Coherent integer actions from commuting equivalences

An autoequivalence supplies inverse functors, but a group action also needs coherent comparison maps between all composites. For several generators, the given commutation maps must survive the construction. Sorting positive words addresses one problem; extending the action to every signed integer vector addresses the other.

We use the tensor structure on endofunctors from Free tensor words and ordered algebra encoding. Composition of functors is strictly associative. Natural transformations are composed from right to left; juxtaposing a functor with a transformation means whiskering. All categories and Hom sets lie in suitable fixed universes.

1. The prescribed action problem

A normalized action of an additive group \(G\) on \(\mathcal C\) consists of functors \(\Psi_g\), with \(\Psi_0=1_{\mathcal C}\), and natural isomorphisms \[ \mu_{g,h}:\Psi_g\Psi_h\longrightarrow\Psi_{g+h}. \] The unit comparisons are identities, and associativity requires \[ \begin{aligned} &\mu_{g+h,k}(\mu_{g,h}\Psi_k)\\ &\qquad=\mu_{g,h+k}(\Psi_g\mu_{h,k}). \end{aligned}\tag{1.1} \] Each \(\Psi_g\) is automatically an equivalence: \(\Psi_{-g}\) is a two-sided inverse up to the comparisons to \(\Psi_0\). This is the canonical monoidal-functor definition of a group action in EGNO, Definition 2.7.1, with its comparison \(J=\mu\). The co-tensor comparison used in the preceding lessons is \(\xi=\mu^{-1}\).

Fix a nonnegative integer \(n\), autoequivalences \(T_1,\ldots,T_n\) of \(\mathcal C\), and natural isomorphisms \[ \varphi_{ij}:T_iT_j\longrightarrow T_jT_i \qquad(1\leq i<j\leq n). \] For each \(i<j<k\), assume the two maps \(T_iT_jT_k\to T_kT_jT_i\) agree: \[ \begin{gathered} (\varphi_{jk}T_i)(T_j\varphi_{ik}) (\varphi_{ij}T_k)\\ {}=(T_k\varphi_{ij})(\varphi_{ik}T_j) (T_i\varphi_{jk}). \end{gathered}\tag{1.2} \]

Theorem 1.1. There is a normalized action of \(\mathbb Z^n\) on \(\mathcal C\) such that, for every standard basis vector \(e_i\), \[ \Psi_{e_i}=T_i \] as the actual given functor, and its induced exchange is the prescribed map: \[ \mu_{e_j,e_i}^{-1}\mu_{e_i,e_j} =\varphi_{ij}.\tag{1.3} \] For \(n=0\), this means the identity action of the trivial group. For \(n=1\), an arbitrary autoequivalence gives an integer action. For \(n=2\), any given \(\varphi_{12}\) works; there is no triple condition to check.

Deligne's Proposition 1.9 extends an action of an Ore monoid by autoequivalences to its group of fractions; its Theorem 1.5 gives the positive coherence construction. Here the monoid is \(\mathbb N^n\): it is cancellative, any two vectors have common upper bound \(p+q\), and its fraction group is \(\mathbb Z^n\). We give the complete construction for this commuting case in §§2–4, from the given pairwise comparisons through every signed vector. Arbitrary transformations between positive actions need not extend to the group, as Exercise 4 shows.

2. Positive words and the triple equation

For \(p=(p_1,\ldots,p_n)\in\mathbb N^n\), choose the ordered product \[ F_p=T_1^{p_1}\cdots T_n^{p_n}, \qquad F_0=1_{\mathcal C}. \] These are autoequivalences. In a word of generator indices, replace an adjacent inversion \(ji\), with \(j>i\), by \(ij\), using \(\varphi_{ij}^{-1}\) at that position. Each replacement reduces the inversion number by one, so every sorting sequence terminates at the same ordered word.

The resulting natural isomorphism to the ordered word is independent of the sorting sequence. Here is the full local check. Two disjoint replacements commute by the interchange law for natural transformations. Two replacements sharing a position can occur only in a strictly descending triple \(kji\), with \(k>j>i\). The two three-step sorts of that triple agree by the inverse of (1.2). Equal adjacent indices are never replaced, so supply no further overlap.

Induct on the inversion number to pass from these local checks to any two sorting sequences. If their first replacements agree, apply induction to the resulting word. Otherwise the local square or triple diagram gives a common word after two or three replacements, with equal composites from the original word. Each intermediate word has smaller inversion number. Induction identifies each remaining sorting path with a path through that common word and then to the ordered word. Consequently both original composites agree. This proves independence as an equality of natural transformations, rather than only an equality of final words.

