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Globular theories and their models

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

In Pursuing Stacks Grothendieck proposed an algebraic definition of \(\infty\)-groupoids: globular sets equipped with operations — compositions, units, inverses and all higher coherence cells — chosen by freely adjoining fillers to globular pasting diagrams. A coherator records these choices. The homotopy hypothesis asserts that the resulting \(\infty\)-groupoids describe the homotopy theory of spaces. Henry reduced this hypothesis to a single statement: freely adjoining to a cellular \(\infty\)-groupoid a new cell parallel to a given one, together with a cell joining the two, does not change the homotopy type. This course proves that statement for every coherator, following OpenAI's 2026 proof, derives the canonical semi-model structure, and explains how the comparison with spaces follows.

This lesson sets up the algebra. Section 1 introduces globular sets and pasting schemes, Section 2 globular theories and coherators, and Section 3 their models. Section 4 proves that the models form a complete and cocomplete category with free objects, and Section 5 proves the small object argument used throughout the course.

Prerequisites. Basic category theory: limits and colimits of presheaves are computed objectwise, and the Yoneda lemma \(\operatorname{Nat}(\mathcal C(-,c),X)\cong X(c)\) (Theorem 1.1 of The functor of points and representability, stated there for schemes; the proof applies to presheaves on any category). Left Kan extensions: Theorem 2.1 of Kan extensions and the derivator axioms.

1. Globular sets and pasting schemes

Let \(\mathbb G\) be the category with objects \(d_0,d_1,d_2,\ldots\), generated by arrows \(\sigma_k,\tau_k:d_k\to d_{k+1}\) subject to \[ \sigma_{k+1}\sigma_k=\tau_{k+1}\sigma_k,\qquad \sigma_{k+1}\tau_k=\tau_{k+1}\tau_k . \] A globular set is a presheaf \(X\) on \(\mathbb G\): sets \(X_k\) of \(k\)-cells with source and target maps \(s,t:X_{k+1}\to X_k\) satisfying \(ss=st\) and \(ts=tt\). For a cell \(x\) of dimension greater than \(i\), \(s_i(x)\) and \(t_i(x)\) are its iterated sources and targets in dimension \(i\). Two \(k\)-cells are parallel if \(k=0\), or if \(k\ge1\) and they have the same source and the same target. A morphism of globular sets is a natural transformation; the category is \(\mathbf{Glob}\).

The globe \(d_j\) is the representable presheaf \(\mathbb G(-,d_j)\). It has one \(j\)-cell, the identity, and two \(k\)-cells for every \(k<j\), its iterated faces. Its boundary \(\partial d_j\) is the subpresheaf without the \(j\)-cell; \(\partial d_0=\varnothing\), and \(\partial d_j\) consists of two parallel \((j-1)\)-globes when \(j\ge1\). By the Yoneda lemma, globular maps \(d_j\to X\) are the \(j\)-cells of \(X\), and maps \(\partial d_j\to X\) are pairs of parallel \((j-1)\)-cells.

A table of dimensions is a sequence \((h_1,b_1,h_2,b_2,\ldots,b_{l-1},h_l)\) of nonnegative integers with \(l\ge1\) and \(b_a<h_a\), \(b_a<h_{a+1}\). Its pasting scheme is the colimit in \(\mathbf{Glob}\) \[ S=d_{h_1}\amalg_{d_{b_1}}d_{h_2}\amalg_{d_{b_2}}\cdots\amalg_{d_{b_{l-1}}}d_{h_l} \] of the zigzag in which \(d_{b_a}\to d_{h_a}\) is the iterated target inclusion and \(d_{b_a}\to d_{h_{a+1}}\) the iterated source inclusion. Its height \(\dim S\) is \(\max_ah_a\). A globular map \(S\to X\) is a tuple of shape \(S\) in \(X\); by the universal property of the colimit, \[ \begin{aligned} X(S)&=\operatorname{Hom}(S,X)\\ &=X_{h_1}\times_{X_{b_1}}X_{h_2}\times_{X_{b_2}}\cdots\\ &\qquad\times_{X_{b_{l-1}}}X_{h_l}, \end{aligned} \] the iterated fibre product of \(t_{b_a}:X_{h_a}\to X_{b_a}\) and \(s_{b_a}:X_{h_{a+1}}\to X_{b_a}\): a string of cells in which each one is glued to the next along a common face. For instance \(d_1\amalg_{d_0}d_1\) parametrizes composable pairs of arrows, and \(d_2\amalg_{d_1}d_2\) vertically composable pairs of \(2\)-cells.

