1. Introduction
It is often convenient to encode weak categorical or algebraic structures as certain kinds of presheaves on a shape category that models the situation in question, just as we model the homotopy theory of spaces using simplicial sets. For example, the simplicial indexing category $\boldsymbol{\Delta}$ is used in $(\infty,1)$ category theory [ Reference Joyal and Tierney40, Reference Lurie45, Reference Rezk52 ] and Joyal’s cell category $\boldsymbol{\Theta}_n$ is used for $(\infty,n)$ categories [ Reference Ara1, Reference Rezk53 ]. For $(\infty,1)$ operads, the Moerdijk–Weiss dendroidal category $\boldsymbol{\Omega}$ [ Reference Moerdijk and Weiss48 ], which is a category of rooted trees and an enlargement of $\boldsymbol{\Delta}$ , underlies several of the many equivalent models [ Reference Barwick3, Reference Chu, Haugseng and Heuts14–Reference Cisinski and Moerdijk17, Reference Heuts, Hinich and Moerdijk36 ]. This paper is concerned with shape categories that have been used to model higher versions of certain important types of generalised operads.
The shapes we study will be certain graphs with loose ends (Figure 1), which might come with additional data (e.g. orientations of edges, planar structures) or restrictions (e.g. connected, acyclic). Such graphs are well suited to describing the kinds of operations appearing in generalised operads via the mechanism of ‘graph substitution’ or ‘graph insertion.’ Given two graphs G and H and a bijection between the germs of edges at a vertex of G and the loose ends of H, we can form a new graph $G\{H\}$ where a small neighborhood of v has been replaced by H. The categories we focus on have graph substitution built in at a fundamental level, as is the case for several other useful categories involving such graphs (see, for instance, [ Reference Batanin and Markl4, Reference Kaufmann and Ward41 ]). However, in this paper we shift attention in our definitions away from graph substitution and towards embeddings of graphs.
The two main graph categories we will discuss are related to modular operads [ Reference Getzler and Kapranov27 ] (also known as compact symmetric multicategories [ Reference Joyal and Kock39 ]), and to wheeled properads [ Reference Markl, Merkulov and Shadrin47 ]. The objects of our shape categories are connected graphs with loose ends, which are undirected in the first case, and directed in the second.
The graphical category $\textbf{U}$ from [ Reference Hackney, Robertson and Yau33 ], whose objects are undirected graphs, is intimately connected with modular operads [ Reference Getzler and Kapranov27, Reference Joyal and Kock39 ], in that there is an associated nerve theorem (proved in [ Reference Hackney, Robertson and Yau34 ], but see also [ Reference Raynor51 ]). This theorem says that there is a fullyfaithful inclusion of $\textbf{ModOp}$ into the category of presheaves $\widehat{\textbf{U}} := \textrm{Fun}(\textbf{U}^{\textrm{op}}, \textbf{Set})$ , whose essential image consists of those presheaves satisfying a Segal condition (Definition 6·13). This is the modular operad analogue of the dendroidal nerve theorem from [ Reference Moerdijk and Weiss49, Reference Weber59 ]. One can also utilize the category $\textbf{U}$ to give notions of $(\infty,1)$ modular operads, as in [ Reference Chu and Haugseng13, Reference Hackney, Robertson and Yau33, Reference Strumila56 ]. Our first main theorem is a new description of the graphical category $\textbf{U}$ .
Theorem 1 (Theorem 4·15 and Theorem C6). Let G and G^{′} be undirected connected graphs. A graphical map $\varphi \colon G \to G'$ is the same thing as a pair consisting of an involutive function $\varphi_0\colon A_G \to A_{G^{\prime}}$ and a function $\hat \varphi \colon \operatorname{Emb}\!(G) \to \operatorname{Emb}\!(G')$ , so that this pair is compatible with respect to boundaries and the function $\hat \varphi$ preserves unions, vertex disjoint pairs of embeddings, and edges. Composition of graphical maps is given by composition of pairs of functions.
This theorem holds uniformly for maps in both the graphical category $\textbf{U}$ and the extended graphical category $\widetilde{\textbf{U}}$ from [ Reference Hackney, Robertson and Yau33 ]. This has aesthetic appeal, as the previous definitions of these categories relied on separate adhoc conditions to prevent ‘collapse.’ A key insight of this paper is the correct notion of unions of embeddings that make this theorem hold, and this appears in Section 3. We will state the precise conditions for the maps appearing in Theorem 1 in Definition 4·2.
One upshot of Theorem 1 is that composition becomes easy, and another is that it makes more transparent the activeinert factorisation system on $\textbf{U}$ (see Remark 4·16).
There is a parallel story for directed graphs and wheeled properads. Wheeled properads are a variant of dioperad or polycategory which are capable of modeling both parallel processes and feedback of processes. In [ Reference Hackney, Robertson and Yau29 ], the wheeled properadic graphical category was introduced as a category whose morphisms are certain maps between the free wheeled properads generated by directed graphs. As for $\textbf{U}$ , there is a nerve theorem for wheeled properads, allowing us to regard $\textbf{WPpd}$ as a full subcategory of presheaves on the wheeled properadic graphical category. The following is an analogue of Theorem 1 for directed graphs.
Theorem 1’ (Theorem 5·13 and Theorem C14). Let G and G^{′} be directed connected graphs. A map $\varphi \colon G \to G'$ in the wheeled properadic graphical category is the same thing as a pair consisting of a function $\bar\varphi_0\colon E_G \to E_{G^{\prime}}$ and a function $\hat \varphi \colon \operatorname{Emb}\!(G) \to \operatorname{Emb}\!(G')$ , so that this pair is compatible with respect to inputs/outputs and the function $\hat \varphi$ preserves unions, vertex disjoint pairs of embeddings, and edges. Composition is given by composition of pairs of functions.
This gives a new, purely categorical/combinatorial definition of the wheeled properadic graphical category from [ Reference Hackney, Robertson and Yau29, Reference Hackney, Robertson and Yau31 ] which does not rely on the category of wheeled properads. We identify the wheeled properadic graphical category as a comma category $\textbf{U}_{/\mathfrak{o}}$ for a certain orientation $\textbf{U}$ presheaf $\mathfrak{o}$ .
With this identification, the forgetful functor $\textbf{U}_{/\mathfrak{o}} \to \textbf{U}$ becomes relatively easy to understand and analyse. We use this to establish that the adjunction $\textbf{WPpd} \rightleftarrows \textbf{ModOp}$ between the categories of wheeled properads and modular operads can be well understood at the presheaf level.
Theorem 2 (Proposition 6·17 and Theorem 6·23). Let $f\colon \textbf{U}_{/\mathfrak{o}} \to \textbf{U}$ be the forgetful functor from the category of directed graphs to the category of undirected graphs. At the level of presheaves, both left Kan extension
and restriction
preserve Segal objects.
The first part of this theorem is somewhat unusual, and arises from a new, particularly simple formula for the left Kan extension when passing from directed to undirected graphs. We only discovered this formula because of the relationship in the descriptions from Theorem 1 and Theorem 1’.
We are also interested in several graph categories suitable for modeling other operadic structures: dioperads/symmetric polycategories [ Reference Gan23, Reference Garner24 ], properads/compact symmetric polycategories [ Reference Duncan21, Reference Vallette57 ], and (augmented) cyclic operads/ $\ast$ polycategories [ Reference Getzler and Kapranov26, Reference Hinich and Vaintrob37, Reference Shulman55 ]. In order to streamline the paper, discussion of these categories is mostly deferred to Appendix A and Appendix B. We also have a third appendix, Appendix C, which addresses the extended (oriented) graphical category, where nodeless loops (that is, vertexfree graphs whose geometric realisation is a circle) are present.
In Appendix A we show how to use Theorem 1 to give a description of categories of simplyconnected undirected graphs in a very similar manner to the ‘complete morphisms’ version of the unrooted tree category $\boldsymbol{\Xi}$ from [ Reference Hackney, Robertson and Yau32 ]. In Appendix B, we show how to identify the properadic graphical category as a subcategory of the wheeled properadic graphical category, and we show that the dioperadic graphical category is a full subcategory of the wheeled properadic graphical category. Finally, in Appendix C, we show that the results of Section 4 and Section 5 also hold when nodeless loops are considered. The results in Appendix C are important, and arguably the ‘correct’ versions of the theorems, but due to the inadequacies of the underlying definitions of graphs themselves they are best treated separately.
A diagram relating the graph categories from this paper appears in Figure 2. The categories in the top right are from Appendix A, the categories on the middle left are in Appendix B, and the two categories on the bottom are in Appendix C.
Further directions
One motivation for this work is the hope that an alternative definition of the graph categories could help facilitate comparisons to other, related graph categories in the literature. Of special interest are the operadic categories of graphs appearing in [ Reference Batanin and Markl4–Reference Batanin, Markl and Obradović6 ], which should be closely related to subcategories of active maps by [ Reference Berger9 , proposition 3·2]. We are also interested in the apparent differences with the graph categories arising from a twisted arrow construction in [ Reference Burkin10 ].
Another motivation is to provide a blueprint for work on operadic structures based on disconnected graphs. Such operadic structures include props, wheeled props, and nonconnected modular operads. We expect that the definitions in this paper generalise to yield plausible graph categories for these three structures, once the appropriate notion of embedding is established. Moreover, the activeinert factorisation systems on these categories, which are used to formulate the Segal condition, should arise in a natural and transparent way (see Remark 4·16). This would all play a role in developments related to [ Reference Raynor50 ], which concerns the circuit algebras of Bar–Natan and Dansco [ Reference BarNatan and Dancso2 ]. These circuit algebras are a generalisation of the planar algebras of Jones [ Reference Jones38 ], and turn out to be the same thing as wheeled props [ Reference Dancso, Halacheva and Robertson18 ]. In addition, it would be interesting to compare the kinds of infinityprops arising from this Segal condition with those appearing in [ Reference Haugseng and Kock35 ].
Lastly, the reader will notice that we have almost entirely avoided any homotopical issues, but these should be addressed later. We would like to know how the formulas for left Kan extensions in Section 6·1 behave with respect to Quillen model structures for higher operadic structures. Can the formula from Remark 6·21 be leveraged to resolve [ Reference DrummondCole and Hackney20 , remark 6·8], which seems to be needed for a conditional result of Walde [ Reference Walde58 , remark 5·0·20]? It would also be interesting to know if Theorem 2 can be promoted to the $\infty$ categories of Segal objects in spaces from [ Reference Chu and Haugseng13 ], and also to the enriched context following [ Reference Chu and Haugseng12, Reference Gepner and Haugseng25 ].
Structure of the paper
Section 2 mostly contains preliminary material which has previously appeared elsewhere. Section 3 contains a definition of unions of embeddings, but only Definition 3·1 and Example 3·2 are needed for the main narrative of the paper; the rest of the section is examining questions of existence of unions and may be skipped. Section 4 introduces the new description of graphical maps at the beginning, and the remainder of the section is devoted to the proof of Theorem 1.
Directed graphs appear in Section 5, and the new definition of the wheeled properadic graphical category appears as Definition 5·8 and Proposition 5·10. A reader who is not already familiar with wheeled properads and the wheeled properadic graphical category may wish to stop reading this section after Remark 5·12 (to skip the proof of Theorem 1’).
The final section of the main narrative of the paper is Section 6, which turns its attention to Segal presheaves which (via nerve theorems of the author, Robertson and Yau) are the same thing as modular operads and wheeled properads, respectively. Theorem 2 is proved in Section 6·1. Section 6 also deals with several other graph categories from the appendices, though relatively little knowledge is needed about them to follow this. It is not necessary to read the appendices to understand most of this section.
The paper contains three appendices, which are largely independent and may be read or skipped based on the interest of the reader. Appendix A is about a characterisation of maps in categories of undirected trees, and can be read at any point after Section 4. Appendix B exhibits the properadic graphical category as a particular nonfull subcategory of $\textbf{U}_{/\mathfrak{o}}$ whose objects are acyclic directed graphs, and can be tackled after Proposition 5·10. Finally, Appendix C is about the ‘extended’ versions of graphical categories that include the nodeless loop as an object. The first part can be read after Section 4, whereas Appendix C·1 relies on Section 5·2.
2. Preliminaries
The purpose of this section is to recall the most essential information about the graphical category $\textbf{U}$ from [ Reference Hackney, Robertson and Yau33 ]. The following definition of a combinatorial model for ‘graphs with loose ends’ is due to Joyal and Kock, and appears in [ Reference Joyal and Kock39 , section 3] under the name Feynman graphs.
Definition 2·1 (Graphs). Write $\mathscr{I}$ for the category freely generated by the graph
subject to the relation that the nontrivial endomorphism of $\texttt{a}$ squares to the identity.