Sort the concatenation of the ordered words for \(p,q\). Denote the resulting comparison by \[ \alpha_{p,q}:F_pF_q\longrightarrow F_{p+q}. \tag{2.1} \] Empty concatenations give \(\alpha_{0,p}=\alpha_{p,0}=1\). The two sides of \[ \begin{aligned} &\alpha_{p+q,r}(\alpha_{p,q}F_r)\\ &\qquad=\alpha_{p,q+r}(F_p\alpha_{q,r}) \end{aligned}\tag{2.2} \] are sorting sequences of the same three concatenated words. The independence just proved makes them equal. Thus \(F,\alpha\) is a normalized positive action. For \(i<j\) its generator comparisons are \[ \alpha_{e_i,e_j}=1,\qquad \alpha_{e_j,e_i}=\varphi_{ij}^{-1}. \tag{2.3} \]

The triple equation is also necessary when a group action induces the specified exchanges. For any three generator indices \(a,b,c\), collapse their product using \[ v_{abc} =\mu_{e_a+e_b,e_c} (\mu_{e_a,e_b}\Psi_{e_c}). \] Equation (1.1) gives the same collapse using the last two factors first. Following an adjacent prescribed exchange and then collapsing the new word equals \(v_{abc}\): use (1.3) to cancel the two generator comparisons, and use (1.1) when exchanging the last two factors. All six words collapse to the same \(\Psi_{e_i+e_j+e_k}\). Therefore each exchange is the inverse of the new collapse followed by the old collapse. These factors telescope along either three-step route from \(ijk\) to \(kji\), leaving the same map \(v_{kji}^{-1}v_{ijk}\). This proves (1.2).

3. Checking the Ore extension on an integer lattice

We now use only the normalized positive action \(F,\alpha\) just constructed. The following explicit category supplies the negative powers, without choosing separate commutation maps for every pair of signed powers.

Let \(\mathcal D\) have objects \((u,X)\), with \(u\in\mathbb Z^n\) and \(X\in\mathcal C\). Order integer vectors componentwise. For two objects put \[ \begin{gathered} \operatorname{Hom}_{\mathcal D} ((u,X),(v,Y))\\ =\mathop{\operatorname{colim}}_{w\geq u,v} \operatorname{Hom}_{\mathcal C} (F_{w-u}X,F_{w-v}Y). \end{gathered}\tag{3.1} \] The indices form a nonempty directed poset: coordinatewise maxima give common upper bounds. In particular all the differences used in \(F\) are nonnegative.

For \(w'=w+s\), transport a representative \(f\) at \(w\) to the representative \[ \begin{gathered} K_s(f)= \alpha_{s,w-v}\,F_s(f)\, \alpha_{s,w-u}^{-1}. \end{gathered}\tag{3.2} \] This has source \(F_{w'-u}X\) and target \(F_{w'-v}Y\). The identity transport is the identity. To check two successive transports, substitute (3.2) twice; equation (2.2) combines the two left comparisons and the two right comparisons, while naturality of \(\alpha_{t,s}\) moves it past \(F_tF_s(f)\). The intermediate comparisons cancel. The result is \(K_tK_s(f)=K_{t+s}(f)\), with the current stage understood at each step. Thus (3.1) is an ordinary diagram of sets, not a diagram whose transition compositions hold only up to isomorphism.

Every transition is a bijection: \(F_s\) is fully faithful and the two surrounding comparisons are isomorphisms. Compose two morphisms by representing both at a common upper stage and composing there in \(\mathcal C\). Transports preserve composition and identities by functoriality of \(F_s\) and cancellation of the intermediate comparison. Replacing representatives gives the same composite after a further common transport. Associativity follows by representing three arrows at one common stage. Identity morphisms are represented by identities at any stage. This proves that \(\mathcal D\) is a category. Its Hom sets are small under the stated universe convention.

Define \[ j:\mathcal C\longrightarrow\mathcal D, \qquad j(X)=(0,X). \] Represent \(j(f)\) at stage \(0\) by \(f\). In its Hom system, every other stage is \(w\geq0\); transport from \(0\) is \(f\mapsto F_w(f)\), a bijection by full faithfulness. Hence \(j\) is fully faithful.

It is essentially surjective too. Given \((u,X)\), choose \(w\geq u,0\). Since \(F_w\) is an equivalence, choose \(Y\) and an isomorphism \[ F_wY\simeq F_{w-u}X. \] At stage \(w\) this represents an isomorphism \(j(Y)\to(u,X)\); its inverse represents the inverse in \(\mathcal D\). Thus \(j\) is an equivalence.