Let \(\Theta_0\) be the full subcategory of \(\mathbf{Glob}\) on the pasting schemes, one for each table of dimensions. Each pasting scheme comes with its designated cocone \((d_{h_a}\to S)_a\) from the zigzag; for a globe the cocone is the identity.

2. Globular theories and coherators

A globular extension is a category \(\mathcal E\) with a functor \(\Theta_0\to\mathcal E\) that sends every designated cocone to a colimit cocone in \(\mathcal E\). A morphism of globular extensions is a functor commuting with the functors from \(\Theta_0\). A globular theory is a globular extension \(\Theta_0\to\mathcal C\) that is the identity on objects. Thus the objects of a globular theory are the pasting schemes, and for every pasting scheme \(S\) with table \((h_1,b_1,\ldots,h_l)\) and every object \(T\), \[ \begin{aligned} \mathcal C(S,T)&=\mathcal C(d_{h_1},T)\times_{\mathcal C(d_{b_1},T)}\cdots\\ &\qquad\times_{\mathcal C(d_{b_{l-1}},T)}\mathcal C(d_{h_l},T). \end{aligned} \] Arrows \(d_j\to S\) of \(\mathcal C\) are operations of output dimension \(j\) and arity \(S\). Two operations \(\sigma,\tau:d_{j-1}\to S\) are parallel if \(j=1\), or if \(j\ge2\) and \(\sigma\sigma_{j-2}=\tau\sigma_{j-2}\) and \(\sigma\tau_{j-2}=\tau\tau_{j-2}\). A filler of a parallel pair \((\sigma,\tau)\) is an operation \(\theta:d_j\to S\) with \(\theta\sigma_{j-1}=\sigma\) and \(\theta\tau_{j-1}=\tau\).

Definition 2.1. Let \(\mathcal C\) be a globular theory and \(P\) a set of parallel pairs \(p=(\sigma_p,\tau_p)\) of operations \(d_{j_p-1}\rightrightarrows S_p\), \(j_p\ge1\). A free extension of \(\mathcal C\) by fillers for \(P\) is a globular theory \(\mathcal C'\) with a morphism \(\mathcal C\to\mathcal C'\) and fillers \(\theta_p\) in \(\mathcal C'\) of the images of the pairs \(p\), with the following universal property. For every globular extension \(\mathcal E\), every morphism \(G:\mathcal C\to\mathcal E\) and every family of fillers \(g_p\) in \(\mathcal E\) of the pairs \(G(p)\), there is exactly one morphism \(G':\mathcal C'\to\mathcal E\) extending \(G\) with \(G'(\theta_p)=g_p\).

Definition 2.2. A Grothendieck coherator is a globular theory \(\mathcal C\) with a sequence of globular theories and morphisms \(\Theta_0=\mathcal C_0\to\mathcal C_1\to\mathcal C_2\to\cdots\) such that:

  1. each \(\mathcal C_{r+1}\) is a free extension of \(\mathcal C_r\) by fillers for a set of parallel pairs \(d_{j-1}\rightrightarrows S\) of \(\mathcal C_r\) with \(j\ge1\) and \(\dim S\le j\);
  2. \(\mathcal C\) is the colimit of the sequence: \(\mathcal C(S,T)=\operatorname{colim}_r\mathcal C_r(S,T)\);
  3. (contractibility) every parallel pair \(d_{j-1}\rightrightarrows S\) in \(\mathcal C\) with \(j\ge1\) and \(\dim S\le j\) has a filler in \(\mathcal C\).

The fillers adjoined in (1) are the generating operations, and the sequence is the defining tower. Condition (2) gives a globular theory: the hom-sets of a designated sum are finite iterated fibre products, which commute with the sequential colimit. The bound \(\dim S\le j\) applies to generating operations only; composite operations of output dimension \(j\) can have arities of greater height. We fix one coherator \(\mathcal C\), with its defining tower, for the whole course. This is the convention of Ara and Henry.

