2.1 Abelian group objects

Let ${\mathcal C}$ be a category with finite products and terminal object $*$ . Recall that an abelian group object G of ${\mathcal C}$ comes with maps

$$\begin{align*}e \colon * \to G, \quad m \colon G \times G \to G, \quad \mathrm{inv} \colon G \to G,\end{align*}$$

subject to the expected axioms. We write $\mathrm {Ab}({\mathcal C})$ for the category of abelian group objects in ${\mathcal C}$ .

Let $\mathrm {CRing}$ denote the category of commutative rings. For a fixed commutative ring A, we denote by $\mathrm {CRing}_{/A}$ the category of commutative rings over A – that is, the category whose objects are ring maps $B \to A$ . For an A-module J, we denote by $A \oplus J$ the split square-zero extension of A by J. The following serves as a moral starting point for the approach to the cotangent complex typically taken in the context of derived and higher algebra (see, for example, [Reference Lurie25, Remark 7.3.2.17]):

Remark 2.3 (Abelianization as indecomposables).

Extension of scalars determines a functor $(-)_+ \colon \mathrm {CRing}_{/A} \to \mathrm {CRing}_{A//A}$ from the category of commutative rings augmented over A to that of commutative rings pointed at A – that is, augmented commutative A-algebras. As abelian group objects are already pointed, the category $\mathrm {Ab}(\mathrm {CRing}_{A // A})$ is still equivalent to that of A-modules, and the resulting functor

$$\begin{align*}\mathrm{Ab}(-) \colon \mathrm{CRing}_{A//A} \to \mathrm{Ab}(\mathrm{CRing}_{A//A}) \simeq \mathrm{Mod}_A\end{align*}$$

can be identified with the indecomposables functor $C \mapsto I/I^2$ , where I is defined to be the kernel of the augmentation map $C \to A$ . We observe that there is a canonical isomorphism $\mathrm {Ab}_+(B) \cong \mathrm {Ab}((B)_+) = \mathrm {Ab}(A \otimes _{\Bbb Z} B)$ of A-modules.

The resulting functor $\mathrm {Ab}_+ \colon \mathrm {CRing}_{/A} \to \mathrm {Mod}_A$ admits a non-abelian left derived functor, whose value at A is the (absolute) cotangent complex ${\Bbb L}_A$ of A.

2.4 Log rings

We now aim to discuss Theorem 2.2 in the context of log geometry. Let us recall the basic definitions:

By adjunction, the structure map $\alpha $ determines a unique map $\overline {\alpha } \colon {\Bbb Z}[M] \to A$ of commutative rings. We shall write $\mathrm {PreLog}$ for the resulting category of pre-log rings. Its coproduct is that of the underlying rings and monoids; that is,

$$\begin{align*}(A \otimes_{\Bbb Z} B, M \oplus N, \alpha \oplus \beta)\end{align*}$$

is the coproduct of $(A, M, \alpha )$ and $(B, N, \beta )$ , where $\alpha \oplus \beta $ is adjoint to the map ${\Bbb Z}[M \oplus N] \cong {\Bbb Z}[M] \otimes _{\Bbb Z} {\Bbb Z}[N] \xrightarrow {\overline {\alpha } \otimes \overline {\beta }} A \otimes _{\Bbb Z} B$ of commutative rings.

We shall denote by $\mathrm {GL}_1(A)$ the group of multiplicative units in a commutative ring A, perhaps more commonly denoted $A^\times $ .

We shall write $\mathrm {Log}$ for the resulting category of log rings. The forgetful functor $\mathrm {Log} \to \mathrm {PreLog}$ admits a left adjoint

$$\begin{align*}(-, -, -)^a \colon \mathrm{PreLog} \to \mathrm{Log}, \quad (A, M, \alpha) \mapsto (A, M^a, \alpha^a),\end{align*}$$

where $M^a$ is defined as the pushout of the diagram $M \xleftarrow {} \alpha ^{-1}\mathrm {GL}_1(A) \to \mathrm {GL}_1(A)$ and $\alpha ^a$ is determined by its universal property along $\alpha $ and the inclusion of the units $\mathrm {GL}_1(A) \to (A, \cdot )$ .

2.7 Exactification and repletion

The following notions are, either implicitly or explicitly, a key ingredient in the construction of most invariants of log rings and log schemes:

For fixed $(A, M)$ , we would like this construction to determine a functor

$$\begin{align*}(-, -)^{\mathrm{ex}} \colon \mathrm{PreLog}_{/(A, M)} \to \mathrm{PreLog}_{/(A, M)}^{\mathrm{ex}}, \quad (B, N) \mapsto (B^{\mathrm{ex}}, N^{\mathrm{ex}})\end{align*}$$

from pre-log rings over $(A, M)$ to pre-log rings over $(A, M)$ with exact structure map to M. For this, one should restrict attention to integral monoids – that is, those monoids that inject into their group completion. See [Reference Ogus30, Proposition I.4.2.17].

In log geometry, exactness is thus typically discussed in the context of integral monoids. To circumvent the lack of a notion of integrality in derived contexts, Rognes identified conditions under which the exactification procedure lends itself to a homotopically meaningful generalization:

The condition that $f^{\flat , \mathrm {gp}}$ be surjective is also used in Kato–Saito’s [Reference Kato and Saito18, Section 4]. Writing $\mathrm {PreLog}^{\mathrm {vsur}}_{/(A, M)}$ for the full subcategory of $\mathrm { PreLog}_{/(A, M)}$ consisting of those $(f, f^\flat ) \colon (B, N) \to (A, M)$ with $f^{\flat , \mathrm {gp}}$ surjective, [Reference Rognes34, Lemma 3.8] implies that the exactification construction gives a well-defined functor

$$\begin{align*}(-, -)^{\mathrm{rep}} \colon \mathrm{PreLog}_{/(A, M)}^{\mathrm{vsur}} \to \mathrm{PreLog}^{\mathrm{rep}}_{/(A, M)}\end{align*}$$

to the category of pre-log rings over $(A, M)$ with replete structure map to M. We remark that if $(B, N) \in \mathrm {PreLog}_{(A, M) // (A, M)}$ is an augmented $(A, M)$ -algebra, then $N \to M$ is automatically virtually surjective. Thus, we obtain a functor

$$\begin{align*}\mathrm{PreLog}_{/(A, M)} \xrightarrow{(-, -)_+} \mathrm{PreLog}_{(A, M) // (A, M)} \xrightarrow{(-, -)^{\mathrm{rep}}} \mathrm{PreLog}_{(A, M) // (A, M)}^{\mathrm{rep}},\end{align*}$$

where $(B, N)_+ := (A \otimes _{{\Bbb Z}} B, M \oplus N)$ . This will be used in Construction 2.17.

Example 2.10. Let $(A, M, \alpha )$ be a pre-log ring and let J be an A-module. The split square-zero extension $(A, M) \oplus J := (A \oplus J, M \oplus J)$ has structure map

$$\begin{align*}M \oplus J \to (A \oplus J, \cdot), \quad (m, j) \mapsto (\alpha(m), \alpha(m)j)\end{align*}$$

and is replete over $(A, M)$ via the projection $(A \oplus J, M \oplus J) \to (A, M)$ . The choice of structure map will be explained in Example 2.15.

Let $(B, N)$ be an augmented $(A, M)$ -algebra. This determines a fixed splitting

(2.1) $$ \begin{align}\mathrm{GL}_1(A) \oplus (\mathrm{GL}_1(B) / \mathrm{GL}_1(A)) \xrightarrow{\cong} \mathrm{ GL}_1(B).\end{align} $$

We shall consider the pre-log structure

(2.2) $$ \begin{align}M \oplus (\mathrm{GL}_1(B) / \mathrm{GL}_1(A)) \xrightarrow{} (B, \cdot)\end{align} $$

induced by $M \to (A, \cdot ) \to (B, \cdot )$ and $\mathrm {GL}_1(B) / \mathrm {GL}_1(A) \to \mathrm {GL}_1(B) \to (B, \cdot )$ .

Let us say that a morphism $(B, N) \to (A, M)$ is split replete if it admits a section and if $N \to M$ is exact (and hence replete).

Proof. By [Reference Rognes34, Lemma 3.11], the fixed splitting of $N \to M$ induces a splitting $M \oplus (N^{\mathrm {gp}}/M^{\mathrm {gp}}) \xrightarrow {\cong } N$ over and under M, so that the structure map of $(B, N)$ factors as the lower horizontal composite in the commutative diagram

The left-hand square is cartesian by definition, while the right-hand square is cartesian since the structure map $A \to B$ admits a retraction so that an element of A is a unit precisely when it maps to one in B. The defining pushout square for the logification $N^a$ is thus isomorphic to the outer rectangle

Since the left-hand square is cocartesian, so is the right-hand square. Hence, both squares in the diagram

are cocartesian. The result follows from the splitting (2.1).

2.13 The replete abelianization functor

We now work towards the construction of the replete abelianization functor. Let us first record the following consequence of Lemma 2.12:

Corollary 2.14. Let $(A, M)$ be a log ring. Then the natural forgetful functor $\mathrm {Log}^{\mathrm {rep}}_{(A, M)//(A, M)} \to \mathrm {CRing}_{A//A}$ is an equivalence of categories, with quasi-inverse

$$\begin{align*}\mathrm{CRing}_{A//A} \to \mathrm{Log}^{\mathrm{rep}}_{(A, M) // (A, M)}, \quad B \mapsto (B, M \oplus (\mathrm{GL}_1(B) / \mathrm{GL}_1(A))),\end{align*}$$

where the structure map on the target is described in (2.2).