(i) A graph is a functor $\mathscr{I} \to \textbf{FinSet}$ so that the selfmap of $\texttt{a}$ goes to a fixpointfree involution and the generating map $\texttt{a} \leftarrow \texttt{d}$ goes to a monomorphism.
A graph G will be written as
and we usually behave as though the leftward arrow is a subset inclusion $D_G \subseteq A_G$ .

(ii) If $v\in V_G$ is a vertex, then we write $nb(v) := t^{1}(v) \subseteq D_G$ for the neighbourhood of v.

(iii) The boundary of a graph G is the set $\eth(G) = A_G \setminus D_G$ .

(iv) The set of edges
\begin{align*} E_G = \{ [a,a^\dagger] \mid a \in A_G\} \cong A_G / {\sim}\end{align*}is the set of $\dagger$ orbits, and has half the cardinality of $A_G$ . 
(v) An edge $[a,a^\dagger]$ is an internal edge if neither of $a,a^\dagger$ are elements of $\eth(G)$ , otherwise it is a boundary edge.
In Appendix C we will use a more general (but also more fiddly) version of graph [ Reference Hackney, Robertson and Yau33 , definition 4·1], where the boundary is additional specified data.
Example 2·2. We give two basic examples of graphs that we will use repeatedly (see [ Reference Hackney, Robertson and Yau33 , example 1·4]).

(i) We write $G = {{\updownarrow}}$ for the edge, which has exactly two arcs $A_G = \{ \sharp, \flat \}$ , and $D_G = \varnothing = V_G$ . Note that $\eth(G) = A_G$ .

(ii) If $n\geq 0$ , we write $G = {\unicode{x2606}}_n$ for the n star. This graph has $V_G$ a onepoint set, $D_G = \{1, \dots, n \}$ , and $A_G = \{ 1, 1^\dagger, \dots, n, n^\dagger \}$ . Note that $\eth(G) = \{ 1^\dagger, \dots, n^\dagger \}$ . The 5star is depicted on the left of Figure 3.
From a graph G, one can obtain a topological space as a quotient space of
by identifying, $x \sim 1x$ where $x\in ({1}/{3},{2}/{3})$ is in the acomponent and $1x\in ({1}/{3},{2}/{3})$ is in the $a^\dagger$ component, and letting t identify 0 in the dcomponent with $t(d) \in V_G$ . As vertices may have arity two, in order to not lose any information one should consider this as a pair of spaces $\mathscr{X} = (X,V)$ with V finite. Notice that we can recover all of the data of G from the homeomorphism class of $\mathscr{X}$ (see [ Reference Hackney, Robertson and Yau33 , section 1]).
The following definition of étale map appeared in [ Reference Joyal and Kock39 ]; these are the maps of graphs which preserve arity of vertices.
Definition 2·3. An étale map between graphs is a natural transformation of functors
so that the righthand square is a pullback. An embedding between connected graphs is an étale map which is injective on vertices.
‘Connected’ in the preceding definition means in the usual sense for an object in $\textbf{FinSet}^{\mathscr{I}}$ , namely G is connected if and only if given any coproduct splitting $G \cong G_1 \amalg G_2$ precisely one of $G_1$ or $G_2$ is empty. This is equivalent to G being nonempty and every two elements have a path between them (see Definition A1), and is also equivalent to the associated topological space being connected.
Étale maps from the edge ${\updownarrow}$ classify arcs, as they are determined by where $\sharp$ is sent. Étale maps from stars weakly classify vertices: if nb(v) has cardinality n, then there are precisely $n!$ étale maps ${\unicode{x2606}}_n \to G$ which send the unique vertex to v. See also Example 2·5.
Remark 2·4. Each étale map $G\to G'$ induces a local homeomorphism $\mathscr{X} \to \mathscr{X}'$ of the corresponding topological graphs, and each embedding between connected graphs induces an injective local homeomorphism. If we insist that we can recover the original étale map from the local homeomorphism, then this local homeomorphism is essentially unique up to deformation through a family of local homeomorphisms. In contrast, there are two distinct (orientationpreserving) embeddings $(0,1) \amalg (0,1) \to (0,1)$ up to deformation through families of embeddings, while there is only one when considered up to deformation through families of local homeomorphisms. This hints at why étale maps may be illsuited for describing embeddings with disconnected domain.
Example 2·5 (Examples of embeddings). If G is a graph, then every edge and every vertex determines an embedding.

(i) Suppose that $e = [a,a^\dagger]$ is an edge of G. We can define a graph, denoted ${\updownarrow}_e$ , with $V = D = \varnothing$ , and $A = \eth({\updownarrow}_e) = \{a,a^\dagger\}$ together with the unique fixpointfree involution. Inclusion of arc sets yields a canonical embedding ${{\updownarrow}}_e \rightarrowtail G$ .

(ii) If $f \colon H \rightarrowtail G$ is any embedding with H isomorphic to ${\updownarrow}$ (which happens if and only if $V_H = \varnothing$ ), we say that f is an edge. Such an f will be isomorphic to one of the inclusions ${{\updownarrow}}_e \rightarrowtail G$ .

(iii) Suppose v is a vertex of G. Define ${\unicode{x2606}}_v$ to be the graph with $V_{{\unicode{x2606}}_v} = \{ v \}$ , $D_{{\unicode{x2606}}_v} = nb(v) = \{d_1, \dots, d_n \} \subseteq D_G$ , $\eth({\unicode{x2606}}_v) = \{b_1, \dots, b_n\}$ (which are new, formally defined elements), and $d_i^\dagger := b_i$ . Then ${\unicode{x2606}}_v$ is isomorphic to ${\unicode{x2606}}_n$ , and there is an embedding $\iota_v \colon {\unicode{x2606}}_v \rightarrowtail G$ which sends $d_i$ to $d_i \in D_G$ and $b_i$ to $d_i^\dagger \in A_G$ .