The parallel pairs with \(\dim S\le j\) are called admissible. Contractibility supplies the operations of an \(\infty\)-groupoid. For example, the two maps \(d_0\rightrightarrows d_1\amalg_{d_0}d_1\) picking the first source and the last target form an admissible pair with \(j=1\); a filler is a composition of two arrows. The two maps \(d_0\rightrightarrows d_1\) picking target and source, in this order, form an admissible pair; a filler is an inverse.

3. Models

Definition 3.1. A model of \(\mathcal C\), or \(\mathcal C\)-\(\infty\)-groupoid, is a functor \(X:\mathcal C^{\mathrm{op}}\to\mathbf{Set}\) that sends every designated cocone to a limit cone: \[ \begin{aligned} &X(S)\longrightarrow X(d_{h_1})\times_{X(d_{b_1})}\cdots\\ &\qquad\times_{X(d_{b_{l-1}})}X(d_{h_l})\quad\text{is bijective.} \end{aligned} \] Morphisms are natural transformations. The category is \(\mathcal A=\operatorname{Mod}(\mathcal C)\). The underlying globular set of \(X\) is its restriction along \(\mathbb G\to\Theta_0\to\mathcal C\), and the cells of \(X\) are those of its underlying globular set. Through the bijection above, \(X(S)\) is the set of tuples of shape \(S\) in the underlying globular set, and each operation \(\theta:d_j\to S\) acts as a function \(\theta_X:X(S)\to X_j\).

For a globular set \(X\), let \(\mathcal E_X\) be the category whose objects are the pasting schemes, with \(\mathcal E_X(S,T)\) the set of functions \(X(T)\to X(S)\), composed in the opposite order. Restriction of tuples is a functor \(\Theta_0\to\mathcal E_X\), and \(\mathcal E_X\) is a globular extension: since \(X(S)\) is the fibre product of the \(X(d_{h_a})\), functions \(X(T)\to X(S)\) are compatible families of functions \(X(T)\to X(d_{h_a})\). A model with underlying globular set \(X\) is the same as a morphism of globular extensions \(\mathcal C\to\mathcal E_X\).

Proposition 3.2 (models by generators). Let \(X\) be a globular set. A model structure on \(X\) amounts to the choice, for every generating operation \(\theta:d_j\to S\) adjoined for the pair \((\sigma,\tau)\), of a function \(\theta_X:X(S)\to X_j\) with \[ s\circ\theta_X=\sigma_X,\qquad t\circ\theta_X=\tau_X , \] where \(\sigma_X,\tau_X\) are the actions of the earlier operations \(\sigma,\tau\). The choices are made by induction along the defining tower, and no further equations are imposed. A globular map \(f:X\to Y\) between models is a morphism of models exactly when \(f\circ\theta_X=\theta_Y\circ f\) for every generating operation \(\theta\).

Proof. By induction on \(r\), morphisms of globular extensions \(\mathcal C_r\to\mathcal E_X\) correspond to choices for the generating operations of \(\mathcal C_1,\ldots,\mathcal C_r\). For \(r=0\) there is exactly one, restriction of tuples. For the step, the universal property of Definition 2.1 says that a morphism \(\mathcal C_{r+1}\to\mathcal E_X\) is a morphism on \(\mathcal C_r\) together with fillers in \(\mathcal E_X\) of the images of the adjoined pairs. A filler of \((\sigma_X,\tau_X)\) in \(\mathcal E_X\) is a function \(X(S)\to X_j\) whose composites with \(s,t:X_j\to X_{j-1}\) are \(\sigma_X\) and \(\tau_X\). By Definition 2.2(2), a morphism \(\mathcal C\to\mathcal E_X\) is a compatible family of morphisms \(\mathcal C_r\to\mathcal E_X\).

For morphisms, let \(\mathcal E_f\) be the category of pasting schemes in which an arrow \(S\to T\) is a pair of functions \(X(T)\to X(S)\), \(Y(T)\to Y(S)\) commuting with the maps \(f_T:X(T)\to Y(T)\) and \(f_S\) induced by \(f\). It is a globular extension, by the same fibre product argument. If \(f\) commutes with every generating operation, the same induction gives a morphism \(\mathcal C\to\mathcal E_f\) whose two components are the given models, and this says that \(f\) is natural. The converse is clear. \(\square\)

So a model is a globular set with a choice of each generating operation, subject only to its two face equations; morphisms are the globular maps preserving the generating operations, hence all operations. In particular the forgetful functor \(U:\mathcal A\to\mathbf{Glob}\) is faithful.

