A Note on Finite-to-Infinite Extensions and Homotopy Invariance of Digraph Brown Functors

Abstract

This paper develops extension theory for Brown functors in directed graph homotopy theory. We establish a systematic method for extending Brown functors from finite directed graphs to arbitrary directed graphs using inverse limits over finite subdigraphs. We prove that this extension is well-defined and preserves essential functorial properties. Additionally, we provide an alternative characterization of this extension through the Yoneda lemma, demonstrating how extended Brown functors can be naturally identified with sets of natural transformations from representable functors. This categorical perspective offers deeper theoretical insight into the structure of extended Brown functors and establishes important connections with classical representability theory, providing the technical foundation for Brown representability in directed graph theory.

1. Introduction

Directed graph homotopy theory provides a discrete framework for studying topological properties of combinatorial structures, where continuous deformations are replaced by sequences of directed graph maps. This field emerged from early work by Gianella [1] and Malle [2], reaching maturity through Chen, Yau, and Yeh [3] and the comprehensive development by Grigor’yan, Lin, Muranov, and Yau [4]. The theory has found applications in representation theory, neural networks, and discrete dynamical systems, where directed structures naturally encode asymmetric relationships.

A fundamental challenge in discrete homotopy theory is establishing representability theorems analogous to Brown’s classical results [5,6]. Unlike topological spaces, directed graphs lack essential properties such as the homotopy extension property, making standard results of homotopy theory inapplicable. The category of directed graphs sits in a unique position: sufficiently rich to support meaningful cohomology theories [7,8] yet combinatorially constrained enough that classical methods systematically fail. The authors proved that any digraph Brown functor—a contravariant functor from finite directed graphs to abelian groups satisfying triviality, additivity, and Mayer–Vietoris axioms—is representable. See [9] for more details.

This paper serves as a technical companion to [9], offering further elaboration and background. We develop a systematic finite-to-infinite extension theory using inverse limits over finite subdigraphs, which enables us to adapt Adams’ method [10] through novel combinatorial constructions including modified mapping cones and mapping tubes in [9]. These tools preserve essential homotopy theoretic properties while respecting the discrete nature of directed graphs. We also provide an alternative characterization of this extension through the Yoneda lemma, which offers deeper categorical insight into the structure of extended Brown functors and establishes important connections with representability theory.

This paper is organized as follows. Section 2 establishes the foundational framework of directed graph homotopy theory, introducing the basic structures and operations needed for subsequent constructions. Section 3 develops the extension theory for Brown functors, showing how inverse limits provide a systematic method for extending functors from finite to infinite directed graphs while preserving essential properties. Section 4 presents an alternative characterization of this extension via the Yoneda lemma, demonstrating how extended functors can be naturally identified with sets of natural transformations and providing theoretical foundations for representability results. Section 5 discusses the relationship to Brown representability, particularly with regard to classifying directed graphs.

2. Preliminaries on Digraph Homotopy Theory

In this section, we establish the preliminaries of directed graph homotopy theory. Throughout, we shall closely follow the conventions established by Grigor’yan, Lin, Muranov, and Yau [4].

2.1. The Category of Directed Graphs

Definition 1.

A directed graph (or digraph for short) $\overrightarrow{G}$ is a pair $(V,E)$ consisting of a set V specifying labeled points called vertices and another set E of ordered pairs of distinct vertices in V called edges.

Having an edge $(x,y)\in E$ means that there is a directed arrow from x to y and graphically one draws $x\to y$. Note that from the definition above, loop-edges are excluded from consideration and since E is a set, $(x,y)$ occurs at most once.

When multiple digraphs are involved in the context, we often denote the set of vertices of $\overrightarrow{G}$ by $V_{\overrightarrow{G}}$ and the set of edges by $E_{\overrightarrow{G}}$.

Definition 2.

A point ∗ is a digraph consisting of only one vertex and no edges.

Definition 3.

An n-step line digraph, $n>0$, is a sequence of vertices, 0, 1, 2,…, n, such that either $(i-1,i)$ or $(i,i-1)$, for $1\le i\le n$, is an edge (but not both) and there are no other edges.

Note that an n-step line digraph is also called a path digraph or a linear digraph. Such a directed graph forms a line with n arbitrarily oriented edges between each of the $n+1$ vertices. When $n=1$, there are two possible line digraphs, $I^{+}:=0\to 1$ and $I^{-}:=0\leftarrow 1$.

Notation 1.

We will denote an arbitrary n-step line digraph as $I_n$ for short and let ${\mathcal{I}}_n$ represent the set of all possible n-step line digraphs. The set of all line digraphs of any length will be denoted $\mathcal{I}={\bigcup }_n{\mathcal{I}}_n$ and we will refer to an arbitrary element of $\mathcal{I}$ as a line digraph I, dropping the reference to the number of steps.