Proof. As all objects in the statement are log rings (as opposed to pre-log rings), they are naturally isomorphic to their logifications. The result thus follows from Lemma 2.12, which provides a natural isomorphism

$$\begin{align*}(B, M \oplus (\mathrm{GL}_1(B)/\mathrm{GL}_1(A))) \xrightarrow{\cong} (B, N)\end{align*}$$

for any augmented replete $(A, M)$ -algebra $(B, N) \in \mathrm {Log}^{\mathrm {rep}}_{(A, M)//(A, M)}$ .

We find it conceptually appealing that the replete category is independent of the log structure already before passing to abelian group objects. See Example 6.11 for the example of log (topological) Hochschild homology.

Example 2.15. Let us identify the image of the split square-zero extension $A \oplus J$ under the equivalence of Corollary 2.14. The units $\mathrm {GL}_1(A \oplus J)$ split as $\mathrm {GL}_1(A)$ and a copy $1 + J$ of the underlying abelian group J. This means that there is an isomorphism $M \oplus (\mathrm {GL}_1(A \oplus J)/\mathrm {GL}_1(A)) \cong M \oplus (1 + J)$ , and the resulting structure map (2.2) is given by

$$\begin{align*}M \oplus (1 + J) \to (A \oplus J, \cdot), \quad (m, (1, j)) \mapsto (\alpha(m), 0)(1, j) = (\alpha(m), \alpha(m)j).\end{align*}$$

This recovers Example 2.10.

As a consequence of Theorem 2.2, Corollary 2.14 and Example 2.15, we obtain the following:

Corollary 2.16. Let $(A, M)$ be a log ring. There assignment

$$\begin{align*}\mathrm{Mod}_A \to \mathrm{Ab}(\mathrm{Log}^{\mathrm{rep}}_{(A, M)//(A, M)}), \qquad J \mapsto (A \oplus J, M \oplus J) \end{align*}$$

defines an equivalence of categories.

We now have all necessary ingredients for the following construction, which is the linear variant of one of the main constructions of this paper:

Construction 2.17. Let $(A, M)$ be a pre-log ring. Consider the following diagram

of categories and functors. Here, $(-, -)_+$ is extension of scalars (along the map $({\Bbb Z}, \{1\}) \to (A, M)$ of pre-log rings), $(-, -)^{\mathrm {rep}}$ is the repletion functor, and $(-, -)^a$ is the logification functor. The functor $\mathrm {Log}^{\mathrm {rep}}_{(A, M^a)//(A, M^a)} \to \mathrm {CAlg}_{A//A}$ is the forgetful functor, which is an equivalence by Corollary 2.14. The functor $\mathrm {Ab}(-)$ is the indecomposables functor of Remark 2.3.

Definition 2.18. We define the replete abelianization functor

$$\begin{align*}\mathrm{Ab}_+^{\mathrm{rep}} \colon \mathrm{PreLog}_{/(A, M)} \to \mathrm{Mod}_A\end{align*}$$

to be the functor resulting from Construction 2.17.

By construction, $\mathrm {Ab}_+^{\mathrm {rep}}$ admits the explicit description

(2.3) $$ \begin{align}\mathrm{Ab}_+^{\mathrm{rep}}(B, N) \cong \mathrm{Ab}((A \otimes_{\Bbb Z} B)^{\mathrm{rep}}) = \mathrm{Ab}((A \otimes_{\Bbb Z} B) \otimes_{{\Bbb Z}[M \oplus N]} {\Bbb Z}[(M \oplus N)^{\mathrm{rep}}]),\end{align} $$

where the repletion is taken with respect to the map $M \oplus N \to M$ induced by the identity on M and the structure map $N \to M$ .

2.20 The log differentials as replete abelianization

In classical log geometry, one defines a log derivation $(d, d^\flat ) \colon (A, M) \to J$ to consist of a derivation $d \colon A \to J$ and a monoid map $d^\flat \colon M \to (J, +)$ satisfying $d(\alpha (m)) = \alpha (m)d^\flat (m)$ . Inspired by Theorem 2.2 and the modern approach to the cotangent complex (of, for example, [Reference Lurie25, Chapter 7]), we aim to make this haromize with the following definition:

There is a well-established notion of differentials associated to a pre-log ring, explicitly defined by

$$\begin{align*}\Omega^1_{(A, M)} := \frac{\Omega^1_A \oplus (A \otimes_{\Bbb Z} M^{\mathrm{gp}})}{d\alpha(m) \sim \alpha(m) \otimes [m]},\end{align*}$$

where $[m]$ denotes the image of m under the canonical map $M \to M^{\mathrm {gp}}$ . The module $\Omega ^1_{(A, M)}$ corepresents log derivations.

Proof. By (2.3), we have that

$$\begin{align*}\mathrm{Ab}_+(A, M) \cong \mathrm{Ab}((A \otimes_{\Bbb Z} A) \otimes_{{\Bbb Z}[M \oplus M]} {\Bbb Z}[(M \oplus M)^{\mathrm{rep}}]).\end{align*}$$

This coincides with a description of the log differentials $\Omega ^1_{(A, M)}$ of Kato–Saito [Reference Kato and Saito18, Section 4]: In this linear setting, this is checked in [Reference Lundemo22, Proposition 2.2.1.1].

We now aim to prove Proposition 1.9. Recall the split square-zero extension $(A \oplus J, M \oplus J)$ of Example 2.15. We first record the following consequence of Corollary 2.16:

Corollary 2.24. Let $(A, M)$ be a pre-log ring. There is a natural isomorphism

$$\begin{align*}\mathrm{Hom}_{\mathrm{Mod}_A}(\mathrm{Ab}_+^{\mathrm{rep}}(A, M), J) \cong \mathrm{Hom}_{\mathrm{Log}_{/(A, M^a)}}((A, M^a), (A \oplus J, M^a \oplus J)).\end{align*}$$

Proof. By Corollary 2.16 and Construction 2.17, the first-mentioned $\mathrm {Hom}$ -set is naturally isomorphic to

$$\begin{align*}\mathrm{Hom}_{\mathrm{Log}_{(A, M^a)//(A, M^a)}}(((A \otimes_{\Bbb Z} A)^{\mathrm{rep}}), (M \oplus M)^{\mathrm{rep}, a}), (A \oplus J, M^a \oplus J)).\end{align*}$$

Since logification and repletion are left adjoints and the map $(A \oplus J, M^a \oplus J) \to (A, M^a)$ is a replete map of log rings, this is naturally isomorphic to

$$\begin{align*}\mathrm{Hom}_{\mathrm{PreLog}_{(A, M)//(A, M^a)}}((A \otimes_{\Bbb Z} A, M \oplus M), (A \oplus J, M^a \oplus J)).\end{align*}$$

By restriction of scalars along $({\Bbb Z}, \{1\}) \to (A, M)$ and once again exploiting that logification is a left adjoint, we obtain the description predicted by the corollary.

Corollary 2.24 implies that the set of log derivations $\mathrm {Der}((A, M), J)$ is naturally isomorphic to

$$\begin{align*}\mathrm{Hom}_{\mathrm{Log}_{/(A, M^a)}}((A, M^a), (A \oplus J, M^a \oplus J))\end{align*}$$

of augmented maps $(A, M^a) \to (A \oplus J, M^a \oplus J)$ . This perspective on log derivations is taken as the definition in, for example, [Reference Rognes34, Definition 4.15] and in the context of topological log structures (cf. [Reference Rognes34, Section 11] or [Reference Sagave38]).

Combined with Corollary 2.16, the following gives a strong analog of Theorem 2.2 in the context of log geometry:

Proof. By the Yoneda lemma, it suffices to prove that there is a natural isomorphism relating the two $\mathrm {Hom}$ -sets on top in the diagram

and so it only remains to provide explanations for the two natural isomorphisms above that have yet to receive one. The bottom horizontal map, induced by the augmentation $(B, N^a) \to (A, M^a)$ , is an isomorphism since the square

is a pullback. Finally, we observe that all maps in the composite

are isomorphisms by logification being left adjoint to the inclusion $\mathrm {Log} \to \mathrm {PreLog}$ and cobase-change along the unit map $({\Bbb Z}, \{1\}) \to (A, M)$ , repletion and logification being left adjoints, and the equivalence of Corollary 2.14, respectively. This concludes the proof.

3.1 Commutative ${\mathcal J}$ -space monoids

We give a very brief recollection of the material on commutative ${\mathcal J}$ -space monoids we shall use throughout. We refer to [Reference Rognes, Sagave and Schlichtkrull37, Reference Rognes, Sagave and Schlichtkrull36, Reference Sagave38, Reference Sagave and Schlichtkrull40] for increasingly detailed expositions.

Following [Reference Sagave and Schlichtkrull40, Section 4], let ${\mathcal J}$ denote Quillen’s localization construction $\Sigma ^{-1}\Sigma $ on the category $\Sigma $ of finite sets and bijections. Objects of ${\mathcal J}$ are pairs $(\mathbf {n}, \mathbf {m})$ where $\mathbf {n}$ denotes the finite set $\{1, \dots , n\}$ . A ${\mathcal J}$ -space is a functor from ${\mathcal J}$ to the category ${\mathcal S}$ of simplicial sets. This is a symmetric monoidal category $(S^{\mathcal J}, \boxtimes , U^{\mathcal J})$ , and commutative monoids therein are commutative ${\mathcal J}$ -space monoids. The resulting category ${\mathcal C}{\mathcal S}^{\mathcal J}$ admits a positive ${\mathcal J}$ -model structure by [Reference Sagave and Schlichtkrull40, Proposition 4.10]. Weak equivalences $M \to N$ in this model structure are those maps that induce a weak equivalence $M_{h{\mathcal J}} \to N_{h{\mathcal J}}$ on (Bousfield–Kan) homotopy colimits over ${\mathcal J}$ . We refer to its fibrations as positive fibrations.

