Graded monoids and the nerves of their actions

Written by Claude Opus 5.5 (Anthropic) on 21 October 2026. Self-checked by the writing AI. Public domain (CC0).

The Ext groups of modules over a ring \(A\) of a topos can be computed as the cohomology of a simplicial topos whose simplices are built from the products \(A^p\times M^q\) of the underlying objects of \(A\) and of a module \(M\). Products see only the underlying objects, so the additive structure has to be recorded by maps between objects. The spectrum of an abelian group object, the family of its iterated nerves, does this (Multisimplicial objects and spectral diagrams, Section 3). The spectra of a ring and of a module are a graded monoid and a graded object with an action of it, graded over the monoidal category of finite ordinals.

This lesson develops the language for such graded actions. A grading is a functor \(J\) from a monoidal category of indices to small categories, and a graded object is a functor on the total category of \(J\); graded monoids and their actions are monoids and actions in the category of diagrams. To a grading \(J\) one attaches a simplicial category \(C_\bullet(J)\), whose \(n\)-simplices are strings of \(n\) arrows of indices with an object of \(J\) at the start. Graded objects are exactly the equivariant sheaves on the associated simplicial topos. An action of a graded monoid \(G\) on a graded object \(X\) gives a bisimplicial diagram, the nerve \(\operatorname{Ner}(C_\bullet(G),C_\bullet(X))\), which determines \(X\). When the products of \(J\) are isomorphisms, its simplicial topos is a pseudo-category whose levels are localizations of a single topos. The lesson Graded rings and the spectra of modules applies this to rings and modules.

Prerequisites. Section 2.5 of Décalage, nerves and the homotopy type of a category (nerves of monoid actions, Propositions 2.4 and 2.9); Examples 1.2 (c) and Section 5 of Fibred topoi and their total topoi; Sections 1–3 of Equivariant sheaves on simplicial topoi (equivariant sheaves, their second nerves, base change maps and pseudo-categories); Section 2 of Multisimplicial objects and spectral diagrams (the categories \(\mathbb S_a\)). From the AI-integrated Stacks project: limits and colimits of sheaves [Stacks, Tag 00W2, Tag 00WI] and exactness of sheafification [Tag 00WJ]; localization of a topos [Tag 04GZ] and of a morphism of topoi [Tag 04H1].

Conventions

\(T\) is a topos with final object \(e\). Sums in \(T\) are disjoint and stable under base change. Indeed, for sheaves on a site, limits are computed on sections, colimits are the sheafifications of the colimits of presheaves, and sheafification commutes with finite limits [Tags 00W2, 00WI, 00WJ]. So both properties follow from the same properties of sums of sets.

For a small category \(K\), \(T^K\) is the topos of functors \(K\to T\) (Section 5 of the lesson on fibred topoi). A functor \(\varphi\colon K'\to K\) gives a morphism of topoi \(T^{K'}\to T^K\) with inverse image \(\varphi^*F=F\circ\varphi\). The functor \(\varphi^*\) has the left Kan extension \(\varphi_!\) as a left adjoint and the right Kan extension as a right adjoint; they exist because \(T\) has small colimits and limits.

Diagrams. A diagram of \(T\) is a pair \((K,F)\) of a small category \(K\), its shape, and a functor \(F\colon K\to T\). A morphism \((K',F')\to(K,F)\) is a pair \((\varphi,u)\) of a functor \(\varphi\colon K'\to K\), its type, and a natural transformation \(u\colon F'\to F\circ\varphi\). The composite of \((\psi,v)\) and \((\varphi,u)\) is \((\psi\varphi,(v\varphi)\circ u)\). These form a category \(\operatorname{Diagr}(T)\). It has a final object, the one-point category with value \(e\), and fibre products: the shape of the fibre product of \((K_1,F_1)\to(K,F)\leftarrow(K_2,F_2)\) is the fibre product \(K_1\times_KK_2\) of categories, and its value at \((k_1,k_2)\) is \(F_1(k_1)\times_{F(k)}F_2(k_2)\), where \(k\) is the common image of \(k_1\) and \(k_2\). In particular \((K,F)\times(K',F')=(K\times K',F\boxtimes F')\) with \((F\boxtimes F')(k,k')=F(k)\times F'(k')\). The same definitions apply to any category \(\mathcal C\) with finite products in place of \(T\), and \(\operatorname{Diagr}(\mathcal C)\) has finite products.

Induced morphisms of topoi. A morphism \((\varphi,u)\colon(K',F')\to(K,F)\) of \(\operatorname{Diagr}(T)\) induces a morphism of topoi \(T^{K'}/F'\to T^K/F\) whose inverse image is \[ (E\to F)\longmapsto\bigl(\varphi^*E\times_{\varphi^*F}F'\to F'\bigr). \] It is the composite of the localization of the topos \(T^{K'}/\varphi^*F\) at its object \(u\colon F'\to\varphi^*F\) [Tag 04GZ] and the morphism \(T^{K'}/\varphi^*F\to T^K/F\) induced by \(\varphi\) on slices [Tag 04H1]. Its inverse image has a left adjoint, which sends \(E'\to F'\) to \(\varphi_!E'\), with the map \(\varphi_!E'\to F\) adjoint to the composite \(E'\to F'\to\varphi^*F\). Indeed, maps \(E'\to\varphi^*E\times_{\varphi^*F}F'\) over \(F'\) are the maps \(E'\to\varphi^*E\) over \(\varphi^*F\), and these correspond to the maps \(\varphi_!E'\to E\) over \(F\). For composable morphisms of diagrams, the induced morphisms of topoi compose up to the canonical isomorphisms between iterated fibre products. As in Examples 1.2 (c) of the lesson on fibred topoi, a functor from a small category \(L\) to \(\operatorname{Diagr}(T)\) therefore defines a topos fibred over \(L\), and this fibred topos is good.

Grothendieck constructions. For a functor \(C\colon L\to\mathbf{Cat}\), \(l\mapsto C_l\), \(\lambda\mapsto C_\lambda\), the total category \(\int C\) has the objects \((l,k)\) with \(k\in C_l\), and the arrows \((\lambda,b)\colon(l,k)\to(l',k')\) with \(\lambda\colon l\to l'\) and \(b\colon C_\lambda(k)\to k'\). The composite of \((\lambda',b')\) and \((\lambda,b)\) is \((\lambda'\lambda,\,b'\circ C_{\lambda'}(b))\). A functor \(E\colon L\to\operatorname{Diagr}(T)\) such that \(E(l)\) has shape \(C_l\) and \(E(\lambda)\) has type \(C_\lambda\) is the same as a functor \(\int C\to T\). Its value at \((l,k)\) is \(E(l)(k)\), and the arrow \((\lambda,b)\) goes to \(E(l')(b)\circ u_\lambda(k)\), where \(u_\lambda\colon E(l)\to E(l')\circ C_\lambda\) is the transformation of \(E(\lambda)\). Natural transformations of identity type between two such functors are the natural transformations of functors \(\int C\to T\). Hence, for one such functor \(F\), the functors \(E\) of the same shapes and types with a morphism \(E\to F\) of identity type form the topos \(T^{\int C}/F\). It is the total category \(\operatorname{Top}^{\mathrm o}\) of the fibred topos \(l\mapsto T^{C_l}/F(l)\) defined by \(F\), by the argument of Proposition 5.1 of the lesson on fibred topoi: a map \(E(l)\to C_\lambda^*E(l')\times_{C_\lambda^*F(l')}F(l)\) over \(F(l)\) is a map \(E(l)\to C_\lambda^*E(l')\) lying over \(u_\lambda\).