For \(g\in\mathbb Z^n\), define a translation \[ \sigma_g(u,X)=(u-g,X). \] On a morphism represented by \(f\) at stage \(w\), keep \(f\) and replace its stage by \(w-g\). All stage differences are unchanged, so this is well-defined and respects (3.2) and composition. These translations give a strict group action: \[ \sigma_g\sigma_h=\sigma_{g+h}, \qquad \sigma_0=1_{\mathcal D}. \tag{3.3} \] Their inverses are literally \(\sigma_{-g}\).

The sign is chosen to match the positive generators. For \(p\geq0\), there is a natural isomorphism \[ \kappa_p:\sigma_pj\longrightarrow jF_p. \tag{3.4} \] At \(X\), its source is \((-p,X)\), its target is \((0,F_pX)\), and it is represented at stage \(0\) by \(1_{F_pX}\). Its inverse is represented by the same identity with source and target interchanged. Naturality follows by transporting \(j(f)\) to stage \(0\), which gives \(F_p(f)\).

These identifications preserve the positive comparisons: \[ \begin{gathered} \kappa_{p+q} =(j\alpha_{p,q}) (\kappa_pF_q)(\sigma_p\kappa_q). \end{gathered}\tag{3.5} \] Indeed \(\sigma_p\kappa_q\) is initially represented at stage \(-p\) by the identity of \(F_qX\). Transporting it to \(0\) gives \(\alpha_{p,q}^{-1}\), by (3.2). At \(0\), \(\kappa_pF_q\) is the identity and \(j\alpha_{p,q}\) is \(\alpha_{p,q}\). Their composite is the identity of \(F_{p+q}X\), exactly the representative of \(\kappa_{p+q}\). This checks the positive comparisons before transporting the group action back.

4. Returning to the given category and the given functors

Choose a quasi-inverse \(Q:\mathcal D\to\mathcal C\) normalized so that \[ Qj=1_{\mathcal C}. \] This normalization can be made literally. For each \(d\in\mathcal D\), choose \(Qd\) and an isomorphism \(\epsilon_d:jQd\to d\). At \(d=jX\), choose \(Qd=X\) and \(\epsilon_d=1\). For an arrow \(f:d\to d'\), define \(Q(f)\) as the unique arrow satisfying \[ jQ(f)=\epsilon_{d'}^{-1}f\epsilon_d. \] Existence and uniqueness use full faithfulness of \(j\); cancellation proves identities and composition. These choices give a natural isomorphism \(\epsilon:jQ\to1_{\mathcal D}\) with \[ \epsilon j=1,\qquad Q\epsilon=1. \tag{4.1} \] The second identity follows by applying the definition of \(Q\) to \(\epsilon_d\): both conjugating factors cancel.

Put \[ G_g=Q\sigma_gj. \] For arbitrary signed \(g,h\), define \[ \begin{gathered} b_{g,h}=Q\sigma_g(\epsilon_{\sigma_hj}),\\ G_gG_h\longrightarrow G_{g+h}. \end{gathered} \tag{4.2} \] These are natural isomorphisms; every \(G_g\) is an equivalence. Equation (4.1) gives \(G_0=1\) and both unit comparisons equal to the identity.

Here is the complete associativity check. Fix \(X\) and set \(d=\sigma_kjX\). Before applying \(Q\sigma_g\), the left and right routes in the action equation are the two sides of \[ \begin{gathered} (\sigma_h\epsilon_d)\, \epsilon_{\sigma_hjQd}\\ {}=\epsilon_{\sigma_hd}\, jQ(\sigma_h\epsilon_d). \end{gathered}\tag{4.3} \] This is naturality of \(\epsilon\) for the arrow \(\sigma_h\epsilon_d:\sigma_hjQd\to\sigma_hd\). After applying \(Q\sigma_g\), the left side is \(b_{g+h,k}(b_{g,h}G_k)\), and the right side is \(b_{g,h+k}(G_gb_{h,k})\). Thus (1.1) holds for all integer vectors, including mixed signs.

For \(p\geq0\), let \[ \gamma_p=Q\kappa_p:G_p\longrightarrow F_p. \] These isomorphisms compare the positive actions: \[ \begin{gathered} \gamma_{p+q}b_{p,q}\\ {}=\alpha_{p,q}(\gamma_pF_q)(G_p\gamma_q). \end{gathered}\tag{4.4} \] To verify it, apply \(Q\) to (3.5) and compose with \(Q\sigma_p\epsilon_{\sigma_qj}\). Naturality of \(\epsilon\) for \(\kappa_q\), together with \(\epsilon_{jF_q}=1\), says \[ \kappa_q\,\epsilon_{\sigma_qj}=j\gamma_q. \] The resulting equality is exactly (4.4).