By [Reference Sagave and Schlichtkrull40, Theorem 1.7], the category of commutative ${\mathcal J}$ -space monoids model ${\Bbb E}_{\infty }$ -spaces over $QS^0 = \Omega ^{\infty }({\Bbb S})$ . We therefore think of commutative ${\mathcal J}$ -space monoids as ( $QS^0$ -)graded ${\Bbb E}_{\infty }$ -spaces.

3.2 Group completion

We say that a commutative ${\mathcal J}$ -space monoid M is grouplike if the commutative monoid $\pi _0(M_{h{\mathcal J}})$ is a group. There is a group completion model structure ${\mathcal C}{\mathcal S}^{\mathcal J}_{\mathrm {gp}}$ on the category of commutative ${\mathcal J}$ -space monoids [Reference Sagave39, Theorem 5.5]. It arises as a left Bousfield localization of the positive ${\mathcal J}$ -model structure, and fibrant objects therein are precisely the (positive fibrant) grouplike commutative ${\mathcal J}$ -space monoids. The group completion $M \to M^{\mathrm {gp}}$ of a given commutative ${\mathcal J}$ -space monoid M is a fibrant replacement in this model structure. In the same way that grouplike ${\Bbb E}_{\infty }$ -spaces model connective spectra, grouplike commutative ${\mathcal J}$ -space monoids model connective spectra over the sphere [Reference Sagave39, Theorem 1.6].

3.3 Replete morphisms

The analog of Definition 2.9 in this context reads:

As explained in [Reference Rognes, Sagave and Schlichtkrull36, Lemma 3.17], one convenient way to model the repletion $N^{\mathrm {rep}} \to M$ of a virtually surjective $N \to M$ is as a fibrant replacement of N relative to M in the group completion model structure.

3.5 Mather’s cube lemma

Mather’s second cube lemma [Reference Mather27, Theorem 25] states that, for a commutative cube of spaces in which the vertical faces are homotopy cartesian and the bottom face is homotopy cocartesian, the top face is homotopy cocartesian as well. For more general homotopy theories, one has to assume that pushouts are universal (i.e., that they commute with forming the homotopy pullback along any map) for this to hold.

We will need the following weakened form of Mather’s second cube lemma in the category of commutative ${\mathcal J}$ -space monoids:

Proposition 3.6. Let

be a commutative diagram of cofibrant commutative ${\mathcal J}$ -space monoids in which the vertical faces are homotopy cartesian and the bottom face is homotopy cocartesian. If $M_{12}$ is replete over $N_{12}$ , then the top face is homotopy cocartesian.

Proof. There is no loss of generality in assuming that the vertical arrows are fibrations and that the vertical faces are pullbacks: Indeed, one can fix the bottom square and functorially replace $M_{12} \to N_{12}$ by a fibration, and define the vertical faces by pullback. Since the positive ${\mathcal J}$ -model structure is right proper, the resulting cube is weakly equivalent to that under consideration.

Let $B^\boxtimes (-, -, -)$ denote the two-sided bar construction in ${\mathcal J}$ -spaces. We wish to prove that the canonical map $B^\boxtimes (M_1, M_\emptyset , M_2) \to M_{12}$ is a weak equivalence. Since $B^\boxtimes (N_1, N_\emptyset , N_2) \to N_{12}$ is a weak equivalence by assumption, we find that it suffices to prove that the left-hand square in the commutative diagram

of commutative ${\mathcal J}$ -space monoids is homotopy cartesian. Since $M_{12} \to N_{12}$ is replete and in particular exact, it suffices to prove that the outer rectangle is homotopy cartesian.

By [Reference Sagave and Schlichtkrull40, Corollary 11.4], this occurs precisely when the associated square of Bousfield–Kan homotopy colimits is homotopy cartesian as a square of simplicial sets. Moreover, [Reference Sagave38, Lemma 2.11] allows us to rewrite the resulting square as

This square arises as the realization of a square of bisimplicial sets which is pointwise homotopy cartesian. As the ${\Bbb E}_\infty $ -spaces involved on the right-hand side are grouplike, the resulting bisimplicial sets satisfy the $\pi _*$ -Kan condition, and virtual surjectivity of the map $M_{12} \to N_{12}$ implies that that the right-hand vertical map is a Kan fibration on vertical path components. Hence, the Bousfield–Friedlander theorem [Reference Bousfield and Friedlander4, Theorem B.4] applies to conclude the proof.

The following was pointed out to us by Maxime Ramzi:

3.8 Commutative ${\mathcal J}$ -space monoids and units

To a positive fibrant commutative ${\mathcal J}$ -space monoid M, we may consider the subobject $M^\times $ , where $M^\times (\mathbf {n}, \mathbf {m})$ is defined to consist of those components of $M(\mathbf {n}, \mathbf {m})$ that represent units in $\pi _0(M_{h{\mathcal J}})$ . In [Reference Sagave39, Theorem 5.12], a units model structure is constructed on the category of commutative ${\mathcal J}$ -space monoids, which arises as a right Bousfield localization of the positive ${\mathcal J}$ -model structure. It is Quillen equivalent to the group completion model structure (via the identity functor), and exhibits $M \mapsto M^\times $ as a colocalization.

There is a Quillen adjunction

(3.1) $$ \begin{align}{\Bbb S}^{\mathcal J}[-] \colon {\mathcal C}{\mathcal S}^{\mathcal J} \rightleftarrows {\mathcal C}\mathrm{Sp}^{\Sigma} \colon \Omega^{\mathcal J}(-)\end{align} $$

relating the positive ${\mathcal J}$ -model structure and the positive stable model structure on commutative symmetric ring spectra of [Reference Mandell, May, Schwede and Shipley28]. For positive fibrant symmetric ring spectrum R, we may consider the graded units $\mathrm {GL}_1^{\mathcal J}(R) := \Omega ^{\mathcal J}(R)^\times $ . To any positive fibrant commutative ${\mathcal J}$ -space monoid M, one can associate a graded signed monoid $\pi _{0, *}(M)$ . When applied to the inclusion $\mathrm {GL}_1^{\mathcal J}(R) \to \Omega ^{\mathcal J}(R)$ , this realizes the inclusion of the units $\mathrm {GL}_1(\pi _*(R))$ of the graded commutative ring $\pi _*(R)$ (cf. [Reference Sagave and Schlichtkrull40, Proposition 4.26]).

We now describe a technical result that will be used in some of our arguments. Consider a homotopy cartesian square

(3.2)

of positive fibrant commutative symmetric ring spectra, and let $\widetilde {R} \to \widetilde {A}$ be a map with $\widetilde {R}$ positive fibrant. Let P be a commutative ${\mathcal J}$ -space monoid, and suppose that we are given a map $\mathrm {GL}_1^{\mathcal J}(\widetilde {R}) \to P$ . Use the factorization properties of the positive ${\mathcal J}$ -model structure to build a commutative diagram

with tailed arrows cofibrations and two-headed arrows fibrations, and $U^{\mathcal J}$ the initial commutative ${\mathcal J}$ -space monoid.

Lemma 3.10. In the situation described above, assume further that the right-hand vertical morphism in (3.2) induces a surjection $\mathrm {GL}_1(\pi _*(\widetilde {B})) \to \mathrm {GL}_1(\pi _*(B)).$ Then the square

of commutative ${\mathcal J}$ -space monoids is homotopy cartesian.

Proof. The square in question is modelled by the square

of two-sided bar constructions in ${\mathcal J}$ -spaces. We wish to argue with the Bousfield–Friedlander theorem as in the proof of Proposition 3.6. The cofibrancy hypotheses ensure that we can indeed reduce to checking whether the square

of bisimplicial sets is cartesian after realization. The square of bisimplicial sets is pointwise homotopy cartesian, and the objects on the right-hand side satisfy the $\pi _*$ -Kan condition as all commutative ${\mathcal J}$ -space monoids involved are grouplike. Combining [Reference Sagave and Schlichtkrull40, Corollary 4.17, Proposition 4.26], we find that the condition on the units ensures that the morphism $\mathrm {GL}_1^{\mathcal J}(\widetilde {B}) \to \mathrm {GL}_1^{\mathcal J}(B)$ is virtually surjective so that the right-hand vertical morphism in the square of bisimplicial sets induces a Kan fibration on vertical path components. Hence, the Bousfield–Friedlander theorem [Reference Bousfield and Friedlander4, Theorem B.4] applies to conclude the proof.

3.11 Replete juggling

Let us also record the following consequence of the proof of [Reference Lundemo21, Lemma 3.12], which relies on the Bousfield–Friedlander theorem:

Lemma 3.12. Let $P \to M$ be a cofibration of cofibrant commutative ${\mathcal J}$ -space monoids. The canonical map

$$\begin{align*}P \boxtimes_{(P \boxtimes P)^{\mathrm{rep}}} (M \boxtimes M)^{\mathrm{rep}} \xrightarrow{} (M \boxtimes_P M)^{\mathrm{rep}}\end{align*}$$

is a ${\mathcal J}$ -equivalence, where all repletions are formed with respect to the natural multiplication maps.

Proof. We can argue exactly as in [Reference Lundemo21, Lemma 3.12]. Since there is a natural isomorphism $M \boxtimes _P M \cong P \boxtimes _{(P \boxtimes P)} (M \boxtimes M)$ and group completions commute with homotopy pushouts [Reference Lundemo21, Lemma 2.9], it suffices to prove that the square of two-sided bar constructions

is homotopy cartesian. Using the homotopy cartesian squares for the repletions $(P \boxtimes P)^{\mathrm {rep}}$ and $(M \boxtimes M)^{\mathrm {rep}}$ , we see that we can argue with the Bousfield–Friedlander theorem as in the proof of Proposition 3.6.