Discrete opfibrations. A functor \(\varphi\colon K'\to K\) is a discrete opfibration if for every object \(k'\) of \(K'\) and every arrow \(b\colon\varphi(k')\to k\) of \(K\) there is exactly one arrow \(b'\) of \(K'\) with source \(k'\) and \(\varphi(b')=b\). The facts used about them are proved in Lemma 3.1.

1. Graded categories, graded objects and graded monoids

Let \(I\) be a small strict monoidal category, written additively: a functor \(+\colon I\times I\to I\), strictly associative, with a strict unit object \(o\). It need not be commutative.

Definition 1.1. An \(I\)-graded category is a functor \(J\colon I\to\mathbf{Cat}\), \(i\mapsto J_i\), \(f\mapsto J_f\), into small categories. Its total category \(\int J\) has the objects \((i,x)\) with \(x\in J_i\), and the morphisms \((f,u)\colon(i,x)\to(i',x')\) with \(f\colon i\to i'\) in \(I\) and \(u\colon J_f(x)\to x'\) in \(J_{i'}\). A \(J\)-graded object of a category \(\mathcal C\) is a functor \(\int J\to\mathcal C\).

A \(J\)-graded object \(X\) is the same as a family of functors \(X_i=X(i,-)\colon J_i\to\mathcal C\) with natural transformations \(X_f\colon X_i\to X_{i'}\circ J_f\) for \(f\colon i\to i'\), such that \(X_{\mathrm{id}}=\mathrm{id}\) and \(X_{f'f}=(X_{f'}J_f)\circ X_f\). The component of \(X_f\) at \(x\) is \(X(f,\mathrm{id})\colon X(i,x)\to X(i',J_fx)\), and \(X(f,u)=X(i',u)\circ X(f,\mathrm{id})\).

Example 1.2. Let \(I=\bar\Delta'\), the category of the ordinals \([n-1]=\{0,\ldots,n-1\}\), \(n\geq0\), and the injective increasing maps, with the ordered sum \([m-1]+[n-1]=[m+n-1]\) and the unit \([-1]=\varnothing\). For \(a\in\{0,1\}\) let \(S_a\colon\bar\Delta'\to\mathbf{Cat}\) send \([n-1]\) to \((\Delta^{\mathrm{op}})^n\) and \(\delta\colon[m-1]\to[n-1]\) to the functor \(\delta_a\) of the lesson on multisimplicial objects, which puts the \(k\)-th coordinate in position \(\delta(k)\) and \([a]\) elsewhere. Then \(\int S_a\) is the category \(\mathbb S_a\) of that lesson. So an \(S_0\)-graded object of an abelian category \(\mathcal A\) is a large simplicial diagram, an object of \(\operatorname{SIMPL}(\mathcal A)\), and an \(S_1\)-graded object is a spectral diagram, an object of \(\operatorname{SPEC}(\mathcal A)\) (Section 2 of that lesson).

Definition 1.3. A monoid structure on an \(I\)-graded category \(J\) consists of functors \(\mu_{p,q}\colon J_p\times J_q\to J_{p+q}\), natural in \((p,q)\in I\times I\), that is, \(J_{f+g}\circ\mu_{p,q}=\mu_{p',q'}\circ(J_f\times J_g)\) for \(f\colon p\to p'\) and \(g\colon q\to q'\), associative, and unital for an object \(1\) of \(J_o\). We consider the hypothesis

(H) every \(\mu_{p,q}\) is an isomorphism of categories.

Under (H), \(J_o\) is the one-point category (Exercise 4.1).

The functor \(\tilde\mu\colon\int J\times\int J\to\int J\), \(\tilde\mu((p,x),(q,y))=(p+q,\mu_{p,q}(x,y))\), \(\tilde\mu((f,u),(g,v))=(f+g,\mu(u,v))\), is well defined, because \(\mu_{p',q'}(J_fx,J_gy)=J_{f+g}\mu_{p,q}(x,y)\). It makes \(\int J\) a strict monoidal category with unit \((o,1)\).

Example 1.4. The concatenation of coordinates \((\Delta^{\mathrm{op}})^m\times(\Delta^{\mathrm{op}})^n=(\Delta^{\mathrm{op}})^{m+n}\) is a monoid structure on \(S_a\). It is natural because \((\delta+\varepsilon)_a\) acts on the first \(m\) coordinates by \(\delta_a\) and on the last \(n\) by \(\varepsilon_a\), and it satisfies (H).

Definition 1.5. Let \(\mathcal C\) be a category with finite products, and let \(K\) be a small strict monoidal category with product \(\otimes\) and unit object \(1\). A \(K\)-graded monoid of \(\mathcal C\) is a monoid of \(\operatorname{Diagr}(\mathcal C)\) whose underlying object \(G\) has shape \(K\), whose multiplication has type \(\otimes\), and whose unit has type the inclusion of \(1\). It amounts to maps \[ m_{k,k'}\colon G(k)\times G(k')\to G(k\otimes k'), \] natural in \(k,k'\in K\), and a unit \(e\to G(1)\), satisfying the associativity and unit laws. A \(K\)-graded \(G\)-object is an object \(X\) of shape \(K\) with an action \(G\times X\to X\) of type \(\otimes\): maps \(G(k)\times X(k')\to X(k\otimes k')\), natural in \(k\) and \(k'\), satisfying the associativity and unit laws. Morphisms are the morphisms of identity type compatible with these maps.

If \(J\) is an \(I\)-graded category with a monoid structure, a \(J\)-graded monoid is an \(\int J\)-graded monoid, for the product \(\tilde\mu\) and the unit \((o,1)\), and a \(J\)-graded \(G\)-object is an \(\int J\)-graded \(G\)-object. The constant \(J\)-graded object \(e_J\) with value \(e\) is a \(G\)-object, the final one.

2. Simplicial categories of a grading and nerves of graded actions

In this section \(J\colon I\to\mathbf{Cat}\) is an \(I\)-graded category. Its monoid structure, when one is given, enters from 2.4 on.

Definition 2.1. For \(n\geq0\), let \(C_n(J)\) be the category whose objects are the pairs \((\sigma,x)\) of a string \(\sigma=(i_0\to i_1\to\cdots\to i_n)\) of \(n\) composable arrows of \(I\), an \(n\)-simplex of the nerve of \(I\), and an object \(x\) of \(J_{i_0}\). Its arrows \((\sigma,x)\to(\sigma,x')\) are the arrows \(x\to x'\) of \(J_{i_0}\), and there are no arrows between different strings. So \(C_n(J)\) is the sum of the categories \(J_{\sigma(0)}\) over the \(n\)-simplices \(\sigma\) of the nerve of \(I\). For an increasing map \(\alpha\colon[m]\to[n]\) put \[ C(\alpha)(\sigma,x)=\bigl(\sigma\alpha,\;J_{\sigma(0,\alpha(0))}(x)\bigr), \] and let \(C(\alpha)\) act on arrows by \(J_{\sigma(0,\alpha(0))}\). Here \(\sigma\alpha\) is the \(m\)-simplex with vertices \(i_{\alpha(0)},\ldots,i_{\alpha(m)}\), and \(\sigma(0,j)\colon i_0\to i_j\) is the composite of the arrows of \(\sigma\) between its vertices \(0\) and \(j\), the identity for \(j=0\). Since \(\sigma\alpha(0,\beta(0))\circ\sigma(0,\alpha(0))=\sigma(0,\alpha\beta(0))\), we have \(C(\beta)C(\alpha)=C(\alpha\beta)\), and \(C_\bullet(J)\) is a simplicial category.