Definition 4.

A digraph map, $f:\overrightarrow{G}\to \overrightarrow{H}$, is a function from the vertex set of $\overrightarrow{G}$ to the vertex set of $\overrightarrow{H}$ such that whenever $(x,y)$ is an edge in $\overrightarrow{G}$, either $f(x)=f(y)$ in $\overrightarrow{H}$ or $f(x)\to f(y)$ is an edge in $\overrightarrow{H}$. We denote by $im(f)=f(\overrightarrow{G})$ the image of f which is a digraph.

If for some edge $(x,y)$, $f(x)=f(y)$ in $\overrightarrow{H}$, then we will say that this edge has been collapsed, and if $(f(x),f(y))\in E_{\overrightarrow{H}}$, then we say that the edge has been preserved. Since a digraph map must be a function on the discrete set of vertices, the image of a digraph map has at most as many vertices as the domain.

Definition 5.

The category of directed graphs $\mathcal{D}$ is a category in which the objects are directed graphs, $\overrightarrow{G}$, and the morphisms are digraph maps, $f:\overrightarrow{G}\to \overrightarrow{H}$.

Definition 6.

A graph $\overrightarrow{G}=(V,E)$ is finite if the vertex set V is finite.

Notation 2.

(1) We will use the notation ${\mathcal{D}}_0$ to denote the category whose objects are finite digraphs and morphisms are digraph maps. The category ${\mathcal{D}}_0$ is a subcategory of $\mathcal{D}$.

(2) Let $\mathcal{C}$ be a category. Throughout this paper, the expression $X\in \mathcal{C}$ means X is an object of the category $\mathcal{C}$. We will write $f\in \mathcal{C}(X,Y)$ to say f is a morphism from X to Y in $\mathcal{C}$.

Definition 7.

A subdigraph $\overrightarrow{X}$ of a digraph $\overrightarrow{G}$ denoted $\overrightarrow{X}\subseteq \overrightarrow{G}$ is a digraph for which $V_{\overrightarrow{X}}\subseteq V_{\overrightarrow{G}}$ and $E_{\overrightarrow{X}}\subseteq E_{\overrightarrow{G}}$.

Note that even if $u,v\in V_{\overrightarrow{X}}$ and $(u,v)\in E_{\overrightarrow{G}}$, it is not necessarily the case that $(u,v)\in E_{\overrightarrow{X}}$.

Definition 8.

Let $\overrightarrow{G}$ and $\overrightarrow{H}$ be subdigraphs of a digraph $\overrightarrow{Y}$. The intersection of digraphs $\overrightarrow{G}$ and $\overrightarrow{H}$, denoted by $\overrightarrow{G}\cap \overrightarrow{H}$, is the digraph consisting of $V_{\overrightarrow{G}\cap \overrightarrow{H}}=V_{\overrightarrow{G}}\cap V_{\overrightarrow{H}}$ and $E_{\overrightarrow{G}\cap \overrightarrow{H}}=E_{\overrightarrow{G}}\cap E_{\overrightarrow{H}}$.

Note that $\overrightarrow{G}\cap \overrightarrow{H}$ is not necessarily an induced subdigraph of $\overrightarrow{G}$ and $\overrightarrow{H}$.

Definition 9.

Let $\overrightarrow{G}$ and $\overrightarrow{H}$ be subdigraphs of a digraph $\overrightarrow{Y}$. The union of digraphs $\overrightarrow{G}$ and $\overrightarrow{H}$, denoted by $\overrightarrow{G}\cup \overrightarrow{H}$, is the digraph consisting of $V_{\overrightarrow{G}\cup \overrightarrow{H}}=V_{\overrightarrow{G}}\cup V_{\overrightarrow{H}}$ and $E_{\overrightarrow{G}\cup \overrightarrow{H}}=E_{\overrightarrow{G}}\cup E_{\overrightarrow{H}}$.

Note that $\overrightarrow{G}$ and $\overrightarrow{H}$ are necessarily induced subdigraphs of $\overrightarrow{G}\cup \overrightarrow{H}$.

Definition 10.

The disjoint union of two digraphs $\overrightarrow{G}$ and $\overrightarrow{H}$, denoted by $\overrightarrow{G}\coprod \overrightarrow{H}$, is given by the disjoint union of their respective vertex sets and edge sets as sets.

The disjoint union is the coproduct in the category $\mathcal{D}$.

Definition 11.