(iv) It is not necessary for an embedding to be injective on arcs. As an example, take the quotient G of ${\unicode{x2606}}_5$ (see Example 2·2) by identifying the arcs $2 \sim 5^\dagger$ and $2^\dagger \sim 5$ but keeping the rest of the structure the same. This is depicted in Figure 3. The canonical map ${\unicode{x2606}}_5 \rightarrowtail G$ is an embedding but not a monomorphism. Likewise, if v is an arbitrary vertex of a graph G, then $\iota_v \colon {\unicode{x2606}}_v \rightarrowtail G$ is injective on arcs if and only if there are no loop edges at v.
This last point essentially indicates the only way an embedding can fail to be a monomorphism on arcs and edges — by identifying pairs of boundary edges (Lemma 2·14). Another example appears in Figure 4 below.
Lemma 2·6. If $f \colon H \rightarrowtail G$ is an embedding and $H\cong G$ , then f is an isomorphism.
Proof. The map f provides injections of finite sets $D_H \hookrightarrow D_G$ and $V_H \hookrightarrow V_G$ which must be isomorphisms. To show that $A_H \to A_G$ is an injection (and hence an isomorphism), it is enough to show that the injection $\eth(H) \to A_G$ lands in $\eth(G)$ . This is clear if G is an edge, since then $A_G = \eth(G)$ . Assume that G is not an edge. Given $a\in \eth(G)$ , we know $a^\dagger \in D_G$ , so $a^\dagger = f(d)$ for some unique $d\in D_H$ . Notice that $d^\dagger \in \eth(H)$ , for otherwise $a = f(d^\dagger)$ would be an element of $D_G$ . This shows that $f(\eth(H)) \subseteq \eth(G)$ as desired, so $f \colon A_H \to A_G$ is an isomorphism.
In definition 1·28 of [ Reference Hackney, Robertson and Yau33 ], a set $\operatorname{Emb}\!(G)$ is defined by considering all embeddings with codomain G, and then identifying isomorphic embeddings. That is, $f\sim h$ just when there exists a (unique) isomorphism z with $f=hz$ . We next give an alternative definition of this set which reveals additional structure.
Given a (small) category $\textbf{C}$ , the posetal reflection of $\textbf{C}$ is obtained by identifying objects $c_0$ and $c_1$ if and only if there are arrows $c_0 \to c_1$ and $c_1 \to c_0$ in $\textbf{C}$ , and declaring that on equivalence classes $[c] \leq [c']$ just when there is an arrow $c \to c'$ in $\textbf{C}$ .
Proposition 2·7. Let $\acute{\textbf{E}}$ be the category of graphs and étale maps between them, let G be a connected graph, and let $\textbf{C} \subseteq \acute{\textbf{E}}_{/G}$ be the full subcategory of the slice on the embeddings with codomain G. Then $\operatorname{Emb}\!(G)$ is the underlying set of the posetal reflection of (a skeleton of) $\textbf{C}$ .
Proof. First, notice that for a morphism
from h to k, we have that the étale map f is an embedding. Suppose that we have two morphisms f and g of $\textbf{C}$
then $gf \colon H \to H$ is an isomorphism by Lemma 2·6. Likewise, $fg \colon K \to K$ is an isomorphism. Since fg and gf are isomorphisms, we conclude that f is an isomorphism and hence $[h]=[k]$ in $\operatorname{Emb}\!(G)$ .
If $h = kz$ for some isomorphism z, then h and k are identified in the posetal reflection.
In order to give the definition of the graphical category $\textbf{U}$ , we first need some notation.
Definition 2·8 (Boundary and vertex sum of embeddings). If S is a set, write $\wp(S)$ for the power set, that is, for the set whose elements are the subsets of S. Let $\mathbb{N}S$ denote the free commutative monoid on S, whose elements are finite unordered lists of elements of S. When S is finite, regard $\wp(S)$ as the subset of $\mathbb{N}S$ containing those lists which do not have repeated elements. Now suppose G is a graph.

(i) Write $\varsigma \colon \operatorname{Emb}\!(G) \to \mathbb{N}V_G$ for the map of sets which sends the class of an embedding $f\colon H \rightarrowtail G$ to $\varsigma[f] = \sum_{v\in V_H} f(v) \in \mathbb{N}V_G$ . As embeddings are injective on vertices, this comes from a function $\varsigma \colon \operatorname{Emb}\!(G) \to \wp(V_G)$ sending [f] to $f(V_H)$ .

(ii) Write $\eth \colon \operatorname{Emb}\!(G) \to \wp(A_G) \subseteq \mathbb{N}A_G$ for the map which sends the class of an embedding $f \colon H \rightarrowtail G$ to $\eth([f]) = f(\eth(H)) \subseteq A_G$ .
By [ Reference Hackney, Robertson and Yau33 , lemma 1·20], $\eth(H) \to A_G$ is injective, so $\eth([f])$ is isomorphic to $\eth(H)$ and may be written as $\sum_{b\in \eth(H)} f(b) \in \mathbb{N}A_G$ . See Section 2·1 below for more on embeddings. The following appears as definition 1·31 of [ Reference Hackney, Robertson and Yau33 ].
Definition 2·9 (Graphical map). Let G and G ^{′} be connected (undirected) graphs. A graphical map $\varphi \colon G \to G'$ is a pair $(\varphi_0, \varphi_1)$ consisting of an involutive function $\varphi_0 \colon A_G \to A_{G^{\prime}}$ and a function $\varphi_1 \colon V_G \to \operatorname{Emb}\!(G')$ , so that:

(i) the inequality
\begin{align*} \sum_{v\in V_G} \varsigma(\varphi_1(v)) \leq \sum_{w\in V_{G^{\prime}}} w\end{align*}holds in $\mathbb{N}V_{G^{\prime}}$ (i.e., $\sum \varsigma(\varphi_1(v)) \in \wp(V_{G^{\prime}})$ ); 
(ii) for each $v \in V_G$ , there is a bijection making the diagram
commute;

(iii) if $\eth(G)$ is empty, then there exists $v \in V_G$ so that $\varphi_1(v)$ is not an edge.
Connected graphs and graphical maps form a category denoted by $\textbf{U}$ . The composition, which we will rarely need to deal with explicitly, appears in definition 1·44 of [ Reference Hackney, Robertson and Yau33 ]. See the second paragraph of Remark 2·13 below for a description that will be used in the proof of Lemma 4·11. The purpose of condition (iii) is to avoid the situation where graph substitution would result in a nodeless loop (Example C2).
Definition 2·10 (Inert and active maps). Let H, G, G ^{′} be connected graphs.

(i) If $f\colon H \rightarrowtail G$ is an embedding, then there is an associated graphical map $H \to G$ whose action on arcs agrees with f and whose action on vertices is given by the composite
\begin{align*} V_H \xrightarrow{f} V_G \hookrightarrow \operatorname{Emb}\!(G).\end{align*}We often also call such a map inert and write $\textbf{U}_{\textrm{int}} \subset \textbf{U}$ for the wide subcategory consisting of the inert maps. (In [ Reference Hackney, Robertson and Yau33, Reference Hackney, Robertson and Yau34 ] this category was denoted by $\textbf{U}_{\textrm{emb}}$ .) 
(ii) A graphical map $\varphi \colon G \to G'$ is called active if $\varphi_0$ induces a bijection
between boundary sets. We write $\textbf{U}_{\textrm{act}} \subset \textbf{U}$ for the wide subcategory consisting of all of the active maps, and we use the notation .
Example 2·11. If G is a graph and $\eth(G)$ has cardinality n, then there are $n!$ different active maps , and these are determined solely by a choice of bijection $\eth({\unicode{x2606}}_n) \cong \eth(G)$ . Given any two such active maps , there is a unique automorphism z of ${\unicode{x2606}}_n$ with $\alpha' z = \alpha$ .
The following is theorem 2·15 of [ Reference Hackney, Robertson and Yau33 ].
Proposition 2·12. The pair $(\textbf{U}_{\textrm{act}},\textbf{U}_{\textrm{int}})$ is an orthogonal factorisation system on $\textbf{U}$ . That is, both subcategories contain all isomorphisms, every graphical map $\varphi \colon G \to G'$ factors as an active map followed by an embedding
and this factorisation is unique up to unique isomorphism.
Remark 2·13. If $\varphi \colon G \to G'$ is a graphical map and $\varphi_v \colon H_v \rightarrowtail G'$ represents $\varphi_1(v)$ , condition (ii) of Definition 2·9 provides a bijection between nb(v) and $\eth(H_v)\cong \eth(\varphi_1(v))$ . This is the data necessary to construct the graph substitution $G\{H_v\}$ (see, for instance, [ Reference Yau and Johnson61 , chapter 5], [ Reference Hackney, Robertson and Yau33 , section 1·2], and [ Reference Batanin and Berger7 , section 13]). Condition (iii) is included to guarantee that $G\{H_v\}$ is not a nodeless loop, while condition (i) ensures that there is an injection $V_{G\{H_v\}} \cong \amalg V_{H_v} \hookrightarrow V_{G^{\prime}}$ . This graph substitution comes equipped with an embedding $G\{H_v\} \rightarrowtail G'$ canonically factoring all of the embeddings $\varphi_v \colon H_v \rightarrowtail G'$ , and we can take $K= G\{H_v\}$ in the factorisation from Proposition 2·12.
Composition of graphical maps was formulated in [ Reference Hackney, Robertson and Yau33 , definition 1·44] by first defining precomposition with inert maps, and then utilising the above graph substitutions. This means that if $\psi \colon G' \to G''$ is another graphical map and $v\in V_G$ , then $(\psi \circ \varphi)_1(v) \in \operatorname{Emb}\!(G'')$ is represented by the right vertical embedding in the diagram below.
2·1. Lemmas concerning embeddings
We now give three lemmas governing the behavior of embeddings which will be useful several times in this paper. The first appears as lemma 1·22 of [ Reference Hackney, Robertson and Yau33 ], and the reader should note that it is not possible to have $i=j$ .
Lemma 2·14. Suppose $f \colon G \rightarrowtail G'$ is an embedding and $a_1\neq a_2$ are distinct arcs of G with $f(a_1) = f(a_2)$ . Then there are indices $i,j \in \{1,2\}$ so that $a_i, a_j^\dagger \in \eth(G)$ and $a_i^\dagger, a_j \in D_G$ .
The second, which appears as proposition 1·25 of [ Reference Hackney, Robertson and Yau33 ], says that embeddings are nearly determined by their boundary, up to some ambiguity about whether or not they contain a vertex.
Lemma 2·15. Let G be a connected graph, and $E_G \hookrightarrow \operatorname{Emb}\!(G)$ be the inclusion of edges into embeddings from Example 2·5. Then the composites
are injective.
The third is a generalisation of lemma 1·26 of [ Reference Hackney, Robertson and Yau33 ], which says that an embedding whose domain has empty boundary must be an isomorphism.
Lemma 2·16 If $f \colon H \rightarrowtail G$ is an embedding and $\eth(f) \subseteq \eth(G)$ , then f is an isomorphism. In particular, f must be an isomorphism if $\eth(H) = \varnothing$ .
Proof. If H is an edge, then $\eth(G)$ contains a pair $\{a,a^\dagger\} = \eth(f)$ . Since G is connected, it must be an edge as well.
Now suppose $V_H$ is inhabited. The argument from the proof of [ Reference Hackney, Robertson and Yau33 , lemma 1·26] shows that $V_H \to V_G$ is surjective, hence a bijection. Since f is étale, $D_H \to D_G$ is an isomorphism as well. If there were an element $a\in \eth(G) \setminus \eth(f)$ , then since G is not an edge we would have $a^\dagger \in D_G$ , hence $a^\dagger = f(d)$ for some (unique) $d\in D_H$ . But then $f(d^\dagger) = a \in \eth(G)$ , and only elements of $\eth(H)$ can map to elements of $\eth(G)$ , so $d^\dagger \in \eth(H)$ . This contradicts the assumption on a. It follows that $\eth(H) \hookrightarrow \eth(G)$ is also surjective, hence $A_H \to A_G$ is an isomorphism.
3. Unions of embeddings
In this section we consider a notion of unions of embeddings into a graph G. Despite the fact that $\operatorname{Emb}\!(G)$ is a poset, it is not particularly wellbehaved, and our notion has little to do with any joins that happen to exist. As an example, the embeddings h and k in Figure 4 do not have a least upper bound, since the domain of such would necessarily be disconnected. Even when a least upper bound exists, it may be too big to be a union (see the paragraph following proposition 2·2·8 of [ Reference Chu and Hackney11 ] for one example). See also Example 4·6.
Definition 3·1. Suppose that $h\colon H \rightarrowtail G$ and $k\colon K \rightarrowtail G$ are two embeddings. An embedding $\ell \colon L \rightarrowtail G$ is called a union of h and k if:

(1) $[\ell]$ is an upper bound for both [h] and [k] in the poset $\operatorname{Emb}\!(G)$ (that is, there is a factorisation $H \rightarrowtail L \rightarrowtail G$ of h and likewise for k); and

(2) $\ell (V_L) = h(V_H) \cup k(V_K)$ .
Unions in this sense are frequently not unique, as we can see in Figure 4.
Example 3·2. If $[h] \leq [k]$ in the poset $\operatorname{Emb}\!(G)$ , then k is a union of h and k.
We now show that if two embeddings have any intersection at all, then they have at least one union.
Proposition 3·3. Suppose $h\colon H \rightarrowtail G$ and $k\colon K \rightarrowtail G$ are embeddings. If either $h(A_H) \cap k(A_K)$ or $h(V_H) \cap k(V_K)$ is inhabited, then there exists at least one union $\ell \colon L \rightarrowtail G$ of h and k.
This proposition follows from Lemma 3·4 and Lemma 3·5 below. It is useful if one wishes to directly construct the activeinert factorisations from Proposition 2·12 using only the description of graphical maps from Definition 4·2. Existence of unions isn’t strictly necessary for the rest of this paper, and we recommend skipping ahead to Section 4 on first reading.
Suppose we have two embeddings $h\colon H \rightarrowtail G$ and $k\colon K \rightarrowtail G$ . We can form the pullback
in the category $\textbf{FinSet}^{\mathscr{I}}$ . The object J will still be a graph, though in general it does not need to be connected. In fact, since embeddings are not always monomorphisms, we may have that J is not connected even when $h=k$ . See Example 3·6 below.
Lemma 3·4. Suppose h and k are embeddings. Form an object $L\in \textbf{FinSet}^{\mathscr{I}}$ by taking a pullback followed by a pushout.
The object L is a graph.
Proof. Consider the map $\ell$ induced by the pushout property:
First note that the involution on $A_L$ does not have a fixed point: suppose, to the contrary, that $a\in A_L$ satisfies $a = a^\dagger$ . Then $\ell(a) = \ell(a^\dagger) = \ell(a)^\dagger$ , implying that $\ell(a)$ is a fixed point of $A_G$ , a contradiction.
It remains to show that $D_L \to A_L$ is injective. But this map fits into the diagram
Since $D_H \to D_G$ and $D_K\to D_G$ are injective and $D_J$ is $D_H \times_{D_G} D_K$ , the left vertical map is injective. The bottom map is injective because G is a graph, hence the top map is injective as well.
Lemma 3·5. Consider the diagram
from the previous lemma. If J is inhabited, then L is connected and f,g, and $\ell$ are embeddings.
Proof. Suppose J contains an arc or a vertex, classified by an étale map $E \to J$ from $E = {\updownarrow}$ or $E = {\unicode{x2606}}_n$ . If L splits as a coproduct $L = L_1 \amalg L_2$ , then since H and K are connected, f factors through exactly one $L_i$ and g factors through exactly one $L_j$ . Since both of these factor the composition $E \to J \to L$ and E is connected, we have $i=j$ and hence the other term is empty. Thus L is connected.
We next show that $\ell$ is an embedding. We have $V_J \cong V_H \times_{V_G} V_K \cong h(V_H) \cap k(V_K)$ , so $V_L = V_H \amalg_{V_J} V_K$ maps monomorphically to $h(V_H) \cup k(V_K) \subseteq V_G$ . Similarly, we have $D_L \cong h(D_H) \cup k(D_K)$ . As h and k are embeddings, the bottom square in the diagram
is a pullback, hence $\ell$ is an embedding. Using the pasting law for pullbacks and the fact that $\ell$ and h are embeddings, the top square is also a pullback, hence f is an embedding. A similar proof shows g is an embedding.
The construction from Lemma 3·4 produces a union from embeddings h and k which intersect, but it may not be the one you’d expect. For example, even when $h=k$ it may be that $[\ell]$ is strictly larger than [h].
Example 3·6. Suppose that $h=k \colon H=K={\unicode{x2606}}_2 \rightarrowtail G$ picks out the vertex in the loop with one vertex (see Figure 5). One can compute that the pullback J has
while $D_J = \{1,2\}$ and $V_J = \{ \bullet \}$ . It follows that $\ell \colon L \rightarrowtail G$ is an isomorphism, hence $[h] < [id_G] = [\ell]$ .
4. A new description of graphical maps
In this section we give an alternative definition of graphical map as a pair of functions with certain properties, which makes composition transparent. These new maps encode the same data as those from Definition 2·9, but encode it in a different way. After presenting the definition, we spend the remainder of the section showing how to go back and forth between the two descriptions, and finally give the equivalence in Theorem 4·15.
Definition 4·1. Two embeddings $h \colon H \rightarrowtail G$ and $k \colon K \rightarrowtail G$ are said to be vertex disjoint if $h(V_H) \cap k(V_K) \subseteq V_G$ is empty. In this case we also say the pair $[h],[k]\in \operatorname{Emb}\!(G)$ is vertex disjoint, and we note that this property does not depend on a choice of representatives.
Definition 4·2. Let G and G ^{′} be connected graphs. A new graph map $\varphi \colon G \to G'$ is a pair $(\varphi_0, \hat \varphi)$ consisting of an involutive function $\varphi_0 \colon A_G \to A_{G^{\prime}}$ and a function $\hat \varphi \colon \operatorname{Emb}\!(G) \to \operatorname{Emb}\!(G')$ , so that:

(i) the function $\hat \varphi$ sends edges to edges;

(ii) the function $\hat \varphi$ preserves unions: if $[\ell]$ is a union of [h] and [k] in $\operatorname{Emb}\!(G)$ , then $\hat \varphi[\ell]$ is a union of $\hat \varphi[h]$ and $\hat \varphi[k]$ in $\operatorname{Emb}\!(G')$ ;

(iii) the function $\hat \varphi$ takes vertex disjoint pairs to vertex disjoint pairs;

