 1 , t 2 ) ∈ p such that e = {t 1 , t 2 }. Since (t 1 , t 2 ) ∈ p, (1a) implies Second, since (T, E {e}) is connected, there is a path without e = {t 1 , t 2 } that links t 1 and t 2 . Denote this path 13 by Since this path does not use e = {t 1 , t 2 }, Thus n ≥ 4 and the path has at least three nodes. This paragraph argues that In particular, I will make an inductive argument in which k is decreasing rather than increasing. Consider the initial step (k = n−1). By (39), {t n−1 , t n } ∈ E. Thus by the definition of E, either (t n−1 , t n ) ∈ p or (t n , t n−1 ) ∈ p. Thus, since p is a function by (1a), either p(t n−1 ) = t n or p(t n ) = p n−1 . The second contingency is the first equality in the contradiction where the inequality is (40), the second equality holds because t n = t 1 by (39), and the third equality holds by (38). Thus p(t n−1 ) = p n . Further t n−1 ∈ T {t o } since the domain of p is T {t o } by (1a). Next consider the inductive step (k∈{2, 3, . . . n−2}). By the definition of t k in (39), {t k , t k+1 } ∈ E. Thus by the definition of E, either (t k , t k+1 ) ∈ p or (t k+1 , t k ) ∈ p. Thus since p is a function by (1a), either p(t k ) = t k+1 or p(t k+1 ) = t k . The second contingency is precluded by [a] the inductive hypothesis that p(t k+1 ) = t k+2 and [b] the fact that t k = t k+2 because the nodes of any path are distinct. Thus This paragraph argues that where p 0 is the identity. The case i = 0 is trivial. The case i = 1 holds by (38). For any i ∈ {2, 3, . . . n−1}, where the first equality holds by (38) and the remaining i−1 equalities holds by (41).
Note that where the first equality holds by (42) at i = n−1, and where the second equality holds by (39). Thus where the first equality holds by the previous sentence, where the second equality holds by (42) Step 3. This step shows (10a). In other words, it shows that (T, E, init, ter) is a nontrivial oriented tree. By Step 2, it suffices to show that (T, E, init, ter) is oriented ([6, p. 28]).
As a preliminary observation, this paragraph shows that p is asymmetric in the sense that Suppose there were such t A and t B . Step 4. This step shows (10b). In other words, it shows that (∀e∈E) init(e) = t o and ter(e) is on the path linking init(e) and t o .
Take any e ∈ E. By the definitions of E, init, and ter, there Thus p(t A ) is on the path linking p A and t o . Note p(t A ) = t B = ter(e) by the definition of (t A , t B ) and by [c]. Also note p A = init(e) by [b]. Thus by substitution, the last three sentences imply that ter(e) is on the path linking init(e) and t o . Suppose that [(T, p), (T , p ), τ] is a functioned-tree morphism. Then G 1 ([(T, p), (T , p ), τ]) is a directed-graph morphism, where G 1 is defined in Theorem 2.8. Further, its source and target are nontrivial converging arborescences.
Proof. By the definitions of G 1 and G 0 , where the first equality holds by (44a)-(44b), the second equality is a rearrangement, the set inclusion holds by (8c), and the third equality holds by (44e).
To see the first half of (12c), take any e ∈ E. By (44b), there is a (t , t) ∈ p such that e = {t , t}. Thus by (44c), init(e) = t . Hence where the first equality holds by the second-previous sentence, and where the second equality holds by the previous sentence and (44f). By this paragraph's two centered equations, τ •init(e) = init •ε(e).
where the first equality holds by the second-previous sentence, and where the second equality holds by the previous sentence and (44g). By this paragraph's two centered equations, τ •ter(e) = ter •ε(e).
Lemma B.3. Theorem 2.8's G is a well-defined functor from Tree to Grph ca .
Proof. By Lemma B.1, G 0 maps objects of Tree to objects of Grph ca . By Lemma B.2, G 1 maps arrows of Tree to arrows of Grph ca . It remains to show [1] that G preserves sources and targets, [2] that G preserves identities, and [3] that G preserves compositions.