For a \(J\)-graded object \(X\) of a category \(\mathcal C\), let \(C_n(X)\colon C_n(J)\to\mathcal C\) send \((\sigma,x)\) to \(X(i_0,x)\), and let \(C(\alpha)\colon C_n(X)\to C_m(X)\circ C(\alpha)\) be the transformation with the components \[ \begin{aligned}&X(\sigma(0,\alpha(0)),\mathrm{id}) \colon X(i_0,x)\\&\quad \to X\bigl(i_{\alpha(0)},J_{\sigma(0,\alpha(0))}x\bigr).\end{aligned} \] By functoriality of \(X\), this makes \(C_\bullet(X)\) a simplicial object of \(\operatorname{Diagr}(\mathcal C)\) of type \(C_\bullet(J)\), functorial in \(X\).

Example 2.2. If \(J\) is constant with value the one-point category, \(C_n(J)\) is the set of \(n\)-simplices of the nerve of \(I\), regarded as a discrete category. A \(J\)-graded object is a functor \(X\colon I\to\mathcal C\), and \(C_n(X)\) is the family of the objects \(X(\sigma(0))\), indexed by the \(n\)-simplices \(\sigma\).

Proposition 2.3 (graded objects as equivariant sheaves). Let \(\mathcal X=T^{C_\bullet(J)}\) be the simplicial topos with levels \(\mathcal X_n=T^{C_n(J)}\), whose transition morphisms \(\mathcal X(\alpha)\colon\mathcal X_n\to\mathcal X_m\) are induced by the functors \(C(\alpha)\), so that \(\mathcal X(\alpha)^*=C(\alpha)^*\).

1. A \(J\)-graded object \(X\) of \(T\) defines the sheaf \(L=C_0(X)\) on \(\mathcal X_0\) and the morphism \(a\colon s^*L\to t^*L\) of \(\mathcal X_1\) whose component at \((f,x)\in C_1(J)\), \(f\colon i\to i'\), is \(X(f,\mathrm{id})\colon X(i,x)\to X(i',J_fx)\). This is an equivalence between \(J\)-graded objects of \(T\) and the category \(B\mathcal X\) of equivariant sheaves on \(\mathcal X\), and \(\operatorname{ner}^{\mathrm o}(L)=C_\bullet(X)\).
2. The functor \(X\mapsto C_\bullet(X)\), from \(J\)-graded objects of \(T\) to simplicial objects of \(\operatorname{Diagr}(T)\) of type \(C_\bullet(J)\), is fully faithful. Its essential image consists of the \(E\) such that, for every \(n\), the transformation \(E_n\to E_0\circ C(v_0)\) of the vertex \(v_0\colon[0]\to[n]\) is an isomorphism. Here \(C(v_0)(\sigma,x)=(i_0,x)\), and this transformation is the composite \(d_1\cdots d_n\) of faces.

Proof. (1) The source and the target \(s=\mathcal X(v_0)\) and \(t=\mathcal X(v_1)\) have the inverse images \(C(v_0)^*\) and \(C(v_1)^*\), where \(C(v_0)(f,x)=(i,x)\) and \(C(v_1)(f,x)=(i',J_fx)\). So a morphism \(a\colon s^*L\to t^*L\) is a family of maps \(a_f(x)\colon L(i,x)\to L(i',J_fx)\), natural in \(x\in J_i\). For the surjection \(\rho\colon[1]\to[0]\) (written \(\sigma\) in the lesson on equivariant sheaves), \(C(\rho)(i,x)=(\mathrm{id}_i,x)\), so condition 1 of Definition 1.1 of the lesson on equivariant sheaves says \(a_{\mathrm{id}_i}=\mathrm{id}\). For a \(2\)-simplex \(i_0\xrightarrow{f}i_1\xrightarrow{g}i_2\) and \(x\in J_{i_0}\), the functors \(C(e_{01})\), \(C(e_{12})\) and \(C(e_{02})\) send it to \((f,x)\), \((g,J_fx)\) and \((gf,x)\). So condition 2 says \(a_g(J_fx)\circ a_f(x)=a_{gf}(x)\). These are exactly the conditions under which the functors \(L(i,-)\) and the transformations \(a_f\) form a \(J\)-graded object \(X\) with \(X(f,\mathrm{id})=a_f\) (Definition 1.1). A morphism \(u\colon L\to M\) of \(\mathcal X_0\) is compatible with the structures exactly when it is a natural transformation on \(\int J\). This gives the equivalence. Its inverse sends \(X\) to \((C_0(X),a)\) as in the statement.

By Definition 2.1 of the lesson on equivariant sheaves, \(\operatorname{ner}^{\mathrm o}(L)_n=\mathcal X(v_0)^*L\), whose value at \((\sigma,x)\) is \(L(i_0,x)=C_n(X)(\sigma,x)\). Its map for \(\alpha\) is \(a_{0\,\alpha(0)}=\mathcal X(e_{0\,\alpha(0)})^*(a)\), whose component at \((\sigma,x)\) is the component of \(a\) at \(C(e_{0\,\alpha(0)})(\sigma,x)=(\sigma(0,\alpha(0)),x)\), that is, \(X(\sigma(0,\alpha(0)),\mathrm{id})\). So \(\operatorname{ner}^{\mathrm o}(L)=C_\bullet(X)\).

(2) By the conventions, applied to the functor \([n]\mapsto(C_n(J),e)\), the simplicial objects of \(\operatorname{Diagr}(T)\) of type \(C_\bullet(J)\) form the total category \(\operatorname{Top}^{\mathrm o}(\mathcal X)\). By (1) and Propositions 2.2 (2) and 2.3 (2) of the lesson on equivariant sheaves, \(X\mapsto C_\bullet(X)\) is fully faithful, with essential image the \(E\) whose maps \(E_\alpha\) are isomorphisms whenever \(\alpha(0)=0\). This condition is equivalent to the one in the statement. If every \(E_{v_0}\) is invertible and \(\alpha\colon[m]\to[n]\) has \(\alpha(0)=0\), then \(\alpha v_0=v_0\), and the rule for composites in \(\operatorname{Top}^{\mathrm o}(\mathcal X)\) gives \(E_{v_0}=\mathcal X(\alpha)^*(E_{v_0})\circ E_\alpha\). So \(E_\alpha\) is invertible. The converse is the case \(\alpha=v_0\). \(\square\)

2.4. Monoid structures. Let \(J\) carry a monoid structure \(\mu\). Since \(+\) is a functor \(I\times I\to I\), two \(n\)-simplices \(\sigma,\tau\) of the nerve of \(I\) have a sum \(\sigma+\tau\), the string of the sums of their arrows. Put \[ \begin{aligned}&(\sigma,x)\otimes(\tau,y)\\&\quad =\bigl(\sigma+\tau,\;\mu_{\sigma(0),\tau(0)}(x,y)\bigr),\end{aligned} \] and similarly on arrows. This makes \(C_n(J)\) a strict monoidal category, with unit \((o_n,1)\), where \(o_n\) is the string of identities of \(o\). Each \(C(\alpha)\) is a strict monoidal functor, because \((\sigma+\tau)(0,\alpha(0))=\) \(\sigma(0,\alpha(0))+\tau(0,\alpha(0))\) and \(\mu\) is natural.