(iv) the diagram
commutes.
Composition of new graph maps is given by composition of the two constituent functions.
Remark 4·3. Condition (iv) is equivalent to insisting that, for each $[h] \in \operatorname{Emb}\!(G)$ , there is a (necessarily unique) bijection as displayed to the left:
By Example 3·2, condition (ii) in particular implies that $\hat \varphi \colon \operatorname{Emb}\!(G) \to \operatorname{Emb}\!(G')$ preserves the partial order. Notice that condition (i) is nearly the same as asking that $\hat \varphi$ preserves minimal elements: indeed, for almost all connected graphs G, the set of edges coincides with the set of minimal elements of $\operatorname{Emb}\!(G)$ . The lone exception is the case when $G = {\unicode{x2606}}_0$ is an isolated vertex, which has no edges – $\operatorname{Emb}\!({\unicode{x2606}}_0)$ has a unique element, and Lemma 2·16 implies that this element maps to the maximal element of $\operatorname{Emb}\!(G')$ .
Our aim in this section is to show that new graph maps are equivalent to graphical maps from Definition 2·9. This work culminates in Theorem 4·15, which shows that the following two constructions are inverses.
Construction 4·4. Given a new graph map $\varphi \colon G \to G'$ , we obtain the composite
We write $\mathfrak{O}$ for the assignment that takes a new graph map $\varphi = (\varphi_0, \hat \varphi)$ and produces the pair $\mathfrak{O} \varphi := (\varphi_0, \varphi_1)$ .
Construction 4·5. Suppose that $\varphi = (\varphi_0, \varphi_1) \colon G \to G'$ is a graphical map. Define a function
by declaring that $\hat \varphi [f] := [g]$ , where g sits inside the activeinert factorisation of $\varphi \circ f$ from Proposition 2·12:
As we are working with an orthogonal factorisation system, the element [g] does not depend on the choice of representative for [f] or the chosen factorisation. We write $\mathfrak{N}$ for the assignment that takes a graphical map $\varphi = (\varphi_0, \varphi_1)$ and produces the pair $\mathfrak{N} \varphi := (\varphi_0, \hat \varphi)$ .
With this construction in hand, we give an example to show that graphical maps do not preserve least upper bounds of embeddings.
Example 4·6. Consider the graphs G (on the left) and G ^{′} (on the right) of Figure 6, and let $\varphi \colon G \to G'$ be the embedding. Now $\iota_v \colon {\unicode{x2606}}_v \rightarrowtail G$ and $\iota_w \colon {\unicode{x2606}}_w \rightarrowtail G$ have a least upper bound, namely $id_G$ . But
The element $\hat\varphi [id_G] = [\varphi]$ is not the least upper bound for the two star inclusions in G ^{′}. Indeed, by snipping the edge $e_1$ in G ^{′} we obtain another embedding which factors the two star inclusions, but which does not factor $\varphi$ . (As in Figure 4, the least upper bound does not exist.)
4·1. From new graph maps to graphical maps
Our aim in this section is to show that $\mathfrak{O}$ takes new graph maps to graphical maps.
Theorem 4·7. If $\varphi = (\varphi_0, \hat \varphi)$ is a new graph map, then $\mathfrak{O}\varphi = (\varphi_0,\varphi_1)$ is a graphical map in the sense of Definition 2·9.
The following statement involves the vertex sum function $\varsigma \colon \operatorname{Emb}\!(G) \to \mathbb{N}V_G$ from Definition 2·8.
Lemma 4·8. Given a diagram of embeddings,
we have
in $\mathbb{N}V_G$ . Equality holds if and only if $\ell$ is a union of h and k. If h and k are vertex disjoint and $\ell$ is a union of h and k, then $\varsigma[\ell] = \varsigma[h] + \varsigma[k]$ .
Proof. We compute, using that embeddings are injective on vertices:
The last inequality is an equality if and only if $f(V_H) \cup g(V_K) = V_L$ , that is, if and only if $\ell$ is a union of h and k. If h and k are vertex disjoint then $f(V_H) \cap g(V_K)$ is empty, so the final statement holds.
Lemma 4·9. Given an embedding $\ell \colon L \rightarrowtail G$ where L has an internal edge, there exists a vertex $v \in L$ and an embedding $g \colon K \rightarrowtail L$ with $g(V_K) = V_L \setminus \{v\}$ . In particular, $\ell$ is a union of $h = \ell \iota_v \colon {\unicode{x2606}}_v \rightarrowtail G$ and $k = \ell g \colon K \rightarrowtail G$ .
Proof. If L has only a single vertex v, then since L contains an edge, we can form
where g classifies some arc of L. The conclusion follows.
Now suppose that L has two or more vertices. Consider the classical graph $\textrm{core}(L)$ obtained by deleting the $\dagger$ orbit of $\eth(L)$ ([ Reference Hackney, Robertson and Yau33 , definition 1·9]). As $\textrm{core}(L)$ is connected and has more than one vertex, there exists a vertex v so that deleting v and all edges incident to it from $\textrm{core}(L)$ will still give a connected graph. Setting
we have $S^\dagger \subsetneq nb(v)$ ; also notice that $S \cap S^\dagger = nb(v) \cap nb(v)^\dagger$ . Define a graph K by
We compute the boundary of K as
and use this to show that $\eth(K)^\dagger \subseteq D_K$ . If $a\in \eth(L)\setminus S$ , then $a^\dagger \notin nb(v)$ , hence $a^\dagger \in D_K$ . If $a \in nb(v) \setminus S^\dagger$ , then we can’t have $a^\dagger \in \eth(L)$ for then $a^\dagger \in nb(v)^\dagger \cap \eth(L) \subseteq S$ , and we also can’t have $a^\dagger \in nb(v)$ since then $a \in nb(v)\cap nb(v)^\dagger \subseteq S^\dagger$ ; it follows that $a^\dagger\in D_K$ . We have thus shown that K does not contain any edge components. As $\textrm{core}(K) = \textrm{core}(L)  v$ is connected, we conclude that K is connected. The inclusions of subsets give our desired embedding $g\colon K \rightarrowtail L$ .
Proposition 4·10. Given an embedding $\ell \colon L \rightarrowtail G$ and a new graph map $\varphi \colon G \to G'$ , we have
Proof. If L is an edge, so is $\hat \varphi[\ell]$ by Definition 4·2(i), hence $\varsigma \hat \varphi[\ell] = 0$ as desired. If L is a star, then the formula is immediate. We have thus established the formula whenever L does not contain an internal edge.
For the general result, we induct on the number of vertices of L, which we assume to have an internal edge (in other words, we are investigating what happens when L is not an edge and not a star). Applying Lemma 4·9, we obtain a diagram
with $\ell$ a union of h and k, and h, k vertex disjoint. Applying $\hat \varphi$ , we obtain embeddings
with $\hat \ell$ a union of $\hat h$ and $\hat k$ (by Definition 4·2(ii)) and $\hat h, \hat k$ vertex disjoint (by Definition 4·2(iii)), hence
by Lemma 4·8. As K has one less vertex than L, by the induction hypothesis we have the first equality below:
The result follows by combining the previous two displays.
Proof of Theorem 4·7. Suppose $(\varphi_0, \hat \varphi) \colon G \to G'$ is a new graph map. We verify each of the three conditions from Definition 2·9.

(i) Applying Proposition 4·10 to the identity embedding $id_G \colon G \rightarrowtail G$ , we find
\begin{align*} \sum_{v\in V_G} \varsigma(\varphi_1(v)) = \sum_{v\in V_G} \varsigma \hat \varphi[\iota_v] = \varsigma \hat \varphi[id_G] \leq \varsigma [id_{G^{\prime}}] = \sum_{w\in V_{G^{\prime}}} w. \end{align*} 
(ii) This is the special case of the interpretation of Definition 4·2(iv) from Remark 4·3 where we take $h = \iota_v$ :