We can now retain the original positive functors, rather than only naturally isomorphic replacements. Define \[ \Psi_g= \begin{cases} F_g,&g\in\mathbb N^n,\\ G_g,&g\notin\mathbb N^n. \end{cases} \] Choose \(\theta_g:\Psi_g\to G_g\) to be \(\gamma_g^{-1}\) in the first case and the identity in the second. In particular \(\theta_0=1\). Transport the comparisons: \[ \begin{gathered} \mu_{g,h} =\theta_{g+h}^{-1}\,b_{g,h}\, (\theta_g*\theta_h). \end{gathered}\tag{4.5} \] Here \(*\) is horizontal composition of transformations. Its source is \(\Psi_g\Psi_h\) and target \(G_gG_h\). Naturality and the interchange law cancel the intermediate \(\theta\)'s in the two three-factor routes; associativity of \(b\) therefore gives (1.1), and its unit identities remain identities.

For positive \(g,h\), (4.4) gives \(\mu_{g,h}=\alpha_{g,h}\). Consequently \(\Psi_{e_i}=F_{e_i}=T_i\) literally, and (2.3) gives \[ \mu_{e_j,e_i}^{-1}\mu_{e_i,e_j} =(\varphi_{ij}^{-1})^{-1}=\varphi_{ij}. \] This proves Theorem 1.1 with every prescribed comparison intact. When \(n=0\), all words and lattice vectors are empty, and the construction is simply the identity action. No additional negative-power condition is required.

5. Graded exercises with complete solutions

Exercise 1 (easy). Let \(H\) be a group and \(\tau\in\operatorname{Aut}(H)\). Regard \(H\) as a one-object category \(BH\), with composition \(b\circ a=ba\). Construct its strict integer action induced by \(\tau\). Describe all equivariant objects and the equation for a morphism between two of them.

Solution. Let \(\Psi_r\) act on arrows by \(\tau^r\), for every signed integer \(r\). The equality \(\tau^r\tau^s=\tau^{r+s}\) makes every comparison the identity. An equivariant object is the unique underlying object together with invertible arrows \(u_r\in H\) satisfying \[ u_{r+s}=u_r\,\tau^r(u_s),\qquad u_0=1. \] It is determined by \(u=u_1\). For \(r>0\), \[ \begin{gathered} u_r=u\,\tau(u)\cdots\tau^{r-1}(u),\\ u_{-r}=\tau^{-r}(u_r^{-1}). \end{gathered} \] These formulas satisfy the equation for all signs: they are the first coordinates of \((u,1)^r\) in the group with multiplication \((a,r)(b,s)=(a\tau^r(b),r+s)\). Associativity follows from \(\tau^{r+s}=\tau^r\tau^s\), and the inverse is \((\tau^{-r}(a^{-1}),-r)\). Conversely the equivariance equation successively forces these positive and negative formulas, so every \(u\in H\) gives exactly one equivariant object. A morphism \(h:u\to v\) satisfies \[ h\,u=v\,\tau(h). \] Induction gives the corresponding equation for positive powers. Taking the inverse-power formulas gives it for negative powers; the equation at zero is automatic. Thus the generator equation is the complete morphism condition.

Exercise 2 (moderate). In \(\operatorname{Mod}(k)\), for a commutative ring \(k\), take \(T_1=T_2=1\) and prescribe \(\varphi_{12}=q\,1\), with \(q\in k^\times\). Give an explicit action on every signed pair \((a,b)\in\mathbb Z^2\) realizing this exchange.

Solution. Let every \(\Psi_{(a,b)}\) be the identity functor and put \[ \mu_{(a,b),(c,d)}=q^{-bc}\,1. \] Negative exponents are defined because \(q\) is a unit. These are natural isomorphisms. A zero vector makes the exponent zero, so the unit comparisons are identities. For a third vector \((e,f)\), the two action composites have exponents \[ -bc-(b+d)e=-de-b(c+e), \] proving associativity over the entire integer lattice. The forward generator comparison is \(1\); the reverse one is \(q^{-1}\). Their exchange is therefore \(q\), as required. Replacing every comparison by the identity would give exchange \(1\), so would lose the prescribed data unless \(q=1\). This example shows that even identity generator functors can carry nontrivial comparison data.