The graph Cartesian product of two directed graphs $\overrightarrow{G}$ and $\overrightarrow{H}$ is the directed graph $\overrightarrow{G}\square \overrightarrow{H}$, where the vertices are all ordered pairs $(u,v)$ such that $u\in V_{\overrightarrow{G}}$ and $v\in V_{\overrightarrow{H}}$, and $(u_1,v_1)\to (u_2,v_2)$ is an edge in $\overrightarrow{G}\square \overrightarrow{H}$ if either $u_1=u_2$ and $v_1\to v_2$ in $\overrightarrow{H}$ or $u_1\to u_2$ in $\overrightarrow{G}$ and $v_1=v_2$.

Note that the graph Cartesian product is not a product in the category $\mathcal{D}$. Given a fixed vertex $v_0\in V_{\overrightarrow{H}}$, we will denote by $\overrightarrow{G}\square \left\{v_0\right\}$ the $v_0$-slice of $\overrightarrow{G}\square \overrightarrow{H}$. It is the induced subdigraph where the vertices are all ordered pairs $(u,v_0)$ such that $u\in V_{\overrightarrow{G}}$ and the edges are those resulting from the edges of $\overrightarrow{G}$.

2.2. Homotopy Theory on Directed Graphs

Definition 12.

Two digraph maps $f,g:\overrightarrow{G}\to \overrightarrow{H}$ are homotopic, denoted as $f\simeq g$, if there exists an $n\ge 1$ and a digraph map $F:\overrightarrow{G}\square I_n\to \overrightarrow{H}$ for some line digraph $I_n\in {\mathcal{I}}_n$ (Recall Notation 1), such that $F{\mid }_{\overrightarrow{G}\times \left\{0\right\}}=f$ and $F{\mid }_{\overrightarrow{G}\times \left\{n\right\}}=g$.

Definition 13.

Two digraphs are said to be homotopically equivalent (or of the same homotopy type) if there exist two digraph maps, $g:\overrightarrow{G}\to \overrightarrow{H}$ and $h:\overrightarrow{H}\to \overrightarrow{G}$, such that $h\circ g\simeq {\mathit{id}}_{\overrightarrow{G}}$ and $g\circ h\simeq {\mathit{id}}_{\overrightarrow{H}}$. This g and h are called homotopy equivalences.

Definition 14.

A digraph $\overrightarrow{G}$ is said to be contractible if there exists a homotopy between ${\mathit{id}}_{\overrightarrow{G}}$ and a constant digraph map.

Definition 15.

Digraph maps $f,g:\overrightarrow{G}\to \overrightarrow{H}$ are said to be weakly homotopic, denoted as $f{\simeq }_{w}g$, if for every $\overrightarrow{K}\in {\mathcal{D}}_0$ and every digraph map $h:\overrightarrow{K}\to \overrightarrow{G}$ compositions $f\circ h$ and $g\circ h$ are homotopic. Let ${[\overrightarrow{G},\overrightarrow{H}]}_w$ denote the set of weak homotopy classes of digraph maps from $\overrightarrow{G}$ to $\overrightarrow{H}$.

Note that when $\overrightarrow{G}$ is a finite digraph, ${[\overrightarrow{G},\overrightarrow{H}]}_w=[\overrightarrow{G},\overrightarrow{H}]$.

Definition 16.

The homotopy category of directed graphs, denoted $\mathit{Ho}\mathcal{D}$, is a category in which the objects are directed graphs and the morphisms are homotopy equivalence classes of digraph maps. The homotopy category of finite directed graphs $\mathit{Ho}{\mathcal{D}}_0$ and the weak homotopy category for directed graphs $\mathit{wHo}\mathcal{D}$ are defined in the same manner.

3. Extension of Brown Functors to Infinite Digraphs

In this section, we establish the theoretical framework for extending Brown functors from finite to infinite directed graphs. We first introduce the notion of a digraph Brown functor (Definition 17) as a contravariant functor from the homotopy category of finite digraphs to abelian groups, satisfying the triviality, additivity, and Mayer–Vietoris axioms. The central contribution is the systematic extension of any Brown functor H to arbitrary (possibly infinite) digraphs via inverse limits over finite subdigraphs: $\widehat{\mathrm{H}}(\overrightarrow{G}):=\underset{\alpha \in \Lambda }{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\alpha })$, where the limit is taken over all finite subdigraphs of $\overrightarrow{G}$. We prove that this extension is well-defined and independent of the choice of covering by finite subdigraphs (Lemma 1), preserves the functorial structure (Lemma 2), and maintains the additivity property (Theorem 1). This extension theory provides the essential foundation for applying Brown representability techniques in the context of infinite directed graphs, bridging classical algebraic topology methods with discrete combinatorial structures.

Notation 3.

Consider a diagram $B\stackrel{f}{\to }A\stackrel{g}{\leftarrow }C$ in the category of abelian groups Ab. We will use the notation $B{\times }_{A}C$ to denote the subset of $B\times C$ defined by $\{(b,c):f(b)=g(c)\}$.

Definition 17.