where ε is defined in Theorem 2.8. The first equality holds by the definition of G 1 , the second by the definition of src in Grph ca , and the third by the definition of src in Tree. A symmetric argument shows that G preserves targets. [2]. Take any Tree object (T, p). As a preliminary step, note that the definition of G 0 implies where and where init and ter are also defined in the definition of G 0 but not needed explicitly here. I argue that The first equality in (47) , p ), τ ] be any two Tree morphisms. As a preliminary step, note that the definition of G 0 implies G 0 (T , p ) = (T , E , init , ter ), and and where init, ter, E , init , ter , E , init , and ter are also derived from the definition of G 0 but not needed explicitly here. I argue that where The first equality in (51) holds by the definition of • in Tree. The second equality holds by the definition of G 1 and by (49). The third equality will be proved in the following paragraph. The fourth equality holds by the definition of • in Grph ca . The fifth equality holds by two applications of the definition of G 1 and by (49). For the third equality in (51), it suffices to show that ε * = ε •ε. Toward that end, take any e in E. By (50), there exists (t , t) ∈ p such that e = {t , t}. Thus where the first equality holds by the second-previous sentence, where the second equality holds by the previous sentence and (52c), and where the third equality holds by [ I show that (T, p) is a functioned tree in four steps. During these steps, (53a) is often used implicitly, while (53b) and (53c) are used explicitly.
Step 1 This step proves the following lemma: "Take any t =t o and let (t, t 1 , t 2 , . . . t m = t o ) be the path linking t and t o . Then (t, t 1 )∈ p." To prove this lemma, take any t = t o and let (t, t 1 , t 2 , . . . t m = t o ) be the path linking t and t o . Then {t, t 1 } ∈ E. Thus by (11), The latter, together with the second half of (53b) at e = {t, t 1 }, would imply that t is on the path linking t 1 and t o . But this is impossible because [a] the path linking t 1 and t o is the subpath (t 1 , t 2 , . . . t n = t o ) and [b] t is not on this subpath since it is on the full path (t, t 1 , t 2 , . . . t n = t o ). Thus the former alternative holds. This equality leads to where the set membership follows from {t, t 1 } ∈ E and (53c).
Step This paragraph shows that (∀t = t o )(∃t 1 ) (t, t 1 ) ∈ p. Take any t = t o . Let (t, t 1 , t 2 , . . . t m = t o ) be the path linking t and t o . Then (t, t 1 ) ∈ p by the lemma of Step 1.

It remains to be shown that (∀t = t o )(/ ∃t
, (t, t B )} ⊆ p. I will show this by contradiction. Toward this end, take any t = t o and suppose t A = t B are such that {(t, t A ), (t, t B )} ⊆ p. Then (53c) implies Let (t, t 1 , t 2 , . . . t n = t o ) be the path linking t and t o . Since t = t o by assumption, this path has at least two vertices and thus Further, the remainder of this paragraph shows that both t A and t B are on this path. Without loss of generality, consider t A . By the first half of (54a), Thus by the second half of (53b) at e = {t, t A }, ter({t, t A }) is on the path linking init({t, t A }) and t o . Thus by the second half of (54a), t A is on the path linking t and Since t A = t B by assumption, at least one of them is distinct from t 1 . Without loss of generality, suppose t A = t 1 . Then consider the subpath (t, t 1 , t 2 , . . . t A ). Since t = t 1 by (55), and since t 1 = t A by the second-previous sentence, this subpath has at least three nodes. This and the first half of (54a) imply the existence of a cycle ( [6, p. 8]). This contradicts (T, E) being a tree ([6, p. 13]).
Step 3 This step shows that ( be the path linking t and t o . By the lemma of Step 1, (t, t 1 ) ∈ p. Thus since p is a function by Step 2, Further, the remainder of this paragraph argues that Take any such k. Then (t k , t k+1 , t k+2 , . . . t m = t o ) is the path linking t k and t o . Thus (t k , t k+1 ) ∈ p by the lemma of Step 1. Thus p(t k ) = t k+1 since p is a function by Step 2. Finally, I argue The first equality holds by (56). The last equality holds by the definition of t m . The intervening equalities hold by (57).
Step 4 By Step 2, p is a function from T {t o } onto X, where X is defined to be the range of p. Further, the nontriviality of (53a) implies that E is nonempty. Thus by (53c), p is nonempty. Hence (1a) has been established. (1b) was shown in Step 3.
By substitution, the last three sentences imply (τ (t ), τ(t)) ∈ p .  where the first equality holds by the definition of H 1 , the second by the definition of src in Tree, and the third by the definition of src in Grph ca . A similar argument shows that H preserves targets. [2]. Take any Grph ca object (T, E, init, ter). Then where the first equality holds by the definition of id in Grph ca , the second by the definition of H 1 , and the third by the definition of id in Tree. [3]. Take any two Grph ca arrows [(T, E, init, ter), (T , E , init , ter ), τ, ε] and [(T , E , init , ter ), (T , E , init , ter ), τ , ε ]. Then Proof B.7. (for Theorem 2.8) Lemma B.3 shows that G is a well-defined operator from Tree to Grph ca . Conversely, Lemma B.6 shows that H is a well-defined functor from Grph ca to Tree.