(iii) Suppose the boundary of G is empty, and consider an embedding $h \colon H \rightarrowtail G'$ which represents $\hat \varphi [id_G]$ . By Definition 4·2(iv) we have that $\eth(H)$ is empty, hence h is an isomorphism by Lemma 2·16. By Proposition 4·10 we have
\begin{align*} \sum_{v\in V_G} \varsigma \varphi_1(v) = \varsigma \hat \varphi [id_G] = \varsigma [h] = \varsigma [id_{G^{\prime}}] = \sum_{w\in V_{G^{\prime}}} w.\end{align*}Since $\eth(G')$ is empty, G ^{′} contains at least one vertex, implying that there is at least one $v\in V_G$ with $\varsigma \varphi_1(v) \neq 0$ .
4·2. From graphical maps to new graph maps
Recall Construction 4·5 which takes a graphical map $\varphi = (\varphi_0, \varphi_1) \colon G \to G'$ to the pair $\mathfrak{N} \varphi = (\varphi_0, \hat \varphi)$ . This function $\hat \varphi$ is defined so that $\hat \varphi [f] = [g]$ where g sits inside an activeinert factorisation of $\varphi f$ :
We will show that $\mathfrak{N} \varphi$ is a new graph map in Theorem 4·14, but first we first establish functoriality of the construction. After establishing that $\mathfrak{N} \varphi$ is a new graph map, we will show in Theorem 4·15 that $\mathfrak{N}$ and $\mathfrak{O}$ are inverses.
Lemma 4·11. Given a composable pair of graphical maps
we have that $\hat \psi \circ \hat \varphi = \widehat{\psi \circ \varphi}$ . Further, $\widehat {id_G} = id_{\operatorname{Emb}\!(G)}$ .
Proof. The second statement is clear since we have an activeinert orthogonal factorisation system on $\textbf{U}$ . The first statement also follows from the factorisation system: suppose $\ell \colon L \rightarrowtail G$ is an embedding.
First factor $\varphi \ell$ as $f\circ p$ , and then factor $\psi f$ as $h\circ q$ , and notice that $h \circ (qp)$ is an activeinert factorisation of $\psi \varphi \ell$ . It follows that $\hat \varphi [\ell] = [f]$ , $\hat \psi [f] = [h]$ , and $\widehat {\psi \circ \varphi} [\ell] = [h]$ .
Our next goal is to show that $\mathfrak{N}$ from Construction 4·5 takes graphical maps to new graph maps. For this purpose, consider a graphical map $\varphi \colon G \to G'$ and embeddings $h,k,\ell$ into G with $[h] \leq [\ell]$ and $[k]\leq [\ell]$ . We form an activeinert factorisation $\hat \ell \circ \alpha$ of $\varphi\ell$ , and then activeinert factorisations of $\alpha f$ and $\alpha g$ as displayed belowright.
In particular, we have that $\hat \varphi [\ell] = [\hat \ell]$ is an upper bound for both $\hat \varphi [h]$ and $\hat \varphi [k]$ .
Lemma 4·12. If $\ell$ is a union of h and k, then $\hat \varphi [\ell]$ is a union of $\hat \varphi [h]$ and $\hat \varphi [k]$ .
Proof. Suppose $v\in V_{L^{\prime}}$ . Our goal is to show that $\hat \ell (v)$ is an element of $\hat h(V'_{\!\!H}) \cup \hat k(V_{K^{\prime}})$ . Since $\alpha \colon L\to L'$ is active, by [ Reference Hackney, Robertson and Yau33 , proposition 2·3] there is a unique vertex $w\in V_L$ so that $\alpha_w \colon M_w \rightarrowtail L'$ (representing $\alpha_1(w)$ ) has v in its image. As $V_H \amalg V_K \to V_L$ is surjective, there is a vertex w ^{′} of H or K mapping to w; without loss of generality we assume $w^{\prime}\in V_H$ . Considering the square
of (4·2), we get that the triangle
commutes by the descriptions of composing with embeddings from [ Reference Hackney, Robertson and Yau33 , definition 1·34]. Hence v is in the image of $H' \rightarrowtail L'$ , so $\hat \ell (v)$ is an element of $\hat h(V_{H^{\prime}})$ .
A similar line of argument gives the following.
Lemma 4·13. If h and k are vertex disjoint, then so are $\hat h$ and $\hat k$ .
Proof. We prove the contrapositive. Suppose $\hat h(V_{H^{\prime}}) \cap \hat k(V_{K^{\prime}})$ is inhabited and let $\hat h(v^{\prime}) = \hat k(w^{\prime})$ be a vertex in this intersection. Since $\beta \colon H \to H'$ is active, by [ Reference Hackney, Robertson and Yau33 , proposition 2·3], there exists a unique vertex $v \in V_H$ so that $\beta_{v} \colon M_{v} \rightarrowtail H'$ (with $[\beta_v] = \beta_1(v)$ ) has v ^{′} in its image. Likewise, there exists a unique vertex $w\in V_K$ so that $\gamma_w \colon N_w \rightarrowtail K'$ has w ^{′} in its image. We have $\alpha_1(f(v)) = [f' \circ \beta_v]$ and $\alpha_1(g(w)) = [g' \circ \gamma_w]$ by remark 1·45 of [ Reference Hackney, Robertson and Yau33 ].
Now f ^{′}(v ^{′}) and g ^{′}(w ^{′}) are in the images of both $\alpha_1(f(v))$ and $\alpha_1(g(w))$ ; since $\alpha$ is a graphical map, Definition 2·9 (i) gives $f(v) = g(w)$ . Hence $h(v) = \ell f(v) = \ell g(w) = k(w)$ , so h and k are not vertex disjoint.
Theorem 4·14. If $\varphi = (\varphi_0, \varphi_1) \colon G \to G'$ is a graphical map, then $\mathfrak{N}\varphi = (\varphi_0, \hat\varphi)$ is a new graph map, where $\hat \varphi$ is as defined in Construction 4·5.
Proof. We verify the four conditions from Definition 4·2. Condition (i) follows from lemma 2·2 of [ Reference Hackney, Robertson and Yau33 ]. The preceding two lemmas give (ii) and (iii), respectively. The interpretation of (iv) given in Remark 4·3 is lemma 1·47 of [ Reference Hackney, Robertson and Yau33 ].
Theorem 4·15. Suppose G and G^{′} are connected graphs. The assignments $\mathfrak{N}$ and $\mathfrak{O}$ are inverse bijections between new graph maps from G to G^{′} and graphical maps from G to G^{′}. Moreover, these assignments identify the category of connected graphs and new graph maps with the category $\textbf{U}$ .
Proof. The conclusion will follow once we establish the bijection, as Lemma 4·11 implies that $\mathfrak{N}$ is an identityonobjects functor from $\textbf{U}$ to the category of new graph maps.
It is nearly immediate that $\mathfrak{O}\mathfrak{N}$ is an identity on $\textbf{U}(G,G')$ , by the same reasoning as in the first bullet point of remark 1·45 of [ Reference Hackney, Robertson and Yau33 ]. Namely, if $\varphi_v \colon H_v \rightarrowtail G'$ represents $\varphi_1(v)$ , then we have a factorisation in $\textbf{U}$
so $\mathfrak{O}\mathfrak{N}(\varphi)_1(v) = \hat \varphi([\iota_v]) = [\varphi_v] = \varphi_1(v)$ .
Now suppose that $\varphi = (\varphi_0, \hat \varphi) \colon G \to G'$ is a new graph map, $\mathfrak{O}(\varphi) = (\varphi_0, \varphi_1)$ is the associated graphical map, and $\mathfrak{N}\mathfrak{O}(\varphi) = (\varphi_0, \tilde \varphi)$ , where $\tilde \varphi$ is a function from $\operatorname{Emb}\!(G)$ to $\operatorname{Emb}\!(G')$ . Let $f \colon H \rightarrowtail G$ be an arbitrary embedding; we wish to show that $\tilde \varphi [f] = \hat \varphi [f]$ . Notice that we already have a special case: if the domain of f is a star, then $[f] = [\iota_v]$ for some $v \in V_G$ , and we know $\tilde \varphi [\iota_v] := \mathfrak{O}(\varphi)_1(v) := \hat \varphi [\iota_v]$ . Let us turn the general case.
Since $\varphi$ and $\mathfrak{N}\mathfrak{O}(\varphi)$ are new graph maps, by Remark 4·3, we have the commutative diagram
and we conclude that $\eth(\hat\varphi [f]) = \eth(\tilde\varphi [f])$ . This is nearly enough to conclude that $\hat\varphi [f]$ and $\tilde\varphi [f]$ are equal — the only thing that could go wrong is if one of them is an edge while the other is not. Applying Proposition 4·10 twice we have
where the middle equality comes from the previously established fact that $\hat \varphi [f\iota_v] = \tilde \varphi [f\iota_v]$ . Since an embedding h is an edge if and only if $\varsigma[h] = 0 $ , we conclude that $\hat \varphi [f]$ is an edge if and only if $\tilde \varphi [f]$ is an edge. By Lemma 2·15, we conclude that $\hat \varphi[f] = \tilde \varphi[f]$ . Thus $\varphi = \mathfrak{N}\mathfrak{O}(\varphi)$ .
Remark 4·16. We can describe the activeinert factorisation system on $\textbf{U}$ from Proposition 2·12 in terms of new graph maps. The active maps are precisely those maps so that $\hat \varphi([id_G]) = [id_{G^{\prime}}] \in \operatorname{Emb}\!(G')$ , that is, maps where $\hat \varphi$ preserves the top element. Likewise, the inert maps are those where $\hat \varphi$ admits a dashed arrow making the following diagram commute:
The factorisation of a map $\varphi \colon G \to G'$ can be obtained by taking $h\colon H \rightarrowtail G'$ to be an embedding with $\hat \varphi [id_G] = [h]$ , and explicitly defining an active map so that $\varphi = h \circ \psi$ . This requires a bit of delicacy, since the embedding h might not be injective on arcs.