4.1 The homotopy theory of graded ${\Bbb E}_{\infty }$ -spaces

Let ${\mathcal C}{\mathcal S}^{\mathcal J}_{\infty }$ denote the $\infty $ -category underlying the positive ${\mathcal J}$ -model structure on commutative ${\mathcal J}$ -space monoids. By [Reference Lurie25, Example 4.1.7.6], the fact that $({\mathcal C}{\mathcal S}^{\mathcal J}, \boxtimes , U^{\mathcal J})$ is a symmetric monoidal model category implies that ${\mathcal C}{\mathcal S}^{\mathcal J}_{\infty }$ is a symmetric monoidal $\infty $ -category. We shall refer to its objects as ( $QS^0$ -)graded ${\Bbb E}_{\infty }$ -spaces. Let us record some properties of the category of graded ${\Bbb E}_{\infty }$ -spaces here:

1. (1) The Quillen adjunction (3.1) induces an adjunction

   $$\begin{align*}{\Bbb S}^{\mathcal J}[-] \colon {\mathcal C}{\mathcal S}^{\mathcal J}_{\infty} \rightleftarrows{} {\mathcal C}\mathrm{Sp}^\Sigma_{\infty} \simeq \mathrm{CAlg}(\mathrm{Sp}) \colon \Omega^{\mathcal J}(-)\end{align*}$$
   relating the categories of graded ${\Bbb E}_{\infty }$ -spaces and ${\Bbb E}_{\infty }$ -ring spectra.

2. (2) Similarly, the Quillen adjunction

   $$\begin{align*}\mathrm{colim}_{\mathcal J} \colon {\mathcal S}^{\mathcal J} \rightleftarrows {\mathcal S} \colon \mathrm{const}_{\mathcal J}\end{align*}$$
   of [Reference Sagave and Schlichtkrull40, Proposition 6.23] gives rise to an adjunction of underlying $\infty $ -categories. We shall denote the resulting left adjoint by $(-)_{h{\mathcal J}}$ .

3. (3) As the group completion model structure ${\mathcal C}{\mathcal S}^{\mathcal J}_{\mathrm {gp}}$ is a left Bousfield localization of the positive model structure on ${\mathcal C}{\mathcal S}^{\mathcal J}$ , the resulting category of grouplike graded ${\Bbb E}_{\infty }$ -spaces ${\mathcal C}{\mathcal S}^{\mathcal J}_{\infty , \mathrm {gp}}$ is a localization of ${\mathcal C}{\mathcal S}^{\mathcal J}_{\infty }$ . We model group completions by the corresponding localization functor $(-)^{\mathrm {gp}} \colon {\mathcal C}{\mathcal S}^{\mathcal J}_{\infty } \to {\mathcal C}{\mathcal S}^{\mathcal J}_{\infty , \mathrm {gp}}$ .

4. (4) Associating to a positive fibrant commutative ${\mathcal J}$ -space monoid M its units $M^\times $ is a right adjoint to the inclusion of grouplike objects. In fact, the units model structure ${\mathcal C}{\mathcal S}^{\mathcal J}_{\mathrm {un}}$ in commutative ${\mathcal J}$ -space monoids of [Reference Sagave39, Theorem 5.12] is a right Bousfield localization and is Quillen equivalent to the group completion model structure [Reference Sagave39, Corollary 5.13]. This induces a right adjoint $(-)^\times \colon {\mathcal C}{\mathcal S}^{\mathcal J}_{\infty } \to {\mathcal C}{\mathcal S}^{\mathcal J}_{\infty , \mathrm { un}} \simeq {\mathcal C}{\mathcal S}^{\mathcal J}_{\infty , \mathrm {gp}}$ to the inclusion ${\mathcal C}{\mathcal S}^{\mathcal J}_{\infty , \mathrm {gp}} \to {\mathcal C}{\mathcal S}^{\mathcal J}_{\infty }$ .

All definitions stated and results proved in Section 3 are homotopically meaningful and thus have natural analogs in this setting that we will use throughout.

4.2 Pulling back units

Let R be an ${\Bbb E}_{\infty }$ -ring and let J be an R-module. Recall that a derivation of R with values in J is an augmented morphism of ${\Bbb E}_{\infty }$ -rings $d \colon R \to R \oplus J$ (with ring structure on the target as in [Reference Lurie25, Remark 7.3.4.15]), so that the space of derivations is $\mathrm {Map}_{\mathrm {CAlg}_{/R}}(R, R \oplus J)$ . We say that $\widetilde {R} \to R$ is a square-zero extension by the R-module J if there is a cartesian diagram of ${\Bbb E}_{\infty }$ -rings

where $d_0$ denotes the trivial derivation.

Lemma 4.3. Let $\widetilde {R} \to R$ be a square-zero extension of ${\Bbb E}_{\infty }$ -rings. Then the square

is cartesian.

Proof. Consider the commutative cube

of graded ${\Bbb E}_{\infty }$ -spaces. As the left- and right-hand faces are cartesian, it suffices to show that the back face is cartesian. By the pasting lemma, this follows from the fact that the square

is cartesian (cf. [Reference Rognes34, Proof of Lemma 11.27]).

4.4 Replete morphisms and base-change

One convenient property of split replete morphisms is their behavior under base-change. The analogous statement in the context of integral monoids (but without the split condition) is [Reference Ogus30, Proposition I.4.2.1(6(b))].

Proof. By [Reference Lundemo21, Lemma 2.12], there is an equivalence $M \boxtimes W(\widetilde {M}) \xrightarrow {\simeq } \widetilde {M}$ over and under M, where $W(\widetilde {M})$ is the grouplike graded ${\Bbb E}_{\infty }$ -space defined as the pullback of the diagram $U^{\mathcal J} \xrightarrow {} M^{\mathrm {gp}} \xleftarrow {} \widetilde {M}^{\mathrm {gp}}$ . Pulling back along f, we obtain a cube

of graded ${\Bbb E}_{\infty }$ -spaces. The back and right-hand vertical faces are cartesian, and the front face is readily seen to be so by combining [Reference Sagave and Schlichtkrull40, Corollary 11.4] and [Reference Sagave38, Lemma 2.11]; we demonstrate the style of the argument below. From this, we see that $N \boxtimes W(\widetilde {M}) \xrightarrow {} \widetilde {N}$ is an equivalence over N, and so it remains to prove that $N \boxtimes W(\widetilde {M}) \to N$ is replete. By [Reference Lundemo21, Lemma 2.9] and the fact that $W(\widetilde {M})$ is grouplike, we have equivalences

$$\begin{align*}N^{\mathrm{gp}} \boxtimes W(\widetilde{M}) \xrightarrow{\simeq} N^{\mathrm{gp}} \boxtimes W(\widetilde{M})^{\mathrm{gp}} \xrightarrow{\simeq} (N \boxtimes W(\widetilde{M}))^{\mathrm{gp}}. \end{align*}$$

Thus, we have to show that

(4.1)

is cartesian. By [Reference Sagave and Schlichtkrull40, Corollary 11.4], this can be checked after applying $(-)_{h{\mathcal J}}$ , and [Reference Sagave38, Lemma 2.11] implies that the resulting square is equivalent to one of the form

which is cartesian.

We remark that the square (4.1) being cartesian only depended upon $W(\widetilde {M})$ being a grouplike graded ${\Bbb E}_{\infty }$ -space augmented over the initial object $U^{\mathcal J}$ . This combines with [Reference Lundemo21, Lemma 2.12] to prove the following:

5.4 Mapping spaces of pre-log ring spectra

For pre-log ring spectra $(A, M)$ and $(B, N)$ , the space of maps $\mathrm {Map}_{\mathrm {PreLog}}((A, M), (B, N))$ sits in a cartesian square

(5.2)

We observe that this follows from the description of $\mathrm {PreLog}$ as a Grothendieck construction sketched from Remark 5.2 by [Reference Lurie24, Proposition 2.4.4.3].

5.5 The logification construction

The log condition in this setting reads:

Definition 5.6. A pre-log ring spectrum $(A, M, \alpha )$ is log if the map $\widetilde {\alpha }$ in the cartesian diagram

of graded ${\Bbb E}_{\infty }$ -spaces is an equivalence.

We shall write $\mathrm {Log}$ for the resulting category of log ring spectra. We now describe a left adjoint to the forgetful functor $\mathrm {Log} \to \mathrm {PreLog}$ :

Definition 5.7. If $(A, M, \alpha )$ is a pre-log ring spectrum, its logification is defined by the cocartesian diagram

with structure map $\alpha ^a$ determined by $\alpha $ and the inclusion of the graded units $\mathrm {GL}_1^{\mathcal J}(A) \to \Omega ^{\mathcal J}(A)$ . The resulting pre-log ring spectrum $(A, M^a, \alpha ^a)$ is log by [Reference Sagave38, Lemma 3.12], while the resulting functor

$$\begin{align*}(-)^a \colon \mathrm{PreLog} \to \mathrm{Log}\end{align*}$$

is left adjoint to the forgetful functor by [Reference Sagave38, Lemma 6.4].

5.9 Strict morphisms of log ring spectra

In our setup, we shall work with the following notion of strict morphisms:

The following is an analog of [Reference Ogus30, Proposition III.1.2.5]:

Lemma 5.11. A morphism $(f, f^\flat ) \colon (R, P) \to (A, M)$ of log ring spectra is strict if and only if the square

(5.3)

is cocartesian in the category of log ring spectra.