It remains to be shown [1] that H 0 •G 0 maps any functioned tree to itself, [2] that G 0 •H 0 maps any nontrivial converging arborescence to itself, [3] that H 1 •G 1 maps any functioned-tree morphism to itself, and [4] that G 1 •H 1 maps any directedgraph morphism, whose source and target are nontrivial converging arborescences, to itself. [1]. Take any functioned tree (T, p). By the definitions of G 0 and H 0 ,
The first equality holds by the definition of e, and the second holds by the definition of (t , t). The third equality holds by (c) and (d) because (t , t) ∈ p by definition. The fourth equality holds because (t , t) ∈ p by definition.
This paragraph shows p ⊆ p * . Take any (t , t) ∈ p. By (b), (c), and (d), respectively, Thus by (a), (t , t) = (init({t , t}), ter({t , t})) belongs to p * . [2]. Take any nontrivial converging arborescence (T, E, init, ter). Define E * , init * , and ter * by equalities (b), (f), and (j) below. It suffices to show the remaining ten equalities.  (11). (e) is trivial. (i) and (m) hold because E is the domain of init and ter. [3]. Take any functioned-tree morphism [(T, p), (T , p ), τ]. I argue The first equality follows from the definition of G 1 . The second equality follows from the definition of H 1 . The third equality follows from (8a) and part [1]. [4]. Let [(T, E, init, ter), (T , E , init , ter ), τ, ε] be a directed-graph morphism whose source and target are nontrivial converging arborescences. As a preliminary observation, note that the definition of H 0 implies where p = {(init(e), ter(e)) | e∈E} and Define In the remainder of this paragraph I argue The first equality follows from the definitions of H 1 , the second from (60), and the third from the definition of G 1 . The fourth equality follows from (61), the fifth from (60), and the sixth from part [2].
where F is derived from (T, C, ⊗). The first equality holds by the previous sentence, the second equality holds by the definition of q, and the third equality holds by the definition of F . The ⊆ half of the fourth equality holds simply because every c is in C. The converse holds because each F −1 (c) is nonempty by (13c). Conclusions [2] and [3] imply (t, c, t ) = (p(t ), q(t ), t ). Thus conclusion [1] implies that (t, c, t ) belongs to To show the ⊇ direction, take any t = t o . Then by (13a) there exists (t, c) such that (t, c, t ) ∈ ⊗. By the definition of p, t = p(t ). By the definition of q, c = q(t ). Therefore by the last three sentences, (p(t), q(t), t ) ∈ ⊗.
(b). Part (a) suffices because (13a) assumes that ⊗ is a bijection when viewed as a function from the first two components of its constituent triples to the third component of its constituent triples.
To show the contrapositive of (16b), suppose H ∈ H and H ∈ H satisfy F (H)∩F (H ) = ∅. Then there exists c * , t, and t such that Since H = {F −1 (c)|c} is a partition by (13c), and since t ∈ F −1 (c * ) by the first half of (62a), the second half of (62a) implies that F −1 (c * ) = H. By similar reasoning with (62b), F −1 (c * ) = H . By the last two sentences, H = H .
(18)⇒ (19). Assume (18). (18a) implies the first half of (19a). For the second half of (19a), take any t ∈ T {t o }. Since ⊗ is onto T {t o } by (13a), there exists (t, c) ∈ F such that t = t⊗c. Thus τ (t ) = τ (t⊗c) = τ (t)⊗ δ(c), where the second equality holds by (18b). Thus since ⊗ is onto (19b) is an equation in the category Set. The left-hand side is well-defined because the codomain of ⊗ is T {t o } by (13a). The right-hand side is well-defined because the domain of ⊗ is F by (13a), and because the codomain of (τ, δ)| F is F by the proposition's definition [c]. The codomains of the two sides are both equal to T {t o } by the proposition's definition [d] and by (13a) for Π . Finally, the domains of the two sides are both equal to F by (13a) for Π. The previous four sentences have established that (19b) is a well-defined equation. By the second-previous sentence and (18b), the equation is true.
Proof. (17c)⇒ (20). Assume (17c). For (20a), I argue where the first equality holds by the definition of p, the first set inclusion holds by (17c), the second set inclusion holds by (17b), and the second equality holds by the definition of p . Similarly for (20b), I argue where the first equality holds by the definition of q, the first set inclusion holds by (17c), the second set inclusion holds by (17b), and the second equality holds by the definition of q .