Let \(G\) be a \(J\)-graded monoid and \(X\) a \(J\)-graded \(G\)-object of a category \(\mathcal C\) with finite products. The maps \[ \begin{aligned}&C_n(G)(\sigma,x)\times C_n(G)(\tau,y)\\ &\quad =G(i_0,x)\times G(j_0,y)\\ &\quad \to G\bigl(i_0+j_0,\mu(x,y)\bigr),\end{aligned} \] where \(j_0=\tau(0)\), make \(C_n(G)\) a \(C_n(J)\)-graded monoid (Definition 1.5). The action of \(G\) on \(X\) makes \(C_n(X)\) a \(C_n(J)\)-graded \(C_n(G)\)-object in the same way. The maps \(C(\alpha)\colon C_n(G)\to C_m(G)\) are homomorphisms of monoids of \(\operatorname{Diagr}(\mathcal C)\), and the maps \(C(\alpha)\colon C_n(X)\to C_m(X)\) are equivariant over them. This is the naturality of the multiplication and of the action of \(G\) with respect to the arrows \((\sigma(0,\alpha(0)),\mathrm{id})\) and \((\tau(0,\alpha(0)),\mathrm{id})\) of \(\int J\), whose product under \(\tilde\mu\) is \(((\sigma+\tau)(0,\alpha(0)),\mathrm{id})\).

The category \(\operatorname{Diagr}(\mathcal C)\) has finite products, and fibre products when \(\mathcal C\) does. Section 2.5 of the lesson on décalage then gives, for each \(n\), the nerve \(N(C_n(G),C_n(X))\), a simplicial object of \(\operatorname{Diagr}(\mathcal C)\) with \[ \begin{aligned}&N_p\bigl(C_n(G),C_n(X)\bigr)\\&\quad =C_n(G)^p\times C_n(X).\end{aligned} \] A homomorphism of monoids \(\varphi\colon G'\to G''\) and a \(\varphi\)-equivariant map \(X'\to X''\) induce a functor between the internal categories \([G',X']\to[G'',X'']\), hence a map of nerves. So the maps \(C(\alpha)\) make \[ \begin{aligned}&\operatorname{Ner}\bigl(C_\bullet(G),C_\bullet(X)\bigr) \colon([p],[q])\\&\quad \longmapsto C_q(G)^p\times C_q(X)\end{aligned} \] a bisimplicial object of \(\operatorname{Diagr}(\mathcal C)\). The first index \(p\) is the direction of the nerve and the second index \(q\) that of \(C_\bullet\). The object of bidegree \((p,q)\) has shape \(C_q(J)^{p+1}\). The type of \(\operatorname{Ner}(C_\bullet(G),C_\bullet(X))\) is the bisimplicial category \(\operatorname{Ner}(C_\bullet(J),C_\bullet(J))\) of the nerves of the monoids \(C_q(J)\) acting on themselves by left multiplication. For instance the face \(d_0\) of the nerve sends \((g_1,\ldots,g_p,x)\) to \((g_2,\ldots,g_p,g_1x)\), and its type sends \((k_1,\ldots,k_p,k)\) to \((k_2,\ldots,k_p,k_1\otimes k)\).

Lemma 2.5 (change of monoid). Let \(\mathcal D\) be a category with fibre products and a final object. Let \(\varphi\colon G'\to G''\) be a homomorphism of monoids of \(\mathcal D\), \(X'\) a \(G'\)-object and \(X''\) a \(G''\)-object. Restriction to the level \(0\) is a bijection from the morphisms of simplicial objects \(N(G',X')\to N(G'',X'')\) lying over \(N(\varphi)\colon N(G')\to N(G'')\) to the \(\varphi\)-equivariant morphisms \(X'\to X''\).

Proof. Let \(\varphi^*X''\) be \(X''\) with \(G'\) acting through \(\varphi\). The isomorphisms \(G'^p\times X''\cong G'^p\times_{G''^p}(G''^p\times X'')\), \((g,x)\mapsto(g,(\varphi(g),x))\), identify \(N(G',\varphi^*X'')\) with \(N(G')\times_{N(G'')}N(G'',X'')\). This is checked on the faces: for instance \(d_0\) sends \((g_1,\ldots,g_p,x)\) to \((g_2,\ldots,g_p,\varphi(g_1)x)\) on both sides. So morphisms \(N(G',X')\to N(G'',X'')\) over \(N(\varphi)\) are morphisms \(N(G',X')\to N(G',\varphi^*X'')\) over \(N(G')\). By Proposition 2.9 of the lesson on décalage, these are induced by the \(G'\)-equivariant morphisms \(X'\to\varphi^*X''\), which are the \(\varphi\)-equivariant morphisms \(X'\to X''\). The morphism induced by such a map is the identity of \(G'^p\) times the map, so the map is recovered at the level \(0\). \(\square\)

Proposition 2.6. Let \(G\) be a \(J\)-graded monoid of \(T\). Let \(\mathcal Y_G\) be the category of bisimplicial objects \(Y\) of \(\operatorname{Diagr}(T)\) of type \(\operatorname{Ner}(C_\bullet(J),C_\bullet(J))\) with a morphism of identity type \(Y\to\operatorname{Ner}(C_\bullet(G),C_\bullet(e_J))\); its morphisms are the morphisms of identity type over \(\operatorname{Ner}(C_\bullet(G),C_\bullet(e_J))\). The functor \[ X\longmapsto\operatorname{Ner}\bigl(C_\bullet(G),C_\bullet(X)\bigr), \] from \(J\)-graded \(G\)-objects of \(T\) to \(\mathcal Y_G\), is fully faithful. Its essential image consists of the \(Y\) satisfying the following two conditions.

1. For all \(p\) and \(q\), the morphism of identity type \[ \begin{aligned}&(\pi_{p,q},\;d_1\cdots d_p) \colon Y_{p,q}\\&\quad \to C_q(G)^p\times Y_{0,q}\end{aligned} \] is an isomorphism. Here \(\pi_{p,q}\) is the structure morphism to \(C_q(G)^p\times C_q(e_J)\) followed by the projection to \(C_q(G)^p\), and \(d_1\cdots d_p\) is the restriction to the vertex \(0\) in the first index.
2. For every \(q\), the transformation \(Y_{0,q}\to Y_{0,0}\circ C(v_0)\) of the vertex \(v_0\colon[0]\to[q]\) in the second index is an isomorphism.

The category \(\mathcal Y_G\) is a topos. By the conventions, it is \(T^{\mathfrak L}/F_G\), where \(\mathfrak L\) is the total category of the functor \(([p],[q])\mapsto C_q(J)^{p+1}\) on \((\Delta\times\Delta)^{\mathrm{op}}\) and \(F_G\colon\mathfrak L\to T\) is given by \(\operatorname{Ner}(C_\bullet(G),C_\bullet(e_J))\). The hypothesis (H) is not used in this section.

Proof. Full faithfulness. Let \(X\) and \(X'\) be \(J\)-graded \(G\)-objects and \(w\colon\) \(\operatorname{Ner}(C_\bullet(G),C_\bullet(X))\to\) \(\operatorname{Ner}(C_\bullet(G),C_\bullet(X'))\) a morphism of \(\mathcal Y_G\). The morphism \(C_q(e_J)\to e\) of \(C_q(G)\)-objects of \(\operatorname{Diagr}(T)\) maps \(\operatorname{Ner}(C_q(G),C_q(e_J))\) to the nerve \(N(C_q(G))\) of the monoid, so for fixed \(q\), \(w_{\bullet,q}\) is a morphism of simplicial objects over \(N(C_q(G))\). By Lemma 2.5 with \(\varphi=\mathrm{id}\), it is induced by the \(C_q(G)\)-equivariant morphism \(w_{0,q}\colon C_q(X)\to C_q(X')\), which has identity type. Compatibility with the second simplicial structure says that \((w_{0,q})_q\) is a morphism of simplicial objects \(C_\bullet(X)\to C_\bullet(X')\) of identity type. By Proposition 2.3 (2), \(w_{0,\bullet}=C_\bullet(u)\) for a unique morphism \(u\colon X\to X'\) of \(J\)-graded objects. The action of \(C_0(G)\) on \(C_0(X)\) consists of the maps \(G(i,x)\times X(j,y)\to X(i+j,\mu(x,y))\), so \(u\) is \(G\)-equivariant because \(C_0(u)\) is \(C_0(G)\)-equivariant. Conversely, if \(u\) is \(G\)-equivariant, every \(C_q(u)\) is \(C_q(G)\)-equivariant, for the action of \(C_q(G)\) on \(C_q(X)\) consists of the same maps, and \(\operatorname{Ner}(C_\bullet(G),C_\bullet(u))\) is a morphism of \(\mathcal Y_G\). The uniqueness parts of Lemma 2.5 and Proposition 2.3 show that the two constructions are inverse.