Proof. The square

is cocartesian in the category of pre-log ring spectra. If $(f, f^\flat )$ is strict, the logification of this square is the cocartesian square of log ring spectra predicted by the lemma. Conversely, (5.3) being cocartesian implies that $(f, f^\flat )$ is the cobase-change of the strict morphism $(R, \mathrm {GL}_1^{\mathcal J}(R)) \to (A, \mathrm {GL}_1^{\mathcal J}(A))$ , and hence, it is itself strict (by, for example, the argument of [Reference Sagave, Schürg and Vezzosi42, Lemma 5.5]).

5.12 Repletion

As in the case of ordinary pre-log rings, the repletion construction extends from graded ${\Bbb E}_{\infty }$ -spaces to pre-log ring spectra:

6.1 Recollections on the tangent bundle

Recall from [Reference Lurie24, Definition 5.5.3.2] that a map of simplicial sets is a presentable fibration if it is a (co)Cartesian fibration and its fibers are presentable $\infty $ -categories. Throughout, we shall be concerned with presentable fibrations ${\mathcal E} \to {\mathcal C}$ where ${\mathcal C}$ is a presentable $\infty $ -category.

We invite the reader to keep in mind the example of the ‘evaluation at the codomain’-functor

(6.1) $$ \begin{align}\mathrm{Fun}(\Delta^1, {\mathcal C}) \to \mathrm{Fun}(\{1\}, {\mathcal C}) \simeq {\mathcal C}, \quad (A \to B) \mapsto B,\end{align} $$

for a presentable $\infty $ -category ${\mathcal C}$ , in which case the fibers are equivalent to ${\mathcal C}_{/B}$ .

To any presentable $\infty $ -category ${\mathcal C}$ , one can associate a pointed envelope and a stable envelope [Reference Lurie23, Definition 1.1]. By [Reference Lurie23, Example 1.7], the inclusion of the full subcategory of pointed objects ${\mathcal C}_* \to {\mathcal C}$ is a pointed envelope of ${\mathcal C}$ , while by [Reference Lurie23, Example 1.4], the composite $\mathrm { Stab}({\mathcal C}) \xrightarrow {\Omega ^\infty _{{\mathcal C}_*}} {\mathcal C}_* \xrightarrow {} {\mathcal C}$ is a stable envelope of ${\mathcal C}$ , where $\mathrm {Stab}({\mathcal C})$ is the stabilization of ${\mathcal C}$ . Pointed and stable envelopes are unique up to equivalence [Reference Lurie23, Remark 1.8].

The tangent bundle of a presentable $\infty $ -category ${\mathcal C}$ is an instance of the stable envelope construction:

6.4 Recollections on the ${\Bbb E}_{\infty }$ -cotangent complex

We now specialize to the case where ${\mathcal C} = \mathrm {CAlg}$ is the category of ${\Bbb E}_{\infty }$ -rings, and we model the stabilization of a presentable $\infty $ -category by its category of spectrum objects. Upon taking fibers above an ${\Bbb E}_{\infty }$ -ring A, we thus obtain the stable envelope $\mathrm {Sp}(\mathrm {CAlg}_{/A})$ of $\mathrm {CAlg}_{/A}$ ; we informally depict this as

The analog of Theorem 2.2(1) is [Reference Lurie25, Corollary 7.3.4.14], which states that $\mathrm {Sp}(\mathrm {CAlg}_{/A}) \simeq \mathrm {Mod}_A$ . By [Reference Lurie25, Theorem 7.3.4.18], we can think of an object of the tangent bundle $T_{\mathrm {CAlg}}$ as a pair $(A, J)$ for J an A-module, and the lower horizontal composite above, then, informally reads $(A, J) \mapsto (A \oplus J \to A) \mapsto A$ .

By [Reference Lurie25, Remark 7.3.2.17], the absolute cotangent complex specializes to $\Sigma _A^{\infty }(A) := \Sigma _+^\infty (A) \in \mathrm {Sp}(\mathrm {CAlg}_{/A})$ on each fiber. Under the equivalence $\mathrm { Sp}(\mathrm {CAlg}_{/A}) \simeq \mathrm {Mod}_A$ , this recovers the A-module computing topological André–Quillen homology [Reference Lurie25, Remark 7.3.0.1]. See also Basterra–Mandell [Reference Basterra and Mandell7].

6.6 Towards the replete tangent bundle

We now aim to carry out these constructions in the context of log ring spectra. The category of pre-log ring spectra does not seem to admit a description as the category of algebras over an operad, and as such, it does not immediately fit in the framework of Lurie’s cotangent complex formalism. We will overcome this by following the same strategy as in Section 2 in this framework.

We begin by establishing an analog of Corollary 2.14. For this, we will need the following construction:

Construction 6.7. Let $(A, M)$ be a pre-log ring spectrum and consider an object $(B, N) \in \mathrm {PreLog}_{(A, M)//(A, M)}$ . Applying graded units to the structure maps $A \to B \to A$ , we obtain a diagram $\mathrm {GL}_1^{\mathcal J}(A) \to \mathrm {GL}_1^{\mathcal J}(B) \to \mathrm {GL}_1^{\mathcal J}(A)$ of grouplike commutative ${\mathcal J}$ -space monoids, and so [Reference Lundemo21, Lemma 2.12] applies to obtain an equivalence

(6.2) $$ \begin{align}\mathrm{GL}_1^{\mathcal J}(A) \boxtimes (\mathrm{GL}_1^{\mathcal J}(B)/\mathrm{GL}_1^{\mathcal J}(A)) \xrightarrow{\simeq} \mathrm{GL}_1^{\mathcal J}(B),\end{align} $$

where $\mathrm {GL}_1^{\mathcal J}(B)/\mathrm {GL}_1^{\mathcal J}(A)$ is defined as the pullback of $U^{\mathcal J} \xrightarrow {} \mathrm {GL}_1^{\mathcal J}(A) \xleftarrow {} \mathrm {GL}_1^{\mathcal J}(B)$ , where $U^{\mathcal J}$ is the initial object in ${\mathcal C}{\mathcal S}^{\mathcal J}_{\infty }$ . The composites $M \xrightarrow {} N \xrightarrow {} \Omega ^{\mathcal J}(B)$ and

$$\begin{align*}\mathrm{GL}_1^{\mathcal J}(B)/\mathrm{ GL}_1^{\mathcal J}(A) \xrightarrow{} \mathrm{GL}_1^{\mathcal J}(A) \boxtimes (\mathrm{GL}_1^{\mathcal J}(B)/\mathrm{GL}_1^{\mathcal J}(A)) \xrightarrow{\simeq} \mathrm{GL}_1^{\mathcal J}(B) \xrightarrow{} \Omega^{\mathcal J}(B)\end{align*}$$

induce a pre-log structure $M \boxtimes (\mathrm {GL}_1^{\mathcal J}(B)/\mathrm {GL}_1^{\mathcal J}(A)) \to \Omega ^{\mathcal J}(B)$ on B.

The following is an analog of Lemma 2.12.

Proof. Our proof is an adaptation of that of Lemma 2.12 in this context. By the assumption that the structure map $N \to M$ is replete, [Reference Lundemo21, Lemma 2.12] applies to obtain an equivalence $M \boxtimes W(N) \xrightarrow {\simeq } N$ , where $W(N)$ is defined as the pullback of $U^{\mathcal J} \xrightarrow {} M^{\mathrm {gp}} \xleftarrow {} N^{\mathrm {gp}}$ . The resulting (equivalent) structure map $M \boxtimes W(N) \xrightarrow {\simeq } N \xrightarrow {} \Omega ^{\mathcal J}(B)$ factors as the lower horizontal composite in the diagram

of commutative ${\mathcal J}$ -space monoids. The left-hand square is clearly cartesian, while the right-hand square is cartesian by Lemma 6.9 below. It follows that the logification $N^a$ is determined by the outer cocartesian rectangle

The left-hand square is cocartesian by definition, and so the right-hand square is cocartesian. Since both squares in the rectangle

are cocartesian, the result follows from the splitting (6.2).

In the proof of Lemma 6.8, we used the following:

Lemma 6.9. Let A be an ${\Bbb E}_{\infty }$ -ring and let $B \in \mathrm {CAlg}_{A//A}$ be an augmented A-algebra. The square

is cartesian.

Proof. This is very similar to the argument of Lemmas 4.3 and 4.5. Combining [Reference Sagave and Schlichtkrull40, Corollary 11.4] and [Reference Sagave38, Lemma 2.11] as in the proof of Lemma 4.5, we find that it suffices to prove that the front face of the commutative cube

is cartesian. The right-hand face is cartesian, and as for any commutative ${\mathcal J}$ -space monoid M, an element of the graded signed monoid $\pi _{0, *}(M)$ is a unit if and only if it represents one in $\pi _0(M_{h{\mathcal J}})$ (cf. [Reference Sagave and Schlichtkrull40, Corollary 4.16] and the surrounding discussion). For the same reason, the left-hand face is cartesian. The back face is isomorphic to

Since $A \to B$ admits a retraction, it is precisely the units of $\pi _*(A)$ that map to units in $\pi _*(B)$ , so that this square is cartesian. Hence, the front face of the cube is cartesian, as desired.