A (digraph) Brown functor is a functor $\mathrm{H}:\mathit{Ho}{\mathcal{D}}_0^{\mathit{op}}\to \mathit{Ab}$ satisfying the following axioms:

(1) 

Triviality Axiom. The functor $\mathrm{H}$ sends a singleton to the trivial group.

(2) 

Additivity Axiom. The functor $\mathrm{H}$ sends the coproduct to the product, i.e., $\mathrm{H}({\coprod }_{\alpha \in \Lambda }{\overrightarrow{G}}_{\alpha })={\prod }_{\alpha \in \Lambda }\mathrm{H}({\overrightarrow{G}}_{\alpha })$ for any family of digraphs ${\left\{{\overrightarrow{G}}_{\alpha }\right\}}_{\alpha \in \Lambda }$.

(3) 

Mayer–Vietoris Axiom. For any digraphs ${\overrightarrow{G}}_1,{\overrightarrow{G}}_2\in \mathit{Ho}{\mathcal{D}}_0$, the map $\mathrm{H}({\overrightarrow{G}}_1\cup {\overrightarrow{G}}_2)\to \mathrm{H}({\overrightarrow{G}}_1){\times }_{\mathrm{H}({\overrightarrow{G}}_1\cap {\overrightarrow{G}}_2)}\mathrm{H}({\overrightarrow{G}}_2)$ induced by the inclusions is a surjection.

We refer the readers to [9] for examples of digraph Brown functors.

Although our main interest lies in the study of finite digraphs, to establish the Brown representability theorem, we need to extend a Brown functor to the category $\mathcal{D}$ of arbitrary digraphs, including both finite and infinite ones. More explicitly, given a Brown functor $\mathrm{H}$, it can be extended to a functor on $\mathcal{D}$ as follows.

Let $\overrightarrow{G}\in \mathcal{D}$. Consider ${\left\{{\overrightarrow{G}}_{\alpha }\right\}}_{\alpha \in \Lambda }$, the set of all the finite subdigraphs of $\overrightarrow{G}$. Together with inclusions, it forms a directed set. We extend $\mathrm{H}$ to a functor $\widehat{\mathrm{H}}$ by taking the inverse limit over this directed set:

$$\widehat{\mathrm{H}}(\overrightarrow{G}):=\underset{\alpha \in \Lambda }{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\alpha }).$$

(1)

Explicitly, we have the identification

$$\widehat{\mathrm{H}}(\overrightarrow{G})=\left\{{(x_{\alpha })}_{\alpha \in \Lambda }\in \prod _{\alpha \in \Lambda }\mathrm{H}({\overrightarrow{G}}_{\alpha }):{\mathfrak{i}}_{\alpha \beta }^{*}(x_{\beta })=x_{\alpha },\forall {\overrightarrow{G}}_{\alpha }\subseteq {\overrightarrow{G}}_{\beta }\underset{\mathrm{finite}}{\subseteq }\overrightarrow{G}\right\},$$

(2)

where the notation ${\mathfrak{i}}_{\alpha \beta }$ means the inclusion map ${\mathfrak{i}}_{\alpha \beta }:{\overrightarrow{G}}_{\alpha }\hookrightarrow {\overrightarrow{G}}_{\beta }$, and $\overrightarrow{H}\underset{\mathrm{finite}}{\subseteq }\overrightarrow{G}$ means that $\overrightarrow{H}$ is a finite subdigraph of $\overrightarrow{G}$.

Note that although $\widehat{\mathrm{H}}$ is defined by taking the inverse limit over all the finite subdigraph, it actually can be defined by taking the limit over any directed set ${\left\{{\overrightarrow{G}}_{\alpha }\right\}}_{\alpha \in {\Lambda }^{\prime }}$ of finite subdigraphs satisfying ${\bigcup }_{\alpha \in {\Lambda }^{\prime }}{\overrightarrow{G}}_{\alpha }=\overrightarrow{G}$.

Lemma 1.

If ${\left\{{\overrightarrow{G}}_{\alpha }\right\}}_{\alpha \in {\Lambda }^{\prime }}$ is a directed set of finite subdigraphs such that ${\bigcup }_{\alpha \in {\Lambda }^{\prime }}{\overrightarrow{G}}_{\alpha }=\overrightarrow{G}$, then

$$\widehat{\mathrm{H}}(\overrightarrow{G})\cong \underset{\alpha \in {\Lambda }^{\prime }}{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\alpha }).$$

Proof. 