Necessity. For \(Y=\operatorname{Ner}(C_\bullet(G),C_\bullet(X))\), the morphism of condition 1 is the identity of \(C_q(G)^p\times C_q(X)\). Condition 2 holds by Proposition 2.3 (2).

Sufficiency. Let \(Y\) satisfy conditions 1 and 2. For fixed \(q\), \(Y_{\bullet,q}\) is a simplicial object of \(\operatorname{Diagr}(T)\) over \(N(C_q(G))\), through the structure morphism and \(C_q(e_J)\to e\). Condition 1 is the condition of Proposition 2.9 of the lesson on décalage, applied in \(\operatorname{Diagr}(T)\). So, by that proposition and its proof, the face \(d_0\colon Y_{1,q}\to Y_{0,q}\), composed with the inverse of the isomorphism of condition 1 for \(p=1\), is an action of \(C_q(G)\) on \(Z_q=Y_{0,q}\). The maps \((\pi_{p,q},d_1\cdots d_p)\) form an isomorphism \(Y_{\bullet,q}\cong N(C_q(G),Z_q)\) over \(N(C_q(G))\), equal to the identity at the level \(0\). The type of this action is the type of \(d_0\) in \(\operatorname{Ner}(C_q(J),C_q(J))\), the product of \(C_q(J)\). So \(Z_q\) is a \(C_q(J)\)-graded \(C_q(G)\)-object, and the isomorphism has identity type.

For \(\alpha\colon[m]\to[q]\), the map \(Y_{\bullet,\alpha}\colon Y_{\bullet,q}\to Y_{\bullet,m}\) lies over \(N(C(\alpha))\colon N(C_q(G))\to N(C_m(G))\), because the structure morphism of \(Y\) is a morphism of bisimplicial objects. By Lemma 2.5, transported through the isomorphisms above, it is induced by the \(C(\alpha)\)-equivariant map \(Y_{0,\alpha}\colon Z_q\to Z_m\). So \(Z=Y_{0,\bullet}\) is a simplicial object of \(\operatorname{Diagr}(T)\) of type \(C_\bullet(J)\) with equivariant transition maps, and \(Y\cong\operatorname{Ner}(C_\bullet(G),Z)\) as bisimplicial objects.

By condition 2 and Proposition 2.3, there are a \(J\)-graded object \(X\) with \(C_0(X)=Z_0\) and an isomorphism \(Z\cong C_\bullet(X)\) whose component at the level \(q\) is the vertex map \(Z_q\to Z_0\circ C(v_0)=C_q(X)\); the proof of Proposition 2.3 (2) of the lesson on equivariant sheaves constructs it in this way. The action of \(C_0(G)\) on \(Z_0=C_0(X)\) consists of maps \(G(i,x)\times X(j,y)\to X(i+j,\mu(x,y))\), natural in \(x\in J_i\) and \(y\in J_j\), associative and unital. They are also natural with respect to the arrows \((f,\mathrm{id})\) and \((g,\mathrm{id})\) of \(\int J\), for \(f\colon i\to i'\) and \(g\colon j\to j'\). Indeed, the vertex maps \(Z_1\to Z_0\circ C(v_0)\) and \(Z_1\to Z_0\circ C(v_1)\) are equivariant. The first is the isomorphism \(Z_1\cong C_1(X)\), over the map \(C(v_0)\colon C_1(G)\to C_0(G)\), which is the identity of \(G(i,x)\) on the summand of \((f,x)\). The second becomes \(X(f,\mathrm{id})\) on the summand of \((f,x)\), and lies over \(G(f,\mathrm{id})\). Its equivariance on the summands of \((f,x)\) and \((g,y)\) is the naturality in question. So \(X\) is a \(J\)-graded \(G\)-object. The same equivariance of the vertex maps \(Z_q\to C_q(X)\) shows that they carry the action of \(C_q(G)\) on \(Z_q\) to the action on \(C_q(X)\) defined by \(G\). Hence \(Y\cong\operatorname{Ner}(C_\bullet(G),C_\bullet(X))\). The structure morphism of \(Y\) is determined by its components \(\pi_{p,q}\), since \(C_q(e_J)\) is final among the diagrams of shape \(C_q(J)\) and maps of identity type. These components are preserved by the isomorphisms, so the isomorphism lies in \(\mathcal Y_G\). \(\square\)

3. Localizations and the pseudo-category of a graded action

Lemma 3.1 (discrete opfibrations). Let \(\varphi\colon K'\to K\) be a functor of small categories.

1. Suppose \(K'=\coprod_\lambda K'_\lambda\) and \(K=\coprod_cK_c\) are sums of categories and \(\varphi\) maps each \(K'_\lambda\) isomorphically onto some \(K_c\). Then \(\varphi\) is a discrete opfibration. Discrete opfibrations are stable under composition and under base change along any functor. If \(\varphi\) and \(\varphi\chi\) are discrete opfibrations, so is \(\chi\).
2. Let \(\varphi\) be a discrete opfibration and \(F'\in T^{K'}\). Then \(\varphi_!F'(k)=\coprod_{k'}F'(k')\), the sum over the objects \(k'\) of the fibre \(\varphi^{-1}(k)\), and an arrow \(b\colon k\to l\) maps the summand of \(k'\) to the summand of the target \(l'\) of the lift \(b'\) of \(b\) with source \(k'\), by \(F'(b')\). For maps \(A\to B\) of \(T^K\) and \(C\to\varphi^*B\) of \(T^{K'}\), the canonical map \[ \varphi_!\bigl(\varphi^*A\times_{\varphi^*B}C\bigr)\to A\times_B\varphi_!C \] is an isomorphism, and \(\varphi_!\) preserves fibre products.
3. Let \((\varphi,u)\colon(K',F')\to(K,F)\) be a morphism of \(\operatorname{Diagr}(T)\) whose type \(\varphi\) is a discrete opfibration. Put \(V=\varphi_!F'\), with the map \(V\to F\) adjoint to \(u\). Then \(E'\mapsto\varphi_!E'\) is an equivalence \(T^{K'}/F'\to T^K/V\). It carries the morphism of topoi \(T^{K'}/F'\to T^K/F\) induced by \((\varphi,u)\) to the localization \(T^K/V\to T^K/F\).
4. Let moreover \((\chi,w)\colon(K'',F'')\to(K',F')\) be a morphism such that \(\varphi\chi\) is a discrete opfibration. Under the equivalences of 3 for \((\varphi,u)\) and for \((\varphi,u)\circ(\chi,w)\), the morphism of topoi induced by \((\chi,w)\) becomes the localization of \(T^K/V\) at the map \(V''=(\varphi\chi)_!F''\to V\) induced by \((\chi,w)\).