Proof. Throughout this proof, we shall use the shorthand $G_B$ for $\mathrm {GL}_1^{\mathcal J}(B)/\mathrm {GL}_1^{\mathcal J}(A)$ . The functor is essentially surjective, as for any $B \in \mathrm {CAlg}_{A//A}$ , it lifts to the replete augmented $(A, M)$ -algebra $(B, M \boxtimes G_B)$ by Lemma 6.8. It thus remains to prove that it is fully faithful. For this, it suffices to prove that, given augmented A-algebras B and C, the left-hand vertical map

in the defining cartesian square (5.2) for mapping spaces in $\mathrm {Log}$ is an equivalence. It thus suffices to prove that the right-hand vertical map is an equivalence. We consider the commutative diagram

of mapping spaces, where the arrows decorated $\simeq $ are equivalences for the indicated reason, and we have implicitly used that the category of grouplike commutative ${\mathcal J}$ -space monoids is a full subcategory of all commutative ${\mathcal J}$ -space monoids when utilizing that $\mathrm { GL}_1^{\mathcal J}(-)$ is a right adjoint. This concludes the proof.

Example 6.11. Employing the homotopy invariant notions of logification and repletion of [Reference Sagave, Schürg and Vezzosi42], the equivalence of Corollary 2.14 extends to a Quillen equivalence of simplicial objects, as one can, for instance, see by imitating the proof of Proposition 6.10. For a fixed pre-log ring $(A, M)$ , this means that, for example, the composite functor

$$\begin{align*}\mathrm{sPreLog}_{(A, M) // (A, M)} \xrightarrow{(-, -)^{\mathrm{rep}}} \mathrm{sPreLog}_{(A, M^a) // (A, M^a)}^{\mathrm{rep}} \xrightarrow{(-, -)^a} \mathrm{sLog}^{\mathrm{rep}}_{(A, M^a) // (A, M^a)}\end{align*}$$

from simplicial augmented $(A, M)$ -algebras naturally takes values in simplicial augmented A-algebras (via the equivalence of Corollary 2.14). The simplicial tensor $S^1 \otimes (A, M)$ is an augmented simplicial pre-log $(A, M)$ -algebra, with augmentation induced by the collapse map of the circle. Under the displayed composite functor, this uniquely determines an augmented commutative A-algebra. By construction, this coincides with Rognes’ log Hochschild homology $\mathrm {HH}(A, M)$ of the pre-log ring $(A, M)$ [Reference Rognes34, Definition 3.23]. The argument of [Reference Rognes, Sagave and Schlichtkrull36, Theorem 4.24] (or its conjunction with [Reference Binda, Lundemo, Park and Østvær5, Corollary 3.4]) shows that $\mathrm {HH}(-, -)$ is invariant under the logification construction. By Proposition 6.10, the analogous remark also applies to log topological Hochschild homology $\mathrm {THH}(A, M)$ .

The following is the spectral analog of Example 2.15.

Remark 6.13. Following [Reference Lurie25, Remark 7.3.4.16], let ${\mathcal C}$ be a presentably symmetric monoidal, stable $\infty $ -category, let $A \in \mathrm {CAlg}({\mathcal C})$ be a commutative algebra object, and let $J \in \mathrm {Mod}_A({\mathcal C})$ be an A-module. One defines the split square-zero extension $A \oplus J$ as the image of J under the composite

$$\begin{align*}\mathrm{Mod}_A({\mathcal C}) \simeq \mathrm{Sp}(\mathrm{CAlg}({\mathcal C})_{/A}) \xrightarrow{\Omega^\infty} \mathrm{CAlg}({\mathcal C})_{/A},\end{align*}$$

where the equivalence is [Reference Lurie25, Theorem 7.3.4.13]. While the category $\mathrm {Log}$ does not seem to fit in this framework, Proposition 6.10 and Corollary 6.12 suggest that the ‘infinitesimal theory’ of log ring spectra can largely be ported from that of ordinary ${\Bbb E}_{\infty }$ -rings. More explicitly, let us consider the composite

$$\begin{align*}\mathrm{Mod}_A \simeq \mathrm{Sp}(\mathrm{Log}^{\mathrm{rep}}_{(A, M)//(A, M)}) \xrightarrow{\Omega^\infty} \mathrm{Log}^{\mathrm{rep}}_{(A, M)//(A, M)}.\end{align*}$$

The image of an A-module J identifies with $(A \oplus J, M \boxtimes \mathrm {GL}_1^{\mathcal J}(A \oplus J)/\mathrm {GL}_1^{\mathcal J}(A))$ . By definition, this recovers the split square-zero extensions of log ring spectra used previously in the literature (cf. [Reference Rognes34, Definition 11.6] and [Reference Sagave38, Construction 5.6]), that we shall denote by $(A, M) \oplus J := (A \oplus J, M \oplus J)$ .

6.14 The replete pointed envelope

Let ${\mathcal C}$ be a presentable $\infty $ -category. As explained in [Reference Lurie23, Notation 1.57], one can explicitly model the pointed envelope $P_{\mathcal C} \to \mathrm {Fun}(\Delta ^1, {\mathcal C})$ of the presentable fibration (6.1) as the full subcategory of $\mathrm {Fun}(\Delta ^2, {\mathcal C})$ consisting of those triangles that compose to an equivalence; that is, those commutative diagrams

for which the horizontal map $X \to Z$ an equivalence. The functor $P_{\mathcal C} \to \mathrm {Fun}(\Delta ^1, {\mathcal C})$ informally sends such a diagram to the morphism $Y \to Z$ ; that is, it is induced by the evaluation $\mathrm {Fun}(\Delta ^2, {\mathcal C}) \to \mathrm {Fun}(\Delta ^{\{1, 2\}}, {\mathcal C}) \simeq \mathrm {Fun}(\Delta ^1, {\mathcal C})$ .

Definition 6.15. Let $P_{\mathrm {Log}}$ be the pointed envelope of $\mathrm {Fun}(\Delta ^1, \mathrm {Log}) \to \mathrm {Log}$ . The replete pointed envelope $P_{\mathrm {Log}}^{\mathrm {rep}}$ is the full subcategory of $P_{\mathrm {Log}}$ spanned by those triangles

(6.3)

for which $(B, N) \to (C, K)$ is replete. The functor $P^{\mathrm {rep}}_{\mathrm {Log}} \to \mathrm {Fun}(\Delta ^1, \mathrm {Log})$ is obtained by restriction of the functor $P_{\mathrm {Log}} \to \mathrm {Fun}(\Delta ^1, \mathrm {Log})$ .

We remark that since $(A, M) \to (C, K)$ is an equivalence in the above triangle, the $M \to K$ is necessarily virtually surjective.

By construction and Lemma 6.16, we have a commutative diagram

of Cartesian fibrations over $\mathrm {Log}$ . The theory of relative adjunctions of [Reference Lurie25, Section 7.3.2] provides us with the following ‘globalization’ of the repletion functor:

Objects (6.3) of $P^{\mathrm {rep}}_{\mathrm {Log}}$ have replete structure maps $(B, N) \to (C, K)$ that admit a section up to homotopy, and so they are strict by Corollary 4.6 and [Reference Lundemo22, Lemma 3.4.1.5]. This means that the only additional data necessary to determine the diagram (6.3) is (1) the underlying diagram of ${\Bbb E}_{\infty }$ -rings and (2) the log structure K on the codomain C. This is made formal below, where we denote by $P_{\mathrm {CAlg}} \to \mathrm {CAlg}$ the pointed envelope of the functor $\mathrm {Fun}(\Delta ^1, \mathrm {CAlg}) \to \mathrm {Fun}(\{1\}, \mathrm { CAlg}) \simeq \mathrm {CAlg}$ .

Lemma 6.18. There is an equivalence

$$\begin{align*}P^{\mathrm{rep}}_{\mathrm{Log}} \xrightarrow{\simeq} \mathrm{Log} \times_{\mathrm{CAlg}} P_{\mathrm{CAlg}}\end{align*}$$

of Cartesian fibrations over $\mathrm {Log}$ .

6.20 The replete tangent bundle

We are now ready to give the definition of the replete tangent bundle. By Lemma 6.16, the functor $P^{\mathrm {rep}}_{\mathrm {Log}} \to \mathrm {Log}$ is a presentable fibration, and so we may form its stable envelope:

The definition comes with some immediate pleasant consequences:

Corollary 6.23. There is an equivalence

$$\begin{align*}T^{\mathrm{rep}}_{\mathrm{Log}} \xrightarrow{\simeq} \mathrm{Log} \times_{\mathrm{CAlg}} T_{\mathrm{CAlg}}\end{align*}$$

of Cartesian fibrations over $\mathrm {Log}$ .

6.24 Construction of the log cotangent complex

The following cube summarizes the construction of the replete tangent bundle $T^{\mathrm {rep}}_{\mathrm {Log}}$ in terms of the ordinary tangent bundle $T_{\mathrm {PreLog}}$ of the presentable $\infty $ -category $\mathrm {PreLog}$ :

Here, the bottom square is defined by logification. The middle square is defined by logification, cobase-change and repletion. More formally, we have defined the functor $\mathrm {Fun}(\Delta ^1, \mathrm {PreLog}) \to P^{\mathrm { rep}}_{\mathrm {Log}}$ as the composite

(6.4) $$ \begin{align}\mathrm{Fun}(\Delta^1, \mathrm{PreLog}) \to \mathrm{Fun}(\Delta^1, \mathrm{Log}) \to P_{\mathrm{Log}} \to P^{\mathrm{rep}}_{\mathrm{Log}}\end{align} $$

given by logification, the left adjoint provided by [Reference Lurie25, Proof of Lemma 7.3.3.21], and the left adjoint provided by Lemma 6.17, respectively. The top square is defined to be the stable envelope of the bottom cube.