Exercise 3 (hard). Show that pairwise commutation isomorphisms alone do not suffice for three generators. Use the one-object category \(B\mathbb Z\), with additive composition, the functors \(T_1(a)=-a\), \(T_2(a)=T_3(a)=a\), and components \[ \varphi_{12}=0,\qquad \varphi_{13}=0,\qquad \varphi_{23}=1. \]

Solution. All three functors are automorphisms and commute strictly. Since \(\mathbb Z\) is abelian, any integer is the component of a natural automorphism between each equal composite functor: the naturality equation is equality after adding that integer on either side. Its inverse component is its negative. Thus the three prescribed maps are valid pairwise isomorphisms; zero denotes an identity arrow, whereas \(1\) is a nontrivial arrow.

Right whiskering keeps the component; left whiskering by \(T_1\) negates it. The first route in (1.2) has component \[ 0+T_2(0)+1=1. \] The second has component \[ T_1(1)+0+T_3(0)=-1. \] They are different arrows. The necessity proof in §2 therefore rules out any \(\mathbb Z^3\)-action with these exact generator functors and these prescribed exchanges. Passing only to isomorphism classes of functors would miss the obstruction.

Exercise 4 (expert). Identify the roles of invertibility in the extension. (a) Suppose the normalized positive action in §3 has fully faithful \(F_p\), without assuming essential surjectivity. Prove that \(j\) is fully faithful and is an equivalence exactly when every \(T_i=F_{e_i}\) is essentially surjective. Compute \(\mathcal D\) for the shift \(T(k)=k+1\) on the discrete category \(\mathbb N\). (b) For the positive identity action on \(\operatorname{Mod}(\mathbb Z)\), classify its unit-preserving tensor endomorphisms and decide which extend to the identity action of \(\mathbb Z\).

Solution. (a) Full faithfulness alone makes all Hom transitions bijections, so the construction of \(\mathcal D\) and full faithfulness of \(j\) still work. If every \(T_i\) is essentially surjective, they are equivalences and so are all \(F_p\); the essential-surjectivity proof in §3 applies.

Conversely suppose \(j\) is essentially surjective. For any \(X\), the object \((e_i,X)\) is isomorphic to some \(jY\). Represent the isomorphism and its inverse at a common stage. Equality of their composites to the identities holds there because the Hom transitions are injective. Thus at some \(w\geq e_i,0\) there is an actual isomorphism \[ F_{w-e_i}X\simeq F_wY. \] The comparison \(\alpha_{w-e_i,e_i}\) identifies the right side with \(F_{w-e_i}T_iY\). A fully faithful functor reflects isomorphisms: lift the inverse by fullness and check the two inverse equations by faithfulness. Hence \(X\simeq T_iY\). Every \(T_i\) is essentially surjective.

For the discrete shift, objects of \(\mathcal D\) are \((u,k)\), with \(u\in\mathbb Z\) and \(k\geq0\). At a common stage \(w\), a morphism exists exactly when \[ \begin{gathered} k+w-u=\ell+w-v,\\ \text{or equivalently}\quad k-u=\ell-v. \end{gathered} \] Then there is exactly one morphism, and it is invertible. Thus \(\mathcal D\) is equivalent to the discrete category \(\mathbb Z\), via \((u,k)\mapsto k-u\). The image of \(j\) is the nonnegative integers and is not essentially surjective. Translation \(\sigma_r\) adds \(r\) to this integer label. It extends the shift on the enlarged category, but cannot yield an integer action on the original category with generator \(T\), since a group-action generator must be an equivalence and this shift misses \(0\).

(b) Every natural endomorphism of the identity on \(\operatorname{Mod}(\mathbb Z)\) is multiplication by one integer. Indeed its value on \(\mathbb Z\) is multiplication by \(t\); naturality for the map \(\mathbb Z\to M\) sending \(1\) to any \(x\in M\) forces the value on \(x\) to be \(tx\). Conversely scalar multiplication is natural.

A unit-preserving positive-action endomorphism therefore has scalars \(t_m\), with \[ t_0=1,\qquad t_{r+s}=t_rt_s. \] It is precisely \(t_m=t^m\) for one arbitrary \(t\in\mathbb Z\). It extends to all signed integers exactly when \(t\) is a unit, hence \(t=1\) or \(t=-1\). Necessity follows from \(t_1t_{-1}=t_0=1\); sufficiency follows by using integer powers of the unit. In particular \(2^m\) gives a valid positive transformation between actions whose functors are all equivalences, and cannot extend to a group transformation. The object-level existence theorem proved in §§2–4 remains valid; an equivalence of action categories cannot include all such noninvertible transformations.

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