Proof. (1) For \(k'\in K'_\lambda\), the arrows of \(K\) out of \(\varphi(k')\) lie in \(K_c\), on which \(\varphi\) has an inverse; and no arrow of \(K'\) out of \(k'\) leaves \(K'_\lambda\). So each such arrow lifts uniquely. Lifting in two steps shows that composites of discrete opfibrations are discrete opfibrations. For a functor \(\psi\colon L\to K\), an object \((l,k')\) of \(L\times_KK'\) and an arrow \(c\colon l\to l_1\) of \(L\), the lifts of \(c\) with source \((l,k')\) are the pairs \((c,b')\) with \(b'\) the lift of \(\psi(c)\) with source \(k'\): there is exactly one. Finally let \(\varphi\) and \(\varphi\chi\) be discrete opfibrations, \(k''\) an object of the source of \(\chi\) and \(b'\colon\chi(k'')\to k'\). Let \(b''\) be the lift of \(\varphi(b')\) with source \(k''\) along \(\varphi\chi\). Then \(\chi(b'')\) and \(b'\) both lift \(\varphi(b')\) along \(\varphi\) with source \(\chi(k'')\), so \(\chi(b'')=b'\). Any lift of \(b'\) along \(\chi\) lifts \(\varphi(b')\) along \(\varphi\chi\), so it is \(b''\).

(2) The value \(\varphi_!F'(k)\) is the colimit of \(F'\) over the category of pairs \((k',b\colon\varphi(k')\to k)\). Each pair has exactly one arrow to a pair of the form \((l',\mathrm{id}_k)\), namely the lift of \(b\). So the fibre, as a discrete subcategory, is final, and the colimit is the sum over the fibre, with the stated action of arrows. The canonical map of the statement has the components \(\varphi_!(\varphi^*A\times_{\varphi^*B}C)\to\varphi_!\varphi^*A\to A\) and \(\varphi_!(\ldots)\to\varphi_!C\). At \(k\) it is the map \[ \begin{aligned}&\coprod_{k'}\bigl(A(k)\times_{B(k)}C(k')\bigr)\\&\quad \to A(k)\times_{B(k)}\coprod_{k'}C(k'),\end{aligned} \] an isomorphism because sums in \(T\) are stable under base change. For maps \(E'\to F'\leftarrow W\), the map \(\varphi_!(E'\times_{F'}W)\to\varphi_!E'\times_{\varphi_!F'}\varphi_!W\) is, at \(k\), the map from the sum of the fibre products of the summands to the fibre product of the sums, which is invertible because sums in \(T\) are disjoint and stable under base change.

(3) The functor has a quasi-inverse, which sends \(E\to V\) to the functor \(E'\) with \(E'(k')=E(\varphi k')\times_{V(\varphi k')}F'(k')\), the fibre product being formed with the inclusion of the summand of \(k'\). An arrow \(b'\colon k'\to l'\) acts through \(E(\varphi b')\), which maps \(E'(k')\) into \(E'(l')\) because \(V(\varphi b')\) maps the summand of \(k'\) into that of \(l'\). The two composites are isomorphic to the identities because \(E(k)\) is the sum of its pullbacks to the summands of \(V(k)\), sums in \(T\) being disjoint and stable under base change. The induced morphism has the inverse image \(E\mapsto\varphi^*E\times_{\varphi^*F}F'\), and by (2) with \(A=E\), \(B=F\), \(C=F'\), \[ \varphi_!\bigl(\varphi^*E\times_{\varphi^*F}F'\bigr)\cong E\times_FV, \] naturally in \(E\to F\). This is the inverse image of the localization at \(V\). Morphisms of topoi with isomorphic inverse images are isomorphic.

(4) The functor \(\chi\) is a discrete opfibration by (1). For \(E'\to F'\), the inverse image of the morphism induced by \((\chi,w)\) is \(\chi^*E'\times_{\chi^*F'}F''\). By (2) for \(\chi\) and then for \(\varphi\), \[ \begin{aligned}&(\varphi\chi)_!\bigl(\chi^*E'\times_{\chi^*F'}F''\bigr)\\ &\quad \cong\varphi_!\bigl(E'\times_{F'}\chi_!F''\bigr)\\ &\quad \cong\varphi_!E'\times_V V''.\end{aligned} \] This is the inverse image of the localization at \(V''\to V\), applied to \(\varphi_!E'\). \(\square\)

Lemma 3.2 (cartesian squares). Let \[ \begin{array}{ccc}(K_3,F_3)&\xrightarrow{\ (\varphi',u')\ }&(K_1,F_1)\\ \downarrow{\scriptstyle(\psi',v')}&&\downarrow{\scriptstyle(\psi,v)}\\ (K_2,F_2)&\xrightarrow{\ (\varphi,u)\ }&(K_0,F_0)\end{array} \] be a cartesian square of \(\operatorname{Diagr}(T)\) in which \(\varphi\) is a discrete opfibration. Write \(g\), \(h\), \(f\), \(f'\) for the morphisms of topoi induced by the top, bottom, right and left sides. Then the base change map \(h^*f_*\to f'_*g^*\) of the square of topoi (Section 3.1 of the lesson on equivariant sheaves) is an isomorphism.

Proof. The top type \(\varphi'\) is the base change of \(\varphi\), a discrete opfibration by Lemma 3.1 (1). Since \(K_3=K_2\times_{K_0}K_1\), the projection to \(K_2\) maps the fibre of \(\varphi'\) over \(k_1\) bijectively onto the fibre of \(\varphi\) over \(k_0=\psi(k_1)\). By Lemma 3.1 of the lesson on equivariant sheaves, it suffices that the left base change map \(c'\colon g_!f'^*\to f^*h_!\) is an isomorphism. Let \(E\to F_2\). By the conventions, Lemma 3.1 (2) and \(F_3(k_2,k_1)=F_2(k_2)\times_{F_0(k_0)}F_1(k_1)\), the values at \(k_1\) are \[ \begin{aligned}&g_!f'^*E(k_1)\\ &\quad =\coprod_{k_2}E(k_2)\times_{F_0(k_0)}F_1(k_1),\\ &f^*h_!E(k_1)\\ &\quad =\Bigl(\coprod_{k_2}E(k_2)\Bigr)\times_{F_0(k_0)}F_1(k_1),\end{aligned} \] the sums being over the fibre of \(\varphi\) over \(k_0\). The unit \(E\to h^*h_!E\) has, at \(k_2\), the components given by the inclusion of the summand \(E(k_2)\) and by the structure map \(E(k_2)\to F_2(k_2)\). So \(f'^*\) of the unit, followed by the transitivity isomorphism \(f'^*h^*\cong g^*f^*\), maps \(f'^*E(k_2,k_1)\) to the summand of \(k_2\) in \(g^*f^*h_!E(k_2,k_1)\), by the identity. Its adjoint \(c'\) maps the summand of \(k_2\) of \(g_!f'^*E(k_1)\) to \(f^*h_!E(k_1)\) by the inclusion. So \(c'\) is the canonical map from the sum of fibre products to the fibre product of the sum, an isomorphism because sums in \(T\) are stable under base change. \(\square\)

Proposition 3.3. Assume (H). Let \(G\) be a \(J\)-graded monoid of \(T\) and \(X\) a \(J\)-graded \(G\)-object. For \(p\geq0\), let \(D_p=(\mathfrak K_p,F_p)\) be the diagram defined, as in the conventions, by the simplicial object \(q\mapsto C_q(G)^p\times C_q(X)\) of \(\operatorname{Diagr}(T)\): \(\mathfrak K_p\) is the total category of \(q\mapsto C_q(J)^{p+1}\), and \(F_p(q,k)=(C_q(G)^p\times C_q(X))(k)\). The simplicial structure in \(p\) makes \([p]\mapsto D_p\) a simplicial object of \(\operatorname{Diagr}(T)\), and so defines a simplicial topos \(\mathcal X\) with levels \[ \mathcal X_p=T^{\mathfrak K_p}/F_p, \] the topos of simplicial objects of \(\operatorname{Diagr}(T)\) of type \(C_\bullet(J)^{p+1}\) over \(q\mapsto C_q(G)^p\times C_q(X)\).