5. Directed graphs and the wheeled properadic graphical category
We now turn our attention to the wheeled properadic graphical category. This category controls wheeled properads [ Reference Markl, Merkulov and Shadrin47 ] (which extend the notion of properad [ Reference Vallette57 ] to encode traces) in the sense that there is a relevant nerve theorem. Wheeled properads can be regarded as modular operads with additional directional structure, which was observed in [ Reference Raynor51 , example 1·29].
In this section we show how the wheeled properadic graphical category can be described in a similar manner to that of $\textbf{U}$ above. One drawback of the existing definition is that it relies on the notion of wheeled properad and maps between wheeled properads. Our new definition does not have this requirement, though of course we will need to consider wheeled properads to prove the equivalence. As one caveat: in this paper we consider only the version of the wheeled properadic graphical category from [ Reference Hackney, Robertson and Yau31 ], rather than the one from [ Reference Hackney, Robertson and Yau29 ]. The latter can be obtained from the former by inverting a single map. For brevity, we do not include the original definition of the wheeled properadic graphical category here, except indirectly in the proof of Theorem 5·13.
5·1. Directed graphs
The following encoding of a directed graph with loose ends was introduced as definition 1·1·1 of [ Reference Kock43 ].
Definition 5·1 (Directed graphs). Let $\mathscr{G}$ denote the category
A directed graph G is a functor $\mathscr{G} \to \textbf{FinSet}$ which sends the two maps with codomain $\texttt{e}$ to monomorphisms, that is, a directed graph is a diagram of finite sets of the form
Each vertex $v\in V_G$ has a set of inputs $\operatorname{in}\!(v) \subseteq I_G$ which is the preimage of v under $I_G \to V_G$ , and likewise has a set of outputs $\operatorname{out}\!(v) \subseteq O_G$ . The set of inputs of G is defined to be $\operatorname{in}\!(G) := E_G \setminus O_G$ and the set of outputs of G is $\operatorname{out}\!(G) := E_G \setminus I_G$ .
Directed graphs in this sense turn out to be equivalent to others in the literature (this is mostly due to Batanin–Berger; see Proposition C12 and [ Reference Chu and Hackney11 , remark 2·0·2]), with the caveat that it is not possible to encode graphs with components that are nodeless loops. See Definition C7 below for a ‘hack’ that fixes this problem, which is similar to that appearing in definition 4·1 of [ Reference Hackney, Robertson and Yau33 ].
A key upside to Kock’s definition compared to other formalisms is that it is admits a straightforward notion of étale maps [ Reference Kock43 , 1·1·7] as certain natural transformations, similar to that of Definition 2·3.
Definition 5·2. An étale map $f\colon G \to H$ between directed graphs is a commutative diagram of sets
so that the middle two squares are pullbacks. An embedding is an étale map between connected directed graphs which is injective on vertices.
These embeddings of directed graphs correspond to those for undirected graphs in Definition 2·3. They are different from the ‘open subgraph inclusions’ of [ Reference Kock43 ], since the edge map $f_E$ is not required to be injective and the domain and codomain must be connected.
We now relate the definitions of directed and undirected graphs.
Definition 5·3 (Orientation presheaf). The orientation presheaf, denoted $\mathfrak{o}$ , is the $\textbf{U}$ presheaf so that
consists of those elements $x = (x_a)$ satisfying $x_a = x_{a^\dagger}$ for all $a\in A_G$ . If $\varphi \colon H \to G$ is a graphical map and $x\in \mathfrak{o}_G$ , then $\varphi^*(x) \in \mathfrak{o}_H$ is the element with
for all $a\in A_H$ .
Pairs (G, x) consisting of a connected undirected graph $G \in \textbf{U}$ and an element $x\in \mathfrak{o}_G$ (equivalently a map of presheaves $x \colon G\to \mathfrak{o}$ ) coincide with the connected directed graphs of Definition 5·1. This is a special case of [ Reference Kock43 , 1·1·13], and in order to fix conventions the following construction exhibits one direction of the correspondence; the reverse direction sends a directed graph to an undirected graph with $D = I \amalg O$ and $A = E\amalg E$ . See also Proposition 5·6.
Construction 5·4. If G is a graph and $x\in \mathfrak{o}_G$ , then we have a splitting of $A_G$ as
and likewise for $D_G = D_G^+ \amalg D_G^$ . We thus obtain a diagram of sets
There are isomorphisms $A_G^ \xrightarrow{\cong} E_G \xleftarrow{\cong} A_G^+$ , each sending an arc a to the edge $[a,a^\dagger]$ it spans, so we conclude that (G, x) determines a diagram
as in Definition 5·1.
Remark 5·5. When working with this presentation, we have, for a vertex $v \in V_G$
We prefer to think about $\operatorname{in}\!(G)$ and $\operatorname{out}\!(G)$ as subsets of $A_G^+$ and $A_G^$ , rather than as subsets of $E_G$ :
Notice the discrepancy in signs. This is because arcs in nb(v) point towards the vertex while arcs in $\eth(G)$ point away from the graph. We think about input edges as coming in from above and output edges as coming in from below, so arcs of $\operatorname{in}\!(v)$ point down while arcs of $\operatorname{in}\!(G)$ point up.
Every étale map of directed graphs induces an étale map of the corresponding undirected graphs (this was also mentioned in [ Reference Kock43 , 1·1·13]). This restricts to embeddings, yielding the following equivalence.
Proposition 5·6. There is an equivalence of categories between the category of connected directed graphs and embeddings (in the sense of Definitions 5·1 and 5·2) and the category $\textbf{U}_{\textrm{int}} \times_{\widehat{\textbf{U}}} \widehat{\textbf{U}}_{/\mathfrak{o}} =: \textbf{U}^{\textrm{int}}_{/\mathfrak{o}}$ .
This proposition can’t be promoted as is to the category of graphs and étale maps, since étale maps between connected graphs which are not embeddings do not yield maps in $\textbf{U}$ .
Definition 5·7 (Inputs and outputs of embeddings). Suppose G is a directed graph. Then there are functions $\operatorname{in} \colon \operatorname{Emb}\!(G) \to \wp(A_G^+) \cong \wp(E_G)$ and $\operatorname{out} \colon \operatorname{Emb}\!(G) \to \wp(A_G^) \cong \wp(E_G)$ defined by
Alternatively, we can consider $f\colon (H,f^*x) \rightarrowtail (G,x)$ as an embedding between directed graphs and we have
5·2. Wheeled properads and the oriented graphical category
We now introduce the oriented graphical category $\textbf{U}_{/\mathfrak{o}}$ , which we will show is equivalent to the wheeled properadic graphical category. Thus, by the nerve theorem (theorem 10·33 and proposition 10·35 of [ Reference Hackney, Robertson and Yau29 ]), this category governs wheeled properads. More precisely, the category of (setbased) wheeled properads is equivalent to the category of Segal presheaves on $\textbf{U}_{/\mathfrak{o}}$ (in the sense of Definition 6·13).
Definition 5·8 (Oriented graphical category). The oriented graphical category is the category $\textbf{U}_{/\mathfrak{o}} := \textbf{U} \times_{\widehat{\textbf{U}}} \widehat{\textbf{U}}_{/\mathfrak{o}}$ whose objects are pairs consisting of an undirected graph G and map of presheaves $x \colon G \to \mathfrak{o}$ (equivalently an element $x \in \mathfrak{o}_G$ ).
Remark 5·9. By Definition 5·3, if G is a graph, then an element $x\in \mathfrak{o}_G$ is the same thing as an involutive map from $A_G$ to $\{ +1, 1 \}$ . A map $(G,x) \to (G',x')$ in $\textbf{U}_{/\mathfrak{o}}$ is a just a map $G\to G'$ in $\textbf{U}$ so that the triangle
commutes.
The following uses the two functions appearing in Definition 5·7 and the correspondence from Construction 5·4 to reinterpret Definition 4·2 in the oriented setting.
Proposition 5·10. A map $\varphi \colon (G,x) \to (G',x')$ in the oriented graphical category $\textbf{U}_{/\mathfrak{o}}$ may be described as a pair $(\bar\varphi_0, \hat\varphi)$ consisting of functions $\bar\varphi_0 \colon E_G \to E'_{\!\!G}$ and $\hat \varphi \colon \operatorname{Emb}\!(G) \to \operatorname{Emb}\!(G')$ so that (i), (ii) and (iii) from Definition 4·2 hold and, additionally,