Remark 6.25. Due to our passing to pointed envelopes in log ring spectra, the fiberwise description of (6.4) is more complicated than in the case of Construction 2.17. On each fiber, we have the description

$$\begin{align*}((B, N) \to (A, M)) \mapsto ((A, M^a) \to ((A \otimes B)^{\mathrm{rep}}, (M \boxtimes N)^{a, \mathrm{rep}}) \to (A, M^a)).\end{align*}$$

Here, the repletion is taken with respect to the map $(M^a \boxtimes N^a)^a \to M^a$ induced by the identity on $M^a$ and $N^a \to M^a$ . Observe that we do not mimic Construction 2.17 directly, as we do not know how to construct the relevant (variants of) relative adjunctions when the base category varies. For this reason, we pass to log ring spectra immediately.

By Corollary 6.22 (or Corollary 6.23 combined with [Reference Lurie25, Theorem 7.3.4.18]), we can think of the objects of the replete tangent bundle $T^{\mathrm {rep}}_{\mathrm {Log}}$ as triples $((A, M), J)$ with $(A, M)$ a log ring spectrum and J an A-module. From this point of view, we may describe the composite $T_{\mathrm {Log}}^{\mathrm {rep}} \to P^{\mathrm {rep}}_{\mathrm {Log}} \to \mathrm {Log}$ by the formula

$$\begin{align*}((A, M), J) \mapsto ((A, M) \to (A, M) \oplus J \to (A, M)) \mapsto (A, M),\end{align*}$$

cf. Remark 6.13.

Proof. On each fiber, the functor under consideration is equivalent to

$$\begin{align*}\mathrm{Sp}(\mathrm{Log}^{\mathrm{rep}}_{(A, M) // (A, M)}) \xrightarrow{\Omega^\infty} \mathrm{Log}^{\mathrm{rep}}_{(A, M)//(A, M)},\end{align*}$$

which admits the left adjoint $\Sigma ^\infty $ . We now aim to apply [Reference Lurie25, Proposition 7.3.2.6] to the diagram

For this, we have to show that morphisms in $T^{\mathrm {rep}}_{\mathrm {Log}}$ that are cartesian over $\mathrm {Log}$ are sent to morphisms in $P^{\mathrm {rep}}_{\mathrm { Log}}$ that are cartesian over $\mathrm {Log}$ . But in light of Lemma 6.18 and Corollary 6.23, this is true for the same reason that it is true for the functor $T_{\mathrm { CAlg}} \to P_{\mathrm {CAlg}}$ over $\mathrm {CAlg}$ (cf. [Reference Lurie25, Definition 7.3.2.14]): Given a cartesian morphism $((A, M), J_1) \to ((B, N), J_2)$ over $\mathrm {Log}$ in $T^{\mathrm {rep}}_{\mathrm {Log}}$ , there is an equivalence of A-modules $J_1 \xrightarrow {\simeq } J_2$ . Hence, the square

is cartesian, which concludes the proof.

Let us pause to record that our results assemble to a full proof of Theorem 1.12.

Definition 6.27. The log cotangent complex is the composite functor

$$\begin{align*}{\Bbb L}^{\mathrm{rep}} \colon \mathrm{PreLog} \xrightarrow{(-)^a} \mathrm{Log} \xrightarrow{} \mathrm{Fun}(\Delta^1, \mathrm{Log}) \xrightarrow{} P_{\mathrm{Log}} \xrightarrow{} P^{\mathrm{rep}}_{\mathrm{Log}} \xrightarrow{} T^{\mathrm{rep}}_{\mathrm{Log}}\end{align*}$$

given by logification, the diagonal embedding, the left adjoint provided by [Reference Lurie25, Proof of Lemma 7.3.3.21], the left adjoint provided by Lemma 6.17, and the left adjoint provided by Lemma 6.26, respectively.

6.28 Identifying the log cotangent complex

By Corollary 6.22, for each pre-log ring spectrum $(A, M)$ , we obtain an A-module that we denote by ${\Bbb L}_{(A, M)}^{\mathrm {rep}}$ - the log cotangent complex of $(A, M)$ . A variant $\mathrm {TAQ}(A, M)$ of the log cotangent complex was introduced and studied by Rognes [Reference Rognes34, Sections 11 and 13] and Sagave [Reference Sagave38] – therein referred to as log topological André–Quillen homology. We now aim to prove Theorem 1.14, which states that these A-modules are canonically equivalent. We first simplify the rather complicated description of Remark 6.25. Let $(B, N) \to (A, M)$ be a map of log ring spectra. There is a map

(6.5) $$ \begin{align}(A \otimes B) \otimes_{{\Bbb S}^{\mathcal J}[M \boxtimes N]} {\Bbb S}^{\mathcal J}[(M \boxtimes N)^{\mathrm{rep}}] \to (A \otimes B) \otimes_{{\Bbb S}^{\mathcal J}[(M \boxtimes_{} N)^a]} {\Bbb S}^{\mathcal J}[(M \boxtimes_{} N)^{a, \mathrm{ rep}}].\end{align} $$

The codomain is as in Remark 6.25. In the domain, the repletion is taken with respect to the map induced by the identity on M and the map $N \to M$ , and the map is induced by logification.

Proof. We very closely follow the proof strategy of [Reference Rognes, Sagave and Schlichtkrull36, Theorem 4.24], of which the argument here should be considered a special case. We spell out the argument for the convenience of the reader. Let us write $(C, K, \gamma )$ for the pre-log ring spectrum $(A \otimes B, M \boxtimes _{} N)$ . Let $\gamma ^{-1}\mathrm {GL}_1^{\mathcal J}(C)^{\mathrm {rep}}$ denote the repletion of the morphism $\gamma ^{-1}\mathrm { GL}_1^{\mathcal J}(C) \to \alpha ^{-1}\mathrm {GL}_1^{\mathcal J}(A)$ (which is virtually surjective, as $(C, K) \to (A, M)$ admits a section). For a cocomplete category ${\mathcal C}$ pointed at c, we shall write $S^1 \odot _c -$ for the pointed tensor with $S^1$ . Consider first the pushout diagram

(6.6)

of graded ${\Bbb E}_{\infty }$ -spaces. The codomain of the right-hand vertical map is grouplike, as its domain is grouplike, and the left-hand vertical map is surjective on $\pi _0((-)_{h{\mathcal J}})$ by Lemma 6.30 below. Combining this with the facts that group completions commute with homotopy pushouts [Reference Lundemo21, Lemma 2.9] and that the left-hand vertical map is an equivalence after group completion implies that the right-hand vertical map was an equivalence to begin with. Observe that the map $\gamma ^{-1}\mathrm {GL}_1^{\mathcal J}(C) \to \Omega ^{\mathcal J}(C)$ factors over $\mathrm {GL}_1^{\mathcal J}(C)$ , and hence over $\gamma ^{-1}\mathrm {GL}_1^{\mathcal J}(C)^{\mathrm {gp}}$ . From this, we obtain the diagram

(6.7)

Since the right-hand vertical map in (6.6) is an equivalence, this composite becomes an equivalence after one suspension $S^1 \odot _A -$ in augmented commutative A-algebras.

Consider now the commutative cube

of ${\Bbb E}_{\infty }$ -rings. The front vertical face is cocartesian by the definition of the logification $K^a$ , while the back face is cocartesian by [Reference Rognes, Sagave and Schlichtkrull36, Lemma 4.26] (the cocartesian squares needed for its statement are the defining cocartesian squares of $M^a \simeq M$ and $K^a$ ), and the composite (6.7) being an equivalence after one suspension implies that the left-hand face is cocartesian after base-change along the induced morphism ${\Bbb S}^{\mathcal J}[S^1 \odot _{\mathrm {GL}_1^{\mathcal J}(A)} \mathrm {GL}_1^{\mathcal J}(C)] \to S^1 \odot _A C$ . This implies that the square

is cocartesian, with the top horizontal map an equivalence. Hence, the lower horizontal map is an equivalence, as desired.

Let $S^1 \odot _M -$ denote the pointed tensor with $S^1$ in the category $({\mathcal C}{\mathcal S}^{\mathcal J}_{\infty })_{M // M}$ of graded ${\Bbb E}_{\infty }$ -spaces pointed at M. In the proof of Proposition 6.29, we used the following:

Proof. By [Reference Lundemo21, Lemma 2.12], there is an equivalence $M \boxtimes W(N) \xrightarrow {\simeq } N^{\mathrm {rep}}$ , where $W(N)$ is the pullback of $U^{\mathcal J} \xrightarrow {} M^{\mathrm {gp}} \xleftarrow {} N^{\mathrm {gp}}$ . Imitating [Reference Rognes, Sagave and Schlichtkrull37, Proposition 3.1] gives a natural equivalence $M \boxtimes W(N) \simeq M \times (N^{\mathrm {gp}}_{h{\mathcal J}}/M^{\mathrm {gp}}_{h{\mathcal J}})$ over and under M. Hence, there are equivalences

$$\begin{align*}S^1 \odot_M N^{\mathrm{rep}} \xleftarrow{\simeq} S^1 \odot_M (M \boxtimes W(N)) \simeq S^1 \odot_M (M \times (N^{\mathrm{gp}}_{h{\mathcal J}}/M^{\mathrm{gp}}_{h{\mathcal J}})).\end{align*}$$

Observe now that $S^1 \odot _M (M \times (N^{\mathrm {gp}}_{h{\mathcal J}}/M^{\mathrm {gp}}_{h{\mathcal J}})) \simeq M \boxtimes B(N^{\mathrm {gp}}_{h{\mathcal J}}/M^{\mathrm {gp}}_{h{\mathcal J}})$ since $B(-)$ is the suspension functor on grouplike ${\Bbb E}_{\infty }$ -spaces. This concludes the proof, as the repletion map is one over M, and $B(N^{\mathrm {gp}}_{h{\mathcal J}}/M^{\mathrm {gp}}_{h{\mathcal J}})$ is path-connected.

Proof of Theorem 1.14.