Recall that

$$\underset{\alpha \in {\Lambda }^{\prime }}{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\alpha })=\left\{{(x_{\alpha })}_{\alpha \in {\Lambda }^{\prime }}\in \prod _{\alpha \in {\Lambda }^{\prime }}\mathrm{H}({\overrightarrow{G}}_{\alpha }):{\mathfrak{i}}_{\alpha \beta }^{*}(x_{\beta })=x_{\alpha },\forall {\overrightarrow{G}}_{\alpha }\subseteq {\overrightarrow{G}}_{\beta },\alpha ,\beta \in {\Lambda }^{\prime }\right\}.$$

Since ${\Lambda }^{\prime }$ is a subset of $\Lambda$, we have the map given by projection:

$$\underset{\alpha \in \Lambda }{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\alpha })\to \underset{\alpha \in {\Lambda }^{\prime }}{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\alpha }).$$

(3)

We claim that the map (3) is an isomorphism. For injectivity, assume ${(x_{\alpha })}_{\alpha \in \Lambda }\in \widehat{\mathrm{H}}(\overrightarrow{G})$ maps to zero under (3), i.e., $x_{\alpha }=0$ for all $\alpha \in {\Lambda }^{\prime }$. We would like to show that the other components $x_{\alpha }$ for $\alpha \in \Lambda \setminus {\Lambda }^{\prime }$ also vanish. To do this, consider $x_{\alpha }\in \mathrm{H}({\overrightarrow{G}}_{\alpha })$, $\alpha \in \Lambda \setminus {\Lambda }^{\prime }$. Since $\overrightarrow{G}={\bigcup }_{\zeta \in {\Lambda }^{\prime }}{\overrightarrow{G}}_{\zeta }$ and ${\overrightarrow{G}}_{\alpha }$ is finite, there exist ${\overrightarrow{G}}_1,\cdots ,{\overrightarrow{G}}_n\in {\left\{{\overrightarrow{G}}_{\zeta }\right\}}_{\zeta \in {\Lambda }^{\prime }}$ such that ${\overrightarrow{G}}_{\alpha }\subseteq {\bigcup }_{j=1}^n{\overrightarrow{G}}_j$. Furthermore, since ${\left\{{\overrightarrow{G}}_{\zeta }\right\}}_{\zeta \in {\Lambda }^{\prime }}$ is a directed set, there exists $\beta \in {\Lambda }^{\prime }$ such that ${\overrightarrow{G}}_{\beta }\supseteq {\bigcup }_{j=1}^n{\overrightarrow{G}}_j$. Now, since $\beta \in {\Lambda }^{\prime }$, we have $x_{\beta }=0$, and consequently $x_{\alpha }={\mathfrak{i}}_{\alpha \beta }^{*}(x_{\beta })=0$. This shows the injectivity of (3).

For surjectivity, let ${(x_{\zeta })}_{\zeta \in {\Lambda }^{\prime }}$ be an arbitrary element in the codomain of (3). For any $\alpha \in \Lambda$, we saw in the previous paragraph that there exist ${\overrightarrow{G}}_1,\cdots ,{\overrightarrow{G}}_n,{\overrightarrow{G}}_{\beta }\in {\left\{{\overrightarrow{G}}_{\zeta }\right\}}_{\zeta \in {\Lambda }^{\prime }}$ such that ${\overrightarrow{G}}_{\alpha }\subseteq {\bigcup }_{j=1}^n{\overrightarrow{G}}_j\subseteq {\overrightarrow{G}}_{\beta }$. Since ${\mathfrak{i}}_{j\beta }^{*}(x_{\beta })=x_j$, it follows that $(x_j,x_k)\in \mathrm{H}({\overrightarrow{G}}_j){\times }_{\mathrm{H}({\overrightarrow{G}}_j\cap {\overrightarrow{G}}_k)}\mathrm{H}({\overrightarrow{G}}_k)$, which is the image of some class in $\mathrm{H}({\overrightarrow{G}}_j\cup {\overrightarrow{G}}_k)$ by the Mayer–Vietoris axiom. Now applying the Mayer–Vietoris axiom inductively, we obtain $x\in \mathrm{H}({\overrightarrow{G}}_1\cup \cdots \cup {\overrightarrow{G}}_n)$ such that ${\mathfrak{i}}_j^{*}(x)=x_j$ for $j=1,\cdots ,n$, and this x defines a class in $\mathrm{H}({\overrightarrow{G}}_{\alpha })$ by

$$x_{\alpha }={\mathfrak{i}}_{\alpha }^{*}(x).$$

Here ${\mathfrak{i}}_j$, for $j=1,\cdots ,n,\alpha$, is the inclusion map ${\mathfrak{i}}_j:{\overrightarrow{G}}_j\hookrightarrow {\overrightarrow{G}}_1\cup \cdots \cup {\overrightarrow{G}}_n$. With this choice of $x_{\alpha }\in \mathrm{H}({\overrightarrow{G}}_{\alpha })$, $\alpha \in \Lambda$, it is not difficult to show that ${\mathfrak{i}}_{\alpha \beta }^{*}(x_{\beta })=x_{\alpha },\forall {\overrightarrow{G}}_{\alpha }\subseteq {\overrightarrow{G}}_{\beta }$, $\alpha ,\beta \in \Lambda$. This completes the proof. □