4. Limits, colimits and free models

Let \(\mathcal P\) be the category of all presheaves on \(\mathcal C\), and \(y:\mathcal C\to\mathcal P\) the Yoneda embedding. Limits and colimits in \(\mathcal P\) are computed objectwise. An object \(A\) of a category with filtered colimits is finitely presentable if \(\operatorname{Hom}(A,-)\) preserves filtered colimits.

Lemma 4.1. In \(\mathcal P\), representable presheaves are finitely presentable, and finite colimits of finitely presentable presheaves are finitely presentable.

Proof. \(\operatorname{Hom}(yT,X)=X(T)\) by the Yoneda lemma, and filtered colimits are objectwise. If \(A=\operatorname{colim}_iA_i\) is a finite colimit, \(\operatorname{Hom}(A,-)=\lim_i\operatorname{Hom}(A_i,-)\) is a finite limit of functors preserving filtered colimits, and finite limits of sets commute with filtered colimits. \(\square\)

For a pasting scheme \(S\) with table \((h_1,b_1,\ldots,h_l)\), let \(\Lambda_S\) be the colimit in \(\mathcal P\) of the zigzag \(yd_{h_1}\leftarrow yd_{b_1}\to yd_{h_2}\leftarrow\cdots\), and \(\varphi_S:\Lambda_S\to yS\) the map induced by the designated cocone. By the Yoneda lemma, \(\operatorname{Hom}(\varphi_S,X)\) is the map of Definition 3.1. So a presheaf is a model exactly when it is orthogonal to every \(\varphi_S\): every map \(\Lambda_S\to X\) extends uniquely along \(\varphi_S\). By Lemma 4.1, the domain and codomain of each \(\varphi_S\) are finitely presentable.

Lemma 4.2 (orthogonal reflection). Let \(\Phi\) be a set of maps \(\varphi:A\to B\) in \(\mathcal P\) between finitely presentable presheaves, and \(\mathcal P_\Phi\) the full subcategory of presheaves orthogonal to every \(\varphi\in\Phi\). Then \(\mathcal P_\Phi\) is closed in \(\mathcal P\) under limits and filtered colimits, and its inclusion has a left adjoint.

Proof. Orthogonality says that \(\operatorname{Hom}(B,X)\to\operatorname{Hom}(A,X)\) is bijective. Both sides commute with limits in \(X\), and, since \(A\) and \(B\) are finitely presentable, with filtered colimits in \(X\).

For \(\varphi:A\to B\) let \(\varphi^\nabla:B\amalg_AB\to B\) be the codiagonal, and \(\Phi'=\Phi\cup\{\varphi^\nabla:\varphi\in\Phi\}\); its members have finitely presentable domains, by Lemma 4.1. A presheaf \(X\) is orthogonal to \(\varphi\) exactly when every map \(A\to X\) extends along \(\varphi\) and every map \(B\amalg_AB\to X\) extends along \(\varphi^\nabla\): the second condition says that two extensions agreeing on \(A\) are equal. Let \(X\) be a presheaf. Put \(X_0=X\) and let \(X_{n+1}\) be the pushout of \(X_n\) along the coproduct, over all \(\psi:A'\to B'\) in \(\Phi'\) and all maps \(A'\to X_n\), of the maps \(\psi\). Let \(X_\omega=\operatorname{colim}_nX_n\) and \(\eta:X\to X_\omega\) the canonical map.

\(X_\omega\) belongs to \(\mathcal P_\Phi\): a map \(A'\to X_\omega\) with \(\psi:A'\to B'\) in \(\Phi'\) factors through some \(X_n\), because \(A'\) is finitely presentable, and extends along \(\psi\) into \(X_{n+1}\). If \(Y\) belongs to \(\mathcal P_\Phi\), then \(Y\) is orthogonal also to every \(\varphi^\nabla\): a map \(B\amalg_AB\to Y\) has equal components, since they agree on \(A\), so it extends along the split epimorphism \(\varphi^\nabla\), uniquely. Hence every map \(X_n\to Y\) extends uniquely over the pushout \(X_{n+1}\), and every map \(X\to Y\) extends uniquely along \(\eta\). So \(\eta\) is a reflection of \(X\) into \(\mathcal P_\Phi\). \(\square\)