(20)⇒ (21). Assume (20). To show (21a), take any t = t o . Since p is a function from T {t o } by (17a) and (13b), there exists t such that (t , t) ∈ p. Thus by (20a), (τ (t ), τ(t)) ∈ p . Thus since p is a function from T {t o } by (17a) and (13b), where the first implication holds because p is a function from T {t o } by (17a) and (13b), and where the next two implications are rearrangements. Similarly, I show (20b) implies (21c) by where the first implication holds because q is a function from T {t o } by (17a) and Lemma C.2, and where the next two implications are rearrangements.
Lemma C.9. A quadruple [Π, Π , τ, δ] is a morphism iff it satisfies (17a)-(17b) and (20). The first and second equalities are rearrangements, the third follows from the definition of c, and the fourth is a rearrangement. The first inclusion follows from the first half of (19a) in Proposition 3.3. The fifth equality is a rearrangement. The second inclusion follows from the first half of (17b). The sixth equality is a rearrangement, and the final equality follows from the definition of H .

C.3. The Category
Proof C.13.  (17a) for α imply (17a) for α •α. Second, note that τ :T →T by the first half of (17b) for α, and that τ :T →T by the first half of (17b) for α . Hence τ •τ :T →T , which is the first half of (17b) for α •α. A parallel argument shows δ •δ:C→C , which is the second half of (17b) for α •α. Finally, to show that (17c) holds for α •α, I argue The equality is a rearrangement. The first inclusion holds by (17c) for α, and the second inclusion holds by (17c) for α .
The first paragraph of this proof shows that the identity arrow id Π is well-defined for any preform Π. The second paragraph shows that the composition α •α is welldefined for any arrows α and α . The unit and associative laws are immediate. Thus NCP is a category.
Lemma C.14. Suppose α = [Π, Π , τ, δ] is an isomorphism. Then (a) τ and δ are bijective, and (b) Proof. Let Π = (T, C, ⊗) and let Π = (T , C , ⊗ ). Because α = [Π, Π , τ, δ] is an isomorphism (e.g. [5, p. 12]), its inverse α −1 = [Π , Π, τ * , δ * ] exists. Thus where the first two equalities in both lines follows from the definition of the inverse α −1 , and the third equality in both lines follows from the definition of id. The third component of (63a) implies that τ * •τ = id T . The third component of (63b) implies that τ •τ * = id T . The last two sentences imply that τ is a bijection from T onto T and that Similarly, the fourth components of (63a) and (63b) imply that δ is a bijection from C onto C and that Conclusion (a) holds by the last two sentences. Conclusion (b) holds by where the first equality holds by definition, and the second equality follows from (64)-(65).
This paragraph shows that t = t o . Derive from (T, p) and from (T , p ) (this is possible by [1]). Now suppose t = t o were true. Then [a] t t since t o precedes every element of T , thus [b] τ (t ) τ (t) by γ being a morphism (by [2]) and Proposition 2.4(e), and thus [c] t t by the definitions of t and t. This contradicts t t which follows from p (t ) = t , which in turn follows from the assumption that (t , c , t ) ∈ ⊗ .
Since the range of ⊗ is T {t o }, and since t = t o by the previous paragraph, Proposition 3.1(b) implies that Note that where the first equality holds by (67) and (17c) for α, and the second equality holds by the definition of t . Because of the previous equality, because t ⊗ c = t by assumption, and because ⊗ is a bijection by (17a) for α and (13a) for Π , Hence Now take (67) and replace its three terms by means of [a] the two equalities in the last sentence and [b] the definition of t . The result is as required by (66c). Finally, Thus α is an isomorphism (and α −1 = α * ).
[3] Take any preform (T, C, ⊗). Let F 0 (T, C, ⊗) = (T, p). I argue where the first equality holds by the definition of identity in NCP, the second holds by the definition of F 1 , the third holds by the definition of (T, p), the fourth holds by the definition of the identity in Tree, and the fifth holds by the definition of (T, p). [4] Take any preform morphism [Π, Π , τ, δ]. For sources, where the first equality holds by the definition of • in NCP, the second by the definition of F 1 , the third by the definition of • in Tree, and the fourth by the definition of F 1 .