Given a morphism $f:\overrightarrow{G}\to \overrightarrow{H}$ in $\mathcal{D}$, we the induce homomorphism $f^{*}:\widehat{\mathrm{H}}(\overrightarrow{H})\to \widehat{\mathrm{H}}(\overrightarrow{G})$ given by

$$f^{*}\left({(y_{\beta })}_{\beta }\right)={\left(f^{*}(y_{f(\alpha )})\right)}_{\alpha },$$

(4)

where $f(\alpha )$ is the index for the subdigraph ${\overrightarrow{H}}_{f(\alpha )}=f({\overrightarrow{G}}_{\alpha })$. The following lemma is clear.

Lemma 2.

The assignment $\widehat{\mathrm{H}}:\mathit{Ho}{\mathcal{D}}^{\mathit{op}}\to \mathit{Ab}$, $\overrightarrow{G}\mapsto \underset{\alpha }{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\alpha })$ is a functor which restricts to the functor $\mathrm{H}:\mathit{Ho}{\mathcal{D}}_0^{\mathit{op}}\to \mathit{Ab}$. Also, an assignment $\widehat{\mathrm{H}}:\mathit{wHo}{\mathcal{D}}^{\mathit{op}}\to \mathit{Ab}$ defined similarly is a functor that restricts to the functor $\mathrm{H}:\mathit{Ho}{\mathcal{D}}_0^{\mathit{op}}\to \mathit{Ab}$.

Lemma 1 can be extended to the following theorem.

Theorem 1.

Let $\overrightarrow{G}\in \mathcal{D}$ and ${\left\{{\overrightarrow{G}}_{\alpha }\right\}}_{\alpha \in {\Lambda }^{\prime }}$ be any directed set of (not necessarily finite) subdigraphs of $\overrightarrow{G}$ such that ${\bigcup }_{\alpha \in {\Lambda }^{\prime }}{\overrightarrow{G}}_{\alpha }=\overrightarrow{G}$. Then there is a canonical isomorphism

$$\begin{array}{cc}\hfill \Theta :\widehat{\mathrm{H}}(\overrightarrow{G})& \to \underset{\alpha \in {\Lambda }^{\prime }}{\underleftarrow{lim}}\widehat{\mathrm{H}}({\overrightarrow{G}}_{\alpha })\hfill \\ \hfill x& \mapsto {({\mathfrak{i}}_{\alpha }^{*}x)}_{\alpha \in {\Lambda }^{\prime }}\hfill \end{array}$$

where ${\mathfrak{i}}_{\alpha }:{\overrightarrow{G}}_{\alpha }\hookrightarrow \overrightarrow{G}$ are the inclusion maps.

Proof. 

As before, let ${\mathfrak{i}}_{\alpha \beta }:{\overrightarrow{G}}_{\alpha }\hookrightarrow {\overrightarrow{G}}_{\beta }$ denote the inclusion map if ${\overrightarrow{G}}_{\alpha }\subseteq {\overrightarrow{G}}_{\beta }$. Recall that an element in $\underset{\alpha }{\underleftarrow{lim}}\widehat{\mathrm{H}}({\overrightarrow{G}}_{\alpha })$ is an element $(x_{\alpha })\in {\prod }_{\alpha \in {\Lambda }^{\prime }}\widehat{\mathrm{H}}({\overrightarrow{G}}_{\alpha })$ such that ${\mathfrak{i}}_{\alpha \beta }^{*}(x_{\beta })=x_{\alpha }$ for all ${\overrightarrow{G}}_{\alpha }\subseteq {\overrightarrow{G}}_{\beta }$. Here each $x_{\alpha }\in {\overrightarrow{G}}_{\alpha }$ is of the form $x_{\alpha }={(x_{\alpha \zeta })}_{\zeta }\in {\prod }_{\zeta }\mathrm{H}({\overrightarrow{G}}_{\alpha \zeta })$ such that ${\mathfrak{i}}_{\zeta {\zeta }^{\prime }}^{*}(x_{\alpha {\zeta }^{\prime }})=x_{\alpha \zeta }$ for all ${\overrightarrow{G}}_{\alpha \zeta }\subseteq {\overrightarrow{G}}_{\alpha \zeta }$. The product ${\prod }_{\zeta }$ is over all the finite digraph ${\overrightarrow{G}}_{\alpha \zeta }$ of ${\overrightarrow{G}}_{\alpha }$. Therefore, we have a natural identification

$$\underset{\alpha \in {\Lambda }^{\prime }}{\underleftarrow{lim}}\widehat{\mathrm{H}}({\overrightarrow{G}}_{\alpha })=\underset{\zeta \in {\Lambda }^{\prime \prime }}{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{G}}_{\zeta }),$$

where the right-hand-side limit is taken over the finite subdigraphs ${\overrightarrow{G}}_{\zeta }$ of $\overrightarrow{G}$ with the property ${\overrightarrow{G}}_{\zeta }\subseteq {\overrightarrow{G}}_{\alpha }$ for some $\alpha \in {\Lambda }^{\prime }$. This is isomorphic to $\widehat{\mathrm{H}}(\overrightarrow{G})$ by Lemma 1, and thus the proof is complete. □