(iv’) the diagram
commutes.
Definition 5·11. The Moerdijk–Weiss dendroidal category, denoted $\boldsymbol{\Omega}$ , is the full subcategory of $\textbf{U}_{/\mathfrak{o}}$ consisting of those graphs G so that $\operatorname{out}\!(v)$ is a singleton set for every $v\in V_G$ . Alternatively, it consists of those graphs G so that $\operatorname{out} \colon \operatorname{Emb}\!(G) \to \mathbb{N}E_G$ factors through $E_G \hookrightarrow \mathbb{N}E_G$ .
Remark 5·12. The equivalence of the usual definition of $\boldsymbol{\Omega}$ from [ Reference Moerdijk and Weiss48 , section 3] with Definition 5·11 follows, for instance, from Corollary B21 and [ Reference Hackney, Robertson and Yau29 , remark 6·55]. But this may also be seen more directly by comparing Proposition 5·10 with the presentation of the dendroidal category as the dendroidally ordered broad posets from [ Reference Weiss60 , section 2·1·4]. Indeed, it turns out that for each $G\in \boldsymbol{\Omega}$ , we have $\operatorname{out} \times \operatorname{in} \colon \operatorname{Emb}\!(G) \hookrightarrow E_G \times \mathbb{N}E_G$ is an inclusion, giving a heterogeneous relation between $E_G$ and $\mathbb{N}E_G$ . This relation is the relevant broad poset structure on $E_G$ , and condition (iv') is merely asserting that $\bar\varphi_0 \colon E_G\to E'_{\!\!G}$ is a map of broad posets.
Our aim in this subsection is to prove the following theorem.
Theorem 5·13. The oriented graphical category $\textbf{U}_{/\mathfrak{o}}$ is equivalent to the wheeled properadic graphical category which excludes the nodeless loop.
This version of the wheeled properadic graphical category was called $\mathcal{B}$ in [ Reference Hackney, Robertson and Yau31 , section 2]. The nodeless loop in question shows up as Example C9, but does not appear among the graphs defined in Definition 5·1. An extension of the preceding theorem will be given in Theorem C14.
Remark 5·14 (A change to objects). The objects of the wheeled properadic category from [ Reference Hackney, Robertson and Yau29, Reference Hackney, Robertson and Yau31 ] are (isomorphism classes of) graphs equipped with a listing, that is, a total ordering on each of the subsets $\operatorname{in}\!(v)$ , $\operatorname{out}\!(v)$ (both as v varies), $\operatorname{in}\!(G)$ , and $\operatorname{out}\!(G)$ of $E_G$ . Every graph admits such a listing, and if we consider two choices of listing for the same graph, then there is a canonical isomorphism between them in the wheeled properadic graphical category. In what follows, we consider only the equivalent category whose objects coincide with the objects of $\textbf{U}_{/\mathfrak{o}}$ (resp., with the objects of $\widetilde{\textbf{U}}_{/\mathfrak{o}}$ in Appendix C·1). That is, we are considering all directed graphs rather than just isomorphism classes of such, and we are not imposing a listing. The equivalence can be realised by choosing, for each directed graph $G \in \textbf{U}_{/\mathfrak{o}}$ , some fixed listing and then passing to the associated isomorphism class. The term ‘wheeled properadic graphical category’ will refer to this equivalent category henceforth.
We will need a few preliminary results before tackling the proof of Theorem 5·13. The main one is the following, which is the directed analogue of the construction from section 2·2 of [ Reference Hackney, Robertson and Yau34 ]. As this is just a minor variation of the undirected case, we merely provide a sketch and refer the reader to [ Reference Hackney, Robertson and Yau34 ] for details.
Proposition 5·15. There is a functor $F \colon \textbf{U}_{/\mathfrak{o}} \to \textbf{WPpd}$ from the oriented graphical category to the category of (colored) wheeled properads [ Reference Hackney, Robertson and Yau29 , definition 9·1] which takes a directed graph to the wheeled properad it freely generates.
Proof sketch. On objects the value of F is given in the same was as in [ Reference Hackney, Robertson and Yau29 , section 9·2·1]. Briefly, if G is a directed graph then F(G) is the free $E_G$ colored wheeled properad which has one generator for each vertex $v\in V_G$ . For each vertex v, choose orderings $\operatorname{in}\!(v) = e_1, \dots, e_m$ and $\operatorname{out}\!(v) = e'_{\!\!1}, \dots, e'_{\!\!n}$ for the inputs and output edges of v, and regard v as a generator in the set
If H is a graph equipped with a total ordering of the sets $\operatorname{in}\!(H) = \{i_1, \dots, i_k\}$ and $\operatorname{out}\!(H) = \{o_1, \dots, o_j\}$ , then an étale map $\mu \colon H \to G$ determines an element in
Given a graphical map $\varphi \colon G \to G'$ , using that F(G) is a free wheeled properad we can specify a map of wheeled properads (see [ Reference Hackney, Robertson and Yau29 , lemma 9·23])
by defining the color map $E_G \to E_{G^{\prime}}$ to be $\bar \varphi_0$ , and value on the generator (5·1),
is given by the embedding $\varphi_1(v)$ . Composition is defined using graph substitution, and associativity of composition is proved as in [ Reference Hackney, Robertson and Yau34 , proposition 2·25].
In Lemma 5·20 we will compare embeddings with the notion of ‘subgraph’ from Definition 9·50 of [ Reference Hackney, Robertson and Yau29 ], and to do so it will be convenient to have the following notion of ‘graph complement’ at hand. We show that such graph complements always exist in Lemma 5·19 (for completeness, we also prove they are unique in Lemma 5·18). See Figure 7 for a pair of examples.
Definition 5·16. Let $f \colon G \rightarrowtail H$ be an embedding of (undirected or directed) connected graphs (in $\textbf{U}_{\textrm{int}}$ or $\textbf{U}^{\textrm{int}}_{/\mathfrak{o}}$ ). A graph complement of f consists of a triple $(K,v^G,\alpha)$ where K is another connected graph, $v^G \in V_K$ is a distinguished vertex, and is an active map. This data must satisfy $\alpha_1(v^G) = [f]$ and, for all other vertices $v \in V_K\setminus \{v^G\}$ , there is a vertex $w\in V_H$ with $\alpha_1(v) = [\iota_w]$ .
It is implicit that if f is an embedding of directed graphs, then K should be directed and $\alpha$ should be an active map in $\textbf{U}_{/\mathfrak{o}}$ . In either case, since $\alpha$ is active, every vertex w of H is either in $f(V_G)$ or comes from a unique $v\in V_K \setminus \{v^G\}$ where $\alpha_1(v) = [\iota_w]$ . Thus $\alpha$ induces a bijection $V_K \setminus \{v^G\} \cong V_H \setminus f(V_G)$ .
Remark 5·17 (Relation to étale complement). By deleting the vertex $v^G$ from K (retaining all of the arcs, but removing $nb(v^G)$ from $D_K$ ), we obtain another graph K^{′} which comes equipped with an étale map $K' \to H$ . The graph K^{′} will frequently be disconnected. If the embedding $f \colon G \rightarrowtail H$ is injective on arcs, then we could ask what the relationship is between K ^{′} and the étale complement of f from [ Reference Kock43 , 1·6·21·6·4]. These turn out to coincide if either G contains at least one vertex or if f classifies an internal edge of H. If f classifies an incoming or outgoing edge of H, then $K' \cong {{\updownarrow}} \amalg H$ , whereas the étale complement is H.
The following lemma is not strictly necessary for the developments below, but is included for completeness.
Lemma 5·18. Graph complements are unique up to isomorphism.
Proof. Suppose $(K, v^G, \alpha)$ and $(J, u^G, \beta)$ are two graph complements of an embedding $f \colon G \rightarrowtail H$ . The isomorphisms
extend to a bijection $z_V \colon V_K \to V_J$ sending $v^G$ to $u^G$ .
By Definition 2·9(ii) we have
and these isomorphisms $nb(v) \to nb(z_V(v))$ assemble into a bijection
over $z_V \colon V_K \cong V_J$ . Finally, since $\alpha$ and $\beta$ are active we have isomorphisms
which gives
If $a \in nb(v)$ (resp. $a \in \eth(K)$ ) then $z_A(a) = z_D(a) \in nb(z_V(v))$ (resp. $z_A(a) \in \eth(J)$ ) is the unique element satisfying $\alpha_0(a) = \beta_0(z_A(a))$ .
The only thing to check is that $z_A\colon A_K \to A_J$ preserves the involution.
If G is not an edge, then $\alpha_0$ and $\beta_0$ are injective by lemma 2·9 of [ Reference Hackney, Robertson and Yau33 ], so $z_A(a)$ is defined by the equation $\alpha_0(a) = \beta_0(z_A(a))$ with no stipulation about where a or $z_A(a)$ live. Using this equation for both a and $a^\dagger$ , we have
hence $z_A(a^\dagger) = z_A(a)^\dagger$ .
If G is an edge, then again using lemma 2·9 of [ Reference Hackney, Robertson and Yau33 ] we have $\alpha_0$ is injective when restricted to $A_K \setminus nb(v^G)$ . Writing $nb(v^G) = \{d_0,d_1\}$ , the only identifications that $\alpha_0$ makes are $\alpha_0(d_0) = \alpha_0(d_1^\dagger)$ and $\alpha_0(d_1) = \alpha_0(d_0^\dagger)$ . A similar situation occurs with $\beta_0$ and $nb(u^G)$ . Thus, as in the previous paragraph, $z_A(a^\dagger) = z_A(a)^\dagger$ so long as $a\notin \{d_0,d_1,d_0^\dagger,d_1^\dagger\}$ . But now $nb(u^G) = \{ z_A(d_0), z_A(d_1) \}$ , and the only way for our bijection to not be involutive would be if $z_A(d_0^\dagger) = z_A(d_1)^\dagger$ and $z_A(d_1^\dagger) = z_A(d_0)^\dagger$ . This cannot happen, as
Lemma 5·19. If $f \colon G \rightarrowtail H$ is an embedding between connected (directed or undirected) graphs, then a graph complement of f exists.
This lemma is closely related to the notion of a ‘shrinkable pasting scheme,’ which includes the pasting scheme for connected directed graphs by proposition 3·3(i) of [ Reference Hackney, Robertson and Yau30 ]. One can iteratively shrink away (in the graph H) all edges coming from internal edges of G and identify their end vertices, and this will yield the graph complement (so long as G is not an edge). See the discussion following remark 3·8 of [ Reference Hackney, Robertson and Yau30 ] for more on this viewpoint; in the following proof we simply give formulas for the resulting graph K.
Proof. We can ignore directionality when constructing the graph complement: if the embedding f is a map in $\textbf{U}_{/\mathfrak{o}}$ , then the produced undirected graph K becomes directed via the composite map of presheaves $K \to H \to \mathfrak{o}$ .
We first address the special case when G does not have a vertex. If G is an edge hitting arcs $a_0, a_1 \in A_H$ (with $a_0^\dagger = a_1$ ), we set $V_K = V_H \amalg \{ v^G \}$ , $nb(v^G)$ consists of two new elements $d_0$ and $d_1$ , $D_K = D_H \amalg nb(v^G)$ , $A_K = A_H \amalg \{d_0,d_1\}$ , and all structure the same except that $a_i^\dagger$ is redefined to be $d_i$ for $i=0,1$ . The definition of $\alpha \colon K \to H$ is forced by the requirements in Definition 5·16.
Now suppose that G has at least one vertex. Define the following three sets
The quotient presentation for $V_K$ identifies all of the vertices of $f(V_G) \subseteq V_H$ to a single vertex $v^G$ . We thus have a map $D_K \hookrightarrow D_H \to V_H \to V_H/{\sim} \cong V_K$ . The set $A_K$ inherits an involutive structure from $A_H$ . This data defines a graph K. Notice that $\eth(K) = A_K \setminus D_K = A_H \setminus D_H = \eth(H)$ .
There is a clear candidate for a graphical map $\alpha \colon K \to H$ which is an inclusion on arcs and satisfies $\varphi_1(v) = [\iota_v]$ for $v\in V_H \setminus f(V_G)$ and $\varphi_1(v) = [f]$ for $v=v^G$ . This satisfies the conditions from Definition 2·9; the only thing to note is that $nb(v^G) = f(\eth(G)^\dagger)$ which is used in showing (ii). The triple $(K,v^G,\alpha)$ satisfies the conditions of Definition 5·16.
Lemma 5·20. The functor $F \colon \textbf{U}_{/\mathfrak{o}} \to \textbf{WPpd}$ gives a bijection between embeddings $G \rightarrowtail G'$ in $\textbf{U}^{\textrm{int}}_{/\mathfrak{o}}$ and subgraphs $F(G) \to F(G')$ in the sense of [ Reference Hackney, Robertson and Yau29 , definition 9·50].
Proof. Suppose $f \colon G \rightarrowtail G'$ is an embedding, which determines a map of wheeled properads $F(f) \colon F(G)\to F(G')$ by Proposition 5·15. Let be the graph complement of f from Lemma 5·19. Since $\alpha$ is active, it induces an isomorphism $K\{G\} \cong G'$ by definition 1·39 and proposition 2·3 of [ Reference Hackney, Robertson and Yau33 ]; here $K\{G\}$ is the graph substitution of G into the vertex $v^G$ of K (see construction 1·18 of [ Reference Hackney, Robertson and Yau33 ]). By theorem 9·52 of [ Reference Hackney, Robertson and Yau29 ], $F(G) \to F(G')$ is a subgraph.
Now suppose that $f \colon F(G) \to F(G')$ is a map of wheeled properads which is a subgraph. By theorem 9·52 of [ Reference Hackney, Robertson and Yau29 ] there is a graph substitution decomposition $K\{G\} \cong G'$ so that f sends the edges and vertices in G to their corresponding images in $K\{G\}$ through this isomorphism. As we always have an embedding $G \rightarrowtail K\{G\}$ , we see that the composition $G \rightarrowtail K\{G\} \cong G'$ is an embedding in $\textbf{U}_{/\mathfrak{o}}$ . The two indicated constructions are inverses to one another.
Proof of Theorem 5·13. Recall from Remark 5·14 that we are taking the objects of the wheeled properadic graphical category, denoted in this proof by $\textbf{C}$ , to be the same as the objects of $\textbf{U}_{/\mathfrak{o}}$ . This category comes with a faithful (but nonfull) functor $\textbf{C} \to \textbf{WPpd}$ which has the same action on objects as the functor