Theorem 4.3. The category \(\mathcal A\) of models is complete and cocomplete. Limits and filtered colimits of models are computed objectwise, in particular on cells. The forgetful functor \(U:\mathcal A\to\mathbf{Glob}\) has a left adjoint \(F\), and:

  1. for a pasting scheme \(S\), \(F(S)\) is the representable model \(\mathcal C(-,S)\), so the \(j\)-cells of \(F(S)\) are the operations \(d_j\to S\), and morphisms \(F(S)\to X\) are tuples of shape \(S\) in \(X\);
  2. \(F(K)\) is finitely presentable in \(\mathcal A\) for every finite globular set \(K\);
  3. \(F\) preserves colimits.

Proof. By the discussion before Lemma 4.2, \(\mathcal A=\mathcal P_\Phi\) for \(\Phi=\{\varphi_S\}\). Limits and filtered colimits are computed in \(\mathcal P\) by Lemma 4.2, hence objectwise; the colimit of a diagram of models is the reflection of its colimit in \(\mathcal P\). Restriction along \(i:\mathbb G\to\mathcal C\) from \(\mathcal P\) to \(\mathbf{Glob}\) has a left adjoint \(i_!\), the left Kan extension of presheaves, by Theorem 2.1 of the lesson on Kan extensions. Composing it with the reflection \(R\) gives \(F=Ri_!\), left adjoint to \(U\).

(1) Every designated cocone is a colimit in \(\mathcal C\), so \(\mathcal C(-,S)\) is a model, and by the Yoneda lemma morphisms of models \(\mathcal C(-,S)\to X\) correspond to \(X(S)\), the globular maps \(S\to UX\). (2) A finite globular set is a finite colimit of globes, \(i_!\) preserves colimits and sends \(d_j\) to the representable \(yd_j\), so \(i_!K\) is finitely presentable by Lemma 4.1. For a filtered colimit of models, computed in \(\mathcal P\), \[ \begin{aligned} &\operatorname{Hom}_{\mathcal A}(Ri_!K,\operatorname{colim}Y_\alpha)\\ &=\operatorname{Hom}_{\mathcal P}(i_!K,\operatorname{colim}Y_\alpha)\\ &=\operatorname{colim}\operatorname{Hom}_{\mathcal A}(Ri_!K,Y_\alpha). \end{aligned} \] (3) \(F\) is a left adjoint. \(\square\)

The colimits of \(\mathcal A\) other than filtered ones are not computed on cells. For instance the free model \(F(d_0)\) on one vertex \(x\) contains the unit operations applied to \(x\), the arrow \(x\to x\) given by any filler of the admissible pair \(d_0\rightrightarrows d_0\) (Exercise 6.3).

5. Cellular extensions and the small object argument

Put \[ \begin{gathered} D_j=F(d_j),\qquad\partial D_j=F(\partial d_j),\\ I_j=F(\partial d_j\subseteq d_j),\\ J_j=F(\sigma_j):D_j\to D_{j+1}, \end{gathered} \] and \(I=\{I_j:j\ge0\}\), \(J=\{J_j:j\ge0\}\). A morphism \(D_j\to X\) is a \(j\)-cell of \(X\), and a morphism \(\partial D_j\to X\) is a pair of parallel \((j-1)\)-cells, a boundary. Attaching a cell along a boundary is a pushout of \(I_j\); the pushout of \(J_j\) along a \(j\)-cell \(a\) adjoins a new \(j\)-cell \(a'\) parallel to \(a\) and a \((j+1)\)-cell \(a\to a'\), and is called an elementary expansion.

For a set \(K\) of morphisms of \(\mathcal A\), a relative \(K\)-cellular extension is a transfinite composite of pushouts of coproducts of members of \(K\): a chain \(X=X_0\to X_1\to\cdots\to X_\lambda\), indexed by an ordinal, whose successor steps are such pushouts and which is continuous at limit ordinals. A model is cellular if its map from the initial model \(F(\varnothing)\) is a relative \(I\)-cellular extension. Each \(J_j\) is a composite of two boundary attachments, one of \(I_j\) adjoining \(a'\) and one of \(I_{j+1}\) adjoining the joining cell; so relative \(J\)-cellular extensions are relative \(I\)-cellular. For every globular set \(K\), \(F(K)\) is cellular: \(K\) is obtained from \(\varnothing\) by attaching its \(k\)-cells along their boundaries for \(k=0,1,2,\ldots\), and \(F\) preserves these pushouts and the sequential colimit.