4. Alternative Characterization via Yoneda Lemma

Having established the systematic extension of Brown functors from finite to infinite directed graphs through inverse limits in Section 3, we now present an alternative theoretical framework for understanding this extension. Throughout this section, we will use the notation $\mathrm{Nat}(F,G)$ to denote the set of all natural transformations from a functor F to a functor G.

The Yoneda lemma provides a powerful categorical perspective that illuminates the relationship between our extended Brown functor $\widehat{\mathrm{H}}$ and natural transformations. This approach not only offers deeper theoretical insight into the structure of extended Brown functors but also establishes important connections with representability theory. In this section, we demonstrate how the extended functor $\widehat{\mathrm{H}}(\overrightarrow{Y})$ can be naturally identified with the set of natural transformations $\mathrm{Nat}([-,\overrightarrow{Y}],\mathrm{H}(-))$, providing an elegant reformulation of our extension construction and laying the groundwork for Brown representability results in directed graph theory.

In Equation (1), we extend a Brown functor $\mathrm{H}$ for finite digraphs to a functor $\widehat{\mathrm{H}}$ for arbitrary (possibly infinite) digraphs. We now give another description of this extended functor via the Yoneda lemma. Explicitly, let $\mathrm{H}:\mathit{Ho}{\mathcal{D}}_0^{\mathit{op}}\to \mathbf{Set}$ be a functor. To extend $\mathrm{H}$ to infinite digraphs, consider $\overrightarrow{Y}\in \mathcal{D}$ and its associated functor $[-,\overrightarrow{Y}]:\mathit{Ho}{\mathcal{D}}_0^{\mathit{op}}\to \mathbf{Set}$, which sends a finite digraph $\overrightarrow{G}$ to the set $[\overrightarrow{G},\overrightarrow{Y}]$ of homotopy classes of digraph maps from $\overrightarrow{G}$ to $\overrightarrow{Y}$.

Define

$$\overline{\mathrm{H}}(\overrightarrow{Y}):=\mathrm{Nat}([-,\overrightarrow{Y}],\mathrm{H}(-)).$$

(5)

If $\overrightarrow{Y}$ is finite, then by the Yoneda lemma, we have the isomorphism

$$\mathrm{H}(\overrightarrow{Y})\stackrel{\cong }{\to }\overline{\mathrm{H}}(\overrightarrow{Y}),y\mapsto T_y,$$

where $T_y$ is the natural transformation given by

$$T_{y,\overrightarrow{G}}:[\overrightarrow{G},\overrightarrow{Y}]\to \mathrm{H}(\overrightarrow{G}),[f]\mapsto f^{*}(y),$$

(6)

for any finite digraph $\overrightarrow{G}$.

Theorem 2.

There is an isomorphism between sets:

$$\varphi :\overline{\mathrm{H}}(\overrightarrow{Y})\to \widehat{\mathrm{H}}(\overrightarrow{Y})=\underset{\alpha }{\underleftarrow{lim}}\mathrm{H}({\overrightarrow{Y}}_{\alpha }),T\mapsto {\left(T_{{\overrightarrow{Y}}_{\alpha }}([{\mathfrak{i}}_{\alpha }])\right)}_{\alpha },$$

where the limit is taken over all the finite subdigraphs ${\overrightarrow{Y}}_{\alpha }$ of $\overrightarrow{Y}$, $T\in \mathrm{Nat}([-,\overrightarrow{Y}],\mathrm{H}(-))$, and ${\mathfrak{i}}_{\alpha }:{\overrightarrow{Y}}_{\alpha }\hookrightarrow \overrightarrow{Y}$ are the inclusion maps.

Proof. 