1. For every \(p\), the type \(\pi_p\colon\mathfrak K_p\to\mathfrak K_0\) of the last-vertex morphism \(D_p\to D_0\), induced by \(v_p\colon[0]\to[p]\), is a discrete opfibration. Hence \(\mathcal X_p\) is equivalent to the localization \(\mathcal X_0/V_p\) with \(V_p=\pi_{p!}F_p\), and \(\mathcal X(v_p)\) to the localization morphism.
2. \(\mathcal X\) is a good two-sided pseudo-category (Definition 3.2 of the lesson on equivariant sheaves).
3. The décalage \(\operatorname{Dec}^1(\mathcal X)\), which forgets the last face, together with its augmentation to \(\mathcal X_0\) (Definition 4.3 of Derived left adjoints on the second total topos and local inductive limits), is equivalent as a topos fibred over \(\bar\Delta^{\mathrm{op}}\) to the fibred topos of the augmented simplicial object \([p]\mapsto V_{p+1}\), \(p\geq-1\), of the topos \(\mathcal X_0\) (Examples 1.2 (c) of the lesson on fibred topoi). In degree \(-1\) it is \(\mathcal X_0\) itself, \(V_0\) being the final object.

Proof. (1) The last vertex of the string \((g_1,\ldots,g_p,x)\) is \(g_p\cdots g_1x\). So \(\pi_p\) sends \((q,(k_1,\ldots,k_p,k))\) to \((q,k_p\otimes\cdots\otimes k_1\otimes k)\), and an arrow \((\alpha,(b_1,\ldots,b_p,b))\) to \((\alpha,b_p\otimes\cdots\otimes b_1\otimes b)\), since \(C(\alpha)\) is strict monoidal. The category \(C_q(J)^{p+1}\) is the sum of the categories \(J_{\sigma_1(0)}\times\cdots\times J_{\sigma_p(0)}\times J_{\tau(0)}\) over the \((p+1)\)-tuples of \(q\)-simplices of the nerve of \(I\). The total product maps this summand by an iterated \(\mu\), an isomorphism by (H), onto the summand of \(C_q(J)\) indexed by \(\sigma_p+\cdots+\sigma_1+\tau\). By Lemma 3.1 (1) the total product \(C_q(J)^{p+1}\to C_q(J)\) is a discrete opfibration. Let \((q,k)\) be an object of \(\mathfrak K_p\) and \((\alpha,b)\colon\pi_p(q,k)\to(q',l)\) an arrow of \(\mathfrak K_0\), so \(b\colon C(\alpha)k_p\otimes\cdots\otimes C(\alpha)k\to l\). The lifts of \((\alpha,b)\) with source \((q,k)\) are the \((\alpha,b')\) with \(b'\) an arrow out of \(C(\alpha)^{p+1}(k)\) whose total product is \(b\), and there is exactly one. So \(\pi_p\) is a discrete opfibration, and Lemma 3.1 (3) applies.

(2) By the conventions, \(\mathcal X\) is good. Let \(n\geq1\). The morphism \(D_{n+1}\to D_n\times_{D_0}D_1\) given by \(d_{n+1}\) and the last edge \(e_{n,n+1}\), the fibre product being formed with the last vertex of \(D_n\) and the vertex \(0\) of \(D_1\), is an isomorphism. Indeed, for each \(q\) the corresponding morphism of \(\operatorname{Diagr}(T)\) is an isomorphism by Proposition 2.4 (4) of the lesson on décalage, applied to the nerve of the internal category \([C_q(G),C_q(X)]\) of \(\operatorname{Diagr}(T)\). These isomorphisms are compatible with the maps in \(q\), and total categories commute with fibre products of functors into \(\mathbf{Cat}\). So the first square of Definition 3.2 of the lesson on equivariant sheaves is induced by a cartesian square of \(\operatorname{Diagr}(T)\) whose bottom side, the last vertex of \(D_n\), has the type \(\pi_n\). Lemma 3.2 shows that its base change map is an isomorphism. In the same way, \(D_{n+1}\to D_1\times_{D_0}D_n\), given by the first edge \(e_{01}\) and \(d_0\), with the fibre product formed with the vertex \(1\) of \(D_1\) and the vertex \(0\) of \(D_n\), is an isomorphism, by Proposition 2.4 (2) of the lesson on décalage with \(a=1\), \(b=n\) and \(k=0\). In the second square of Definition 3.2, the bottom side \(t\) is induced by the vertex \(1\) of \(D_1\), whose type is \(\pi_1\). Lemma 3.2 gives the second condition.

(3) The maps of \(\operatorname{Dec}^1(\mathcal X)\) are induced by the morphisms \(D(\psi\sqcup\mathrm{id}_{[0]})\) for the arrows \(\psi\) of \(\bar\Delta\). The map \(\psi\sqcup\mathrm{id}_{[0]}\) fixes the last vertex, so these morphisms commute with the last-vertex morphisms to \(D_0\). By Lemma 3.1 (4), under the equivalences of (1) the induced morphisms of topoi are the localizations along the maps \(V_{n+1}\to V_{m+1}\) induced by these morphisms. These maps make \([p]\mapsto V_{p+1}\) an augmented simplicial object of \(\mathcal X_0\), and all the isomorphisms involved are the canonical ones between sums and fibre products, so they are compatible with composition. Finally \(V_0=\pi_{0!}F_0=F_0\) is the final object of \(\mathcal X_0=T^{\mathfrak K_0}/F_0\). \(\square\)

Explicitly, \(V_p\) at \((q,c)\) is the sum of the products \(C_q(G)(c_1)\times\) \(\cdots\times\) \(C_q(G)(c_p)\times\) \(C_q(X)(c')\) over the factorizations \(c=c_p\otimes\cdots\otimes c_1\otimes c'\) in \(C_q(J)\), with its map to \(C_q(X)(c)\) given by the action. The localizations are formed on the total category \(\mathfrak K_0\), not level by level in \(q\): the fibres of the products of \(C_q(J)\) are not preserved by the maps \(C(\alpha)\) (Exercise 4.2).

4. Exercises

4.1. Let \(J\) be an \(I\)-graded category with a monoid structure. Show that if \(\mu_{o,o}\colon J_o\times J_o\to J_o\) is an isomorphism, then \(J_o\) is the one-point category.

4.2. Let \(J=S_a\) on \(I=\bar\Delta'\) (Examples 1.2 and 1.4). Let \(\delta\colon[0]\to[1]\) be the map with \(\delta(0)=0\), and \(z\) an object of \(\Delta^{\mathrm{op}}\). Show that the fibre of the product \(C_1(S_a)\times C_1(S_a)\to C_1(S_a)\) over \((\delta,z)\) has three objects, that the fibre of the product of \(C_0(S_a)\) over \(C(v_0)(\delta,z)=([0],z)\) has two, and that \(C(v_0)\times C(v_0)\) does not map the first fibre bijectively onto the second. Deduce that \(m_!\circ(C(v_0)\times C(v_0))^*\to C(v_0)^*\circ m_!\), for the products \(m\), is not an isomorphism.

4.3. Let \(I\) be the strict monoidal category with one object \(o\) and one arrow, and let \(J_o=[1]=\{0\to1\}\) with \(\mu=\max\) and unit \(0\). Let \(T\) be the topos of sets and \(G=X=e_J\). Show that (H) fails, that the type of the last-vertex morphism \(D_1\to D_0\) is not a discrete opfibration, and that the morphism of topoi \(\mathcal X_1\to\mathcal X_0\) it induces is not equivalent to a localization of \(\mathcal X_0\).