By construction (see Remark 6.25), the log cotangent complex ${\Bbb L}_{(A, M)}^{\mathrm {rep}}$ corresponds to the suspension spectrum

$$\begin{align*}\Sigma^{\infty} ((A \otimes A) \otimes_{{\Bbb S}^{\mathcal J}[(M \boxtimes_{} M)^a]} {\Bbb S}^{\mathcal J}((M \boxtimes_{} M)^{a, \mathrm{rep}}), ((M \boxtimes_{} M)^{a, \mathrm{rep}}) \end{align*}$$

in $\mathrm {Sp}(\mathrm {Log}^{\mathrm { rep}}_{(A, M^a) // (A, M^a)}) \simeq \mathrm {Mod}_A$ . Under the equivalence of categories of Proposition 6.10 and the equivalences of Proposition 6.29, this corresponds to the suspension spectrum

$$\begin{align*}\Sigma^{\infty}((A \otimes A) \otimes_{{\Bbb S}^{\mathcal J}[M \boxtimes M]} {\Bbb S}^{\mathcal J}[(M \boxtimes M)^{\mathrm{rep}}])\end{align*}$$

in $\mathrm {Sp}(\mathrm {CAlg}_{A//A})$ . By [Reference Lundemo21, Section 9], this A-module is equivalent to $\mathrm {TAQ}(A, M)$ .

For a map $(R, P) \to (A, M)$ of pre-log ring spectra, we define the relative log cotangent complex ${\Bbb L}_{(A, M) / (R, P)}^{\mathrm {rep}}$ as the cofiber of $A \otimes _R {\Bbb L}_{(R, P)}^{\mathrm {rep}} \to {\Bbb L}_{(A, M)}^{\mathrm {rep}}$ . By Theorem 1.14, the following definition coincides with the one considered in [Reference Sagave38]:

7.1 Log square-zero extensions

Recall from the discussion of Remark 6.13 that for a presentably symmetric monoidal, stable $\infty $ -category ${\mathcal C}$ with a commutative algebra object A and an A-module J, we define the square-zero extension $A \oplus J$ to be the image of J under

$$\begin{align*}\mathrm{Mod}_A({\mathcal C}) \simeq \mathrm{Sp}(\mathrm{CAlg}({\mathcal C})_{/A}) \xrightarrow{\Omega^\infty} \mathrm{ CAlg}({\mathcal C})_{/A}.\end{align*}$$

The heuristics of Section 6 suggest that the correct analogous object $(A, M) \oplus J$ in the context of log ring spectra is the image of J under

(7.1) $$ \begin{align}\mathrm{Mod}_A \simeq \mathrm{Sp}(\mathrm{Log}^{\mathrm{rep}}_{(A, M)//(A, M)}) \xrightarrow{\Omega^{\infty}} \mathrm{Log}^{\mathrm{rep}}_{(A, M)//(A, M)},\end{align} $$

where the equivalence is Corollary 6.12. In Remark 6.13, we identified this with the square-zero extension $(A \oplus J, M \oplus J)$ considered in [Reference Sagave38, Construction 5.6].

By Remark 6.13, the definitions above recover those considered in [Reference Rognes34, Section 11 and 13], [Reference Sagave38] and [Reference Lundemo22, Chapter 4]. The following is a consequence of our construction (or Theorem 1.14):

Lemma 7.3. The log cotangent complex corepresents log derivations; that is, there is a canonical equivalence

$$\begin{align*}\mathrm{Map}_{\mathrm{Mod}_A}({\Bbb L}_{(A, M)}^{\mathrm{rep}}, J) \simeq \mathrm{ Der}((A, M), J)\end{align*}$$

for log ring spectra $(A, M)$ and A-modules J.

7.4 Starting from a strict square-zero extension

We now aim to prove Theorem 1.16. With notation as in its statement, we shall first show that if $(p, p^\flat )$ is strict, then $(p, p^\flat )$ is a log square-zero extension by J. Since p is a square-zero extension by J, we obtain a cartesian diagram

(7.3)

of pre-log ring spectra. Observe that it is essential that we use the inverse image pre-log structure $\widetilde {P} \to P \to \Omega ^{\mathcal J}(R)$ , so that the pre-log structure on the square-zero extension $R \oplus J[1]$ is unambiguously defined. The following should be considered a higher variant of [Reference Sagave, Schürg and Vezzosi42, Theorem 4.17], as we explain in Remark 7.8.

Proposition 7.5. Let $(p, p^\flat )$ be strict. The logification of the cartesian diagram (7.3) is equivalent to a cartesian diagram of the form

of log ring spectra. In particular, $(p, p^\flat )$ is a log square-zero extension.

The remainder of this subsection is dedicated to the proof of Proposition 7.5. We first give a convenient description of the log structure on R in the presence of a strict map $(\widetilde {R}, \widetilde {P}) \to (R, P)$ with $\widetilde {R} \to R$ square-zero.

Lemma 7.6. The diagram

of commutative ${\mathcal J}$ -space monoids is cocartesian.

Proof. Consider the cartesian squares

of graded ${\Bbb E}_{\infty }$ -spaces. We have that $p^{-1}\mathrm {GL}_1^{\mathcal J}(R) \simeq \mathrm {GL}_1^{\mathcal J}(\widetilde {R})$ by Lemma 4.3, so that $(p \circ \widetilde {\beta })^{-1}\mathrm {GL}_1^{\mathcal J}(R) \simeq \mathrm {GL}_1^{\mathcal J}(\widetilde {R})$ since $(\widetilde {R}, \widetilde {P})$ is log. Hence, the pushout of the diagram

$$\begin{align*}\widetilde{P} \xleftarrow{} \mathrm{GL}_1^{\mathcal J}(\widetilde{R}) \xrightarrow{} \mathrm{ GL}_1^{\mathcal J}(R)\end{align*}$$

of graded ${\Bbb E}_{\infty }$ -spaces models the underlying graded ${\Bbb E}_{\infty }$ -space of the logification of $(R, \widetilde {P})$ , which is equivalent to P by the assumption that $(\widetilde {R}, \widetilde {P}) \to (R, P)$ is strict. This concludes the proof.

Let us now identify the logification of $(R \oplus J[1], \widetilde {P})$ .

Lemma 7.7. The logification of the pre-log structure

(7.4) $$ \begin{align}\widetilde{P} \xrightarrow{\widetilde{\beta}} \Omega^{\mathcal J}(\widetilde{R}) \xrightarrow{\Omega^{\mathcal J}(p)} \Omega^{\mathcal J}(R) \xrightarrow{d} \Omega^{\mathcal J}(R \oplus J[1])\end{align} $$

is equivalent to the split square-zero extension $(R, P) \oplus J[1]$ of Remark 6.13.

Proof. Pulling back the pre-log structure (7.4) along the canonical inclusion morphism $\mathrm {GL}_1^{\mathcal J}(R \oplus J[1]) \to \Omega ^{\mathcal J}(R \oplus J[1])$ , we obtain a commutative diagram

in which every square is a pullback: The left-hand square is so by definition, while the middle and right-hand squares are pullbacks by Lemma 4.3. Hence, the logification of the pre-log structure (7.4) is obtained by cobase-change of the map $\mathrm {GL}_1^{\mathcal J}(\widetilde {R}) \to \widetilde {P}$ along the composite

$$\begin{align*}\mathrm{ GL}_1^{\mathcal J}(\widetilde{R}) \xrightarrow{\mathrm{GL}_1^{\mathcal J}(p)} \mathrm{GL}_1^{\mathcal J}(R) \xrightarrow{\mathrm{GL}_1^{\mathcal J}(d)} \mathrm{GL}_1^{\mathcal J}(R \oplus J[1]),\end{align*}$$

which is equivalent to the composite

$$\begin{align*}\mathrm{GL}_1^{\mathcal J}(\widetilde{R}) \xrightarrow{\mathrm{GL}_1^{\mathcal J}(p)} \mathrm{GL}_1^{\mathcal J}(R) \xrightarrow{\mathrm{GL}_1^{\mathcal J}(d_0)} \mathrm{GL}_1^{\mathcal J}(R \oplus J[1]).\end{align*}$$

By Lemma 7.6, we hence obtain that the logification of the pre-log structure (7.4) can be described as the pushout of the diagram

$$\begin{align*}P \xleftarrow{} \mathrm{GL}_1^{\mathcal J}(R) \xrightarrow{\mathrm{GL}_1^{\mathcal J}(d_0)} \mathrm{GL}_1^{\mathcal J}(R \oplus J[1])\end{align*}$$

of commutative ${\mathcal J}$ -space monoids. The result follows from the equivalence

(7.5) $$ \begin{align}\mathrm{GL}_1^{\mathcal J}(R) \boxtimes (1 + J[1]) \xrightarrow{\simeq} \mathrm{GL}_1^{\mathcal J}(R \oplus J[1])\end{align} $$

of [Reference Sagave38, Lemma 5.4].

Remark 7.8. An analog of Proposition 7.5 appears in [Reference Sagave, Schürg and Vezzosi42, Theorem 4.17] for integral log rings $(R, P)$ . In its proof, the authors consider the projection

$$\begin{align*}\delta \colon R \xrightarrow{d} R \oplus J[1] \xrightarrow{} J[1]\end{align*}$$

for a derivation d, and they define a monoid map

$$\begin{align*}\mathrm{GL}_1(R) \to P \oplus J[1], \quad x \mapsto (x, x^{-1}\delta(x)),\end{align*}$$

where we have used the log condition on $(R, P)$ to identify $x \in \mathrm {GL}_1(R)$ with its image in P. In the presence of a strict square-zero extension $(\widetilde {R}, \widetilde {