Let $T_1,T_2\in \overline{\mathrm{H}}(\overrightarrow{Y})$, $\overrightarrow{G}\in {\mathcal{D}}_0$, and $[f]\in [\overrightarrow{G},\overrightarrow{Y}]$. Since $\overrightarrow{G}$ is finite, the image $f(\overrightarrow{G})$ is also finite, and we have the following commutative diagram:

Here, the map $\tilde{f}$ is the corestriction of f, and ${\mathfrak{i}}_f$ is the inclusion map. Suppose that $\varphi (T_1)=\varphi (T_2)$, i.e., $T_{1,{\overrightarrow{Y}}_{\alpha }}([{\mathfrak{i}}_{\alpha }])=T_{2,{\overrightarrow{Y}}_{\alpha }}([{\mathfrak{i}}_{\alpha }])$ for any finite subdigraphs ${\overrightarrow{Y}}_{\alpha }$ of $\overrightarrow{Y}$. Since

$$T_{1,\overrightarrow{G}}([f])=T_{1,\overrightarrow{G}}({\tilde{f}}^{*}[{\mathfrak{i}}_f])={\tilde{f}}^{*}{T}_{1,f(\overrightarrow{G})}([{\mathfrak{i}}_f])={\tilde{f}}^{*}{T}_{2,f(\overrightarrow{G})}([{\mathfrak{i}}_f])=T_{2,\overrightarrow{G}}([f]),$$

we have $T_1=T_2$. This shows that $\varphi$ is injective.

For the surjectivity, given any $y\in \widehat{\mathrm{H}}(\overrightarrow{Y})$, it can be expressed as $y={(y_{\alpha })}_{\alpha }$, where $y_{\alpha }\in \mathrm{H}({\overrightarrow{Y}}_{\alpha })$ and ${\mathfrak{i}}_{\alpha \beta }^{*}(y_{\beta })=y_{\alpha }$ for any ${\overrightarrow{Y}}_{\alpha }\subseteq {\overrightarrow{Y}}_{\beta }\underset{\mathrm{finite}}{\subseteq }\overrightarrow{Y}$. In particular, we have the component $y_f\in \mathrm{H}(f(\overrightarrow{G}))$ (considering ${\overrightarrow{Y}}_{\alpha }=f(\overrightarrow{G})$ and $y_f=y_{\alpha }$) if $\overrightarrow{G}$ is a finite digraph and $f:\overrightarrow{G}\to \overrightarrow{Y}$ is a digraph map. Define $T_y\in \mathrm{Nat}([-,\overrightarrow{Y}],\mathrm{H}(-))$ by

$$T_{y,\overrightarrow{G}}:[\overrightarrow{G},\overrightarrow{Y}]\to \mathrm{H}(\overrightarrow{G}),[f]\mapsto {\tilde{f}}^{*}(y_f).$$

It is easy to see that $T_y$ is indeed a natural transformation and $\varphi (T_y)=y$. This completes the proof. □

Let $\mathrm{H}$ be a Brown functor which induces $\widehat{\mathrm{H}}:\mathit{wHo}{\mathcal{D}}^{\mathit{op}}\to \mathbf{Set}$ by Equation (1). By the Yoneda lemma, for each digraph $\overrightarrow{Y}\in \mathcal{D}$, we have

$$\widehat{\mathrm{H}}(\overrightarrow{Y})\cong \mathrm{Nat}({[-,\overrightarrow{Y}]}_w,\widehat{\mathrm{H}}(-)).$$

Combining with Theorem 2, we have

$$\mathrm{Nat}({[-,\overrightarrow{Y}]}_w,\widehat{\mathrm{H}}(-))\cong \widehat{\mathrm{H}}(\overrightarrow{Y})\cong \mathrm{Nat}([-,\overrightarrow{Y}],\mathrm{H}(-)),$$

(7)

where ${[-,\overrightarrow{Y}]}_w$ and $\widehat{\mathrm{H}}(-)$ are functors from $\mathit{wHo}{\mathcal{D}}^{\mathit{op}}$ to $\mathbf{Set}$, and $[-,\overrightarrow{Y}]$ and $\mathrm{H}(-)$ are functors from $\mathit{Ho}{\mathcal{D}}_0^{\mathit{op}}$ to $\mathbf{Set}$.

5. Discussion

The extension theory developed in Section 3 provides the essential technical foundation for proving Brown representability theorems in digraph homotopy theory. As demonstrated in the companion work [9], this extension is crucial for establishing the Mayer–Vietoris property for unions of infinite directed graphs, which forms a key step in constructing classifying objects for Brown functors. The inverse limit construction ensures that Brown functors initially defined on finite digraphs can be meaningfully extended to arbitrary digraphs while preserving the axioms necessary for representability. Without this extension theory, the classical Adams–Brown approach to representability would not be applicable to directed graph homotopy theory.

The alternative characterization presented in Section 4 through the Yoneda lemma provides theoretical insight into the nature of these extensions, revealing their connections to category-theoretic foundations. This Yoneda perspective validates our inverse limit approach and establishes important connections with classical representability theory. Together, these complementary approaches provide a complete theoretical foundation for Brown representability in directed graph homotopy theory, enabling the adaptation of classical algebraic topology methods to discrete combinatorial structures.