Proof. Derive F from (T, C, ⊗). Note that where the first equality follows from the definition of q, the second follows from the definition of F , the third holds because ⊗ is a function by (13a), and the fourth and the fifth are rearrangements. Suppose (T, C, ⊗) has perfect information. By Lemma C.2, q is a function onto C. Thus it remains to show that each q −1 (c) is a singleton. This follows from (70) because perfect information means that each F −1 (c) is a singleton.
Conversely, suppose q is bijective. By the definition of perfect information, it suffices to show that each F −1 (c) is a singleton. Take any c. Because q is bijective and because q is onto C by Lemma C.2, q −1 (c) is a singleton. Thus, by (70), { t⊗c | t∈F −1 (c) } is a singleton. Thus, since ⊗ is injective by (13a), F −1 (c) is a singleton.
I will show that (T, C, ⊗) satisfies (13a)-(13b) and (23). This suffices because (23) is both the definition of perfect information and a sufficient condition for (13c).
It must be shown [a] that the above quadruple satisfies (17a)-(17c) and [b] that its source and target have perfect information. (17a) and [b] follow from Lemma D.3. The first half of (17b) follows from (8b). To see the second half of (17b), recall that The first equality holds by (73b). The second equality holds because the domain of p is T {t o } by (8a) and (1a). The first set inclusion holds by (8c). The second set inclusion holds by (8b). The final equality is (73d).
Lemma D.5. E is a well-defined functor from Tree to NCP p , where E is defined in Theorem 3.13.
Proof. By Lemma D.3, E 0 maps objects of Tree to objects of NCP p . By Lemma D.4, E 1 maps arrows of Tree to arrows of NCP p . Thus it remains to show [1] that E preserves identities, [2] that E preserves sources and targets, and [3] that E preserves compositions. [1]. Take any functioned tree (T, p) with its t o . Note   Lemma D.6. Define the functors F p and E as in Theorem 3.13. Then E•F p is naturally isomorphic to the identity functor for NCP p . In particular, for every object Π in NCP p , define the quadruple where q is the previous-choice function of Π = (T, C, ⊗). Then (a) for every object Π in NCP p , η Π is an isomorphism in NCP p . Further (b) for every arrow α = [Π, Π , τ, δ] in NCP p , Proof. The functors F p and E are well-defined by Theorem 3.9 and Lemma D.5.
(a). Take any perfect-information preform Π = (T, C, ⊗) with its t o , p, and q. Importantly, Lemma C.2, Lemma D.2, and the perfect information of Π imply Further, by the definitions of F p0 and E 0 ,

{(t, t , t )|(t , t)∈p}).
By the definition of η Π and the previous sentence, This paragraph shows that η Π is a morphism in NCP p . By definition (17) and the equalities of (76), this is equivalent to showing Π and E 0 •F p0 (Π) are perfect-information preforms, id T : T → T, q −1 : C → T {t o }, and (77b) (77a) holds since Π is a perfect-information preform by assumption and since E 0 •F p0 (Π) is a perfect-information preform by Lemma D.3. The first half of (77b) is obvious. The second half of (77b) follows from (75). Finally, for (77c), I argue To see the set inclusion, take any (t, c, t )∈⊗. By the definition of q, c = q(t ). Thus by (75), q −1 (c) = t . The equality follows from the definition of p. Lastly, this paragraph shows that η Π is an isomorphism in NCP p . By the previous paragraph and Theorem 3.7, it suffices to show that the transformations id T and q −1 are bijective. Obviously, id T is bijective. Further, q −1 is bijective by (75).
First, I argue Finally, I argue The first equality holds by (78) and the lemma's definition of η Π . The second equality holds by the definition of • in NCP p . The third equality holds by (79). The fourth equality holds by the definition of • in NCP p . The fifth equality holds by the definitions of η Π and α.
Lemma D.7. Define the functors F p and E as in Theorem 3.13. Then F p •E equals the identity functor for Tree.
Proof. The functors F p and E are well-defined by Theorem 3.9 and Lemma D.5.
Step 0 will show that F p0 •E 0 maps each functioned tree to itself.
Step 1 will show that F p1 •E 1 maps each functioned-tree morphism to itself.
Step 0 Take any functioned tree (T, p) with its t o . By the definition of E 0 , Hence it suffices to show that p * = p. I argue where the first equality is the definition of p * , the second follows from the definition of ⊗, and the third holds by the equivalence of the sets' predicates.
Step 1 Take any functioned-tree morphism [(T, p), (T , p ), τ], and let (T, p) determine t o . Then where the first equality follows from the definition of E 1 , the second equality follows from the definition of F p1 , and the third equality follows from two applications of Step 0.