4.4. Let \(I\) be as in 4.3 and let \(J_o\) be the one-point category. Let \(G\) be a monoid of \(T\). Identify the \(J\)-graded \(G\)-objects, the functor \(C_\bullet\) and the category \(\mathcal Y_G\) of Proposition 2.6, and show that condition 2 of that proposition cannot be omitted.

5. Complete solutions

4.1. The unit law says \(\mu(1,x)=x=\mu(x,1)\) for every object \(x\) of \(J_o\), so \(\mu\) sends \((1,x)\) and \((x,1)\) to the same object. If \(\mu_{o,o}\) is injective on objects, \((1,x)=(x,1)\), so \(x=1\). For an arrow \(u\) of \(J_o\), the unit law gives \(\mu(\mathrm{id}_1,u)=u=\mu(u,\mathrm{id}_1)\), and injectivity on arrows gives \(u=\mathrm{id}_1\).

4.2. An object of \(C_1(S_a)\) is a pair \((\varepsilon,y)\) of an arrow \(\varepsilon\colon[m-1]\to[n-1]\) of \(\bar\Delta'\) and \(y\in(\Delta^{\mathrm{op}})^m\), and the product is \((\varepsilon_1,y_1)\otimes(\varepsilon_2,y_2)=(\varepsilon_1+\varepsilon_2,(y_1,y_2))\). The fibre over \((\delta,z)\) consists of the decompositions \(\delta=\varepsilon_1+\varepsilon_2\), with \(z\) split accordingly. If \(\varepsilon_1\colon[a-1]\to[a'-1]\) and \(\varepsilon_2\colon[b-1]\to[b'-1]\), then \(a+b=1\) and \(a'+b'=2\), and \(\varepsilon_1+\varepsilon_2\) sends \(k<a\) to \(\varepsilon_1(k)\) and \(k\geq a\) to \(a'+\varepsilon_2(k-a)\). If \(a=1\), then \(\varepsilon_1(0)=0\) and \(a'\in\{1,2\}\): either \(\varepsilon_1=\mathrm{id}_{[0]}\) and \(\varepsilon_2\colon\varnothing\to[0]\), or \(\varepsilon_1=\delta\) and \(\varepsilon_2=\mathrm{id}_\varnothing\). If \(a=0\), then \(a'+\varepsilon_2(0)=0\) forces \(a'=0\) and \(\varepsilon_2=\delta\). So the fibre has three objects. The fibre over \(([0],z)\) in \(C_0(S_a)\) consists of the decompositions \([0]=[a-1]+[b-1]\), that is, \((a,b)=(1,0)\) or \((0,1)\): two objects. The functor \(C(v_0)\) sends \((\varepsilon,y)\) to the source of \(\varepsilon\) with \(y\), so both decompositions with \(a=1\) go to \((([0],z),([-1],()))\). The canonical map in question is the mate of the equality \(m\circ(C(v_0)\times C(v_0))=C(v_0)\circ m\). By Lemma 3.1 (2), at \((\delta,z)\) and applied to the final object, it is the map from a sum of three copies of \(e\) to a sum of two copies that sends each summand to the summand of its image under this map of fibres, by the identity. Two summands go to the same summand. Since sums are disjoint, the two coprojections of these summands have an initial fibre product, while their composites with the map agree; so the map is not a monomorphism unless \(e\) is initial, that is, unless \(T\) is the trivial topos.

4.3. Here \(\mu=\max\colon[1]\times[1]\to[1]\) is not injective on objects, so (H) fails. The arrow \(0\to1\) of \([1]\) has three lifts with source \((0,0)\): \((0\to1,\mathrm{id}_0)\), \((\mathrm{id}_0,0\to1)\) and \((0\to1,0\to1)\). So \(\max\) is not a discrete opfibration. Every \(C_q(J)\) is \([1]\) and every \(C(\alpha)\) is the identity, so \(\mathfrak K_p=\Delta^{\mathrm{op}}\times[1]^{p+1}\), \(F_p\) is the final object, \(\mathcal X_p\) is the topos of functors \(\Delta^{\mathrm{op}}\times[1]^{p+1}\to\mathbf{Set}\), and the type of the last vertex of \(D_1\) is \(\varphi=\mathrm{id}\times\max\). Suppose that the induced morphism \(f\) were isomorphic to a localization \(\mathcal X_0/V\to\mathcal X_0\) composed with an equivalence \(\mathcal X_1\simeq\mathcal X_0/V\). The left adjoint of the inverse image of a localization sends the final object to \(V\), so \(V\cong f_!(e)=\varphi_!(e)\). The value of \(\varphi_!(e)\) at \((q,c)\) is the set of connected components of the category of pairs \((k,\beta\colon\varphi(k)\to(q,c))\). This category has a final object: \(((q,(0,0)),\mathrm{id})\) for \(c=0\), and \(((q,(1,1)),\mathrm{id})\) for \(c=1\). So \(\varphi_!(e)\) is the final object, the localization is an equivalence, and \(\mathcal X_1\simeq\mathcal X_0\). But the subobjects of the final object of the topos of functors \(K\to\mathbf{Set}\) correspond to the sets of objects of \(K\) closed under arrows. In \(\Delta^{\mathrm{op}}\) there are arrows in both directions between any two objects, so for \(K=\Delta^{\mathrm{op}}\times P\), with \(P\) a poset, these sets are the products with the upward closed subsets of \(P\). There are \(3\) of them for \(P=[1]\) and \(6\) for \(P=[1]^2\). An equivalence preserves subobjects of the final object, so \(\mathcal X_1\not\simeq\mathcal X_0\), a contradiction.

4.4. Here \(\int J\) is the one-point category, so a \(J\)-graded monoid is a monoid \(G\) of \(T\), and a \(J\)-graded \(G\)-object is a \(G\)-object \(X\) of \(T\). Every \(C_q(J)\) is the one-point category and every \(C(\alpha)\) is the identity, so \(C_\bullet(X)\) is the constant simplicial object with value \(X\). An object of \(\mathcal Y_G\) is a bisimplicial object \(Y\) of \(T\) with a map to the bisimplicial object \((p,q)\mapsto G^p\), the nerve of \(G\) in the first index. Condition 1 says that each \(Y_{\bullet,q}\) is isomorphic to the nerve \(N(G,Z_q)\) of a \(G\)-object \(Z_q\), and condition 2 says that the vertex maps \(Z_q\to Z_0\) are isomorphisms.

Suppose that \(G\) is not the final object. Let \(Z\) be the simplicial object with \(Z_q=G^{q+1}\), whose faces delete factors and whose degeneracies repeat them, with \(G\) acting on each \(Z_q\) by left multiplication on every factor. The faces and degeneracies are equivariant, so \(Y=\operatorname{Ner}(G,Z)\), with \(Y_{p,q}=G^p\times G^{q+1}\), is an object of \(\mathcal Y_G\) satisfying condition 1. Its vertex map \(Z_1=G\times G\to Z_0=G\) is the first projection. This is not an isomorphism: otherwise the diagonal of \(G\), a section of it, would be an isomorphism, so \(G\to e\) would be a monomorphism, and then the unit \(e\to G\), a section of \(G\to e\), would be an isomorphism. So \(Y\) fails condition 2, and it is not isomorphic to any \(\operatorname{Ner}(C_\bullet(G),C_\bullet(X))\).

Sources

Graded monoids over a monoid in categories, the simplicial categories \(C_\bullet(J)\), the nerves of graded actions and the pseudo-category attached to them are due to Illusie, in his work on the cotangent complex. Nerves of monoid actions and internal categories are standard; the base change arguments use the localization of topoi of SGA 4, Exposé IV, in the form of the Stacks project.