A morphism \(p\) has the right lifting property against \(k:A\to B\) if every commutative square with \(k\) on the left and \(p\) on the right has a diagonal \(B\to\) (domain of \(p\)) making both triangles commute.

Proposition 5.1 (small object argument). Let \(K\) be a set of morphisms of \(\mathcal A\) with finitely presentable domains. Every morphism \(f:X\to Y\) factors as \(X\xrightarrow{a}Z\xrightarrow{p}Y\), where \(a\) is a countable relative \(K\)-cellular extension and \(p\) has the right lifting property against every member of \(K\).

Proof. Put \(Z_0=X\) and \(p_0=f\). Given \(p_n:Z_n\to Y\), let \(L_n\) be the set of commutative squares from members \(k:A\to B\) of \(K\) to \(p_n\). Let \(Z_{n+1}\) be the pushout of \(Z_n\) along \(\coprod_{L_n}A\to\coprod_{L_n}B\), with \(p_{n+1}\) induced by the squares. Let \(Z=\operatorname{colim}_nZ_n\), with \(a\) and \(p\) induced. For a square from \(k:A\to B\) to \(p\), the map \(A\to Z\) factors through some \(Z_n\) since \(A\) is finitely presentable, and the square at stage \(n\) belongs to \(L_n\); its pushout supplies a diagonal \(B\to Z_{n+1}\to Z\). \(\square\)

By Theorem 4.3(2) the domains of \(I\) and \(J\) are finitely presentable, so Proposition 5.1 applies to every subset of \(I\) or of \(J\). These are algebraic constructions; no homotopical statement is made about them.

6. Exercises

6.1 (easy). Describe the tuples of shape \(d_1\amalg_{d_0}d_1\amalg_{d_0}d_1\) and of shape \(d_2\amalg_{d_0}d_1\) in a globular set.

6.2 (easy). Show that a morphism of models is an isomorphism if and only if its underlying globular map is bijective in every dimension.

6.3 (easy). Show that the free model \(F(d_0)\) on one vertex has at least one \(1\)-cell, so \(U F(d_0)\ne d_0\).

6.4 (medium). Show that a model \(X\) is terminal in \(\mathcal A\) if and only if it has exactly one cell in each dimension.

6.5 (medium). Show that a retract, in \(\mathcal A\), of a morphism with the right lifting property against a morphism \(k\) again has that property.

7. Solutions

6.1. A tuple of shape \(d_1\amalg_{d_0}d_1\amalg_{d_0}d_1\) is a string \((f,g,h)\) of arrows with \(t f=s g\) and \(t g=s h\). A tuple of shape \(d_2\amalg_{d_0}d_1\) is a \(2\)-cell \(\alpha\) and an arrow \(g\) with \(t_0\alpha=s g\): a \(2\)-cell followed by a whisker.

6.2. If the underlying map \(f\) is bijective, its inverse is a globular map, and it preserves every operation \(\theta\), because \(f\theta_Xf^{-1}=\theta_Yff^{-1}=\theta_Y\) on tuples; so it is a morphism of models by Proposition 3.2. The converse is clear.

6.3. By contractibility, the admissible pair \(d_0\rightrightarrows d_0\) consisting of the identity twice has a filler \(e:d_1\to d_0\). It is a \(1\)-cell of \(F(d_0)=\mathcal C(-,d_0)\), by Theorem 4.3(1). Its source and target are the vertex.

6.4. The terminal presheaf has one element at every object, and it is a model; limits are objectwise, so the terminal model is it, with one cell in each dimension. Conversely, a model with one cell in each dimension has one tuple of each shape, so it is the terminal presheaf.

6.5. Let \(p'\) be a retract of \(p\): maps \(p'\to p\to p'\) of arrows composing to the identity. Given a square from \(k\) to \(p'\), compose it with \(p'\to p\), lift against \(p\), and compose the lift with the map back from the domain of \(p\) to the domain of \(p'\). Both triangles commute because the composite of the two maps of arrows is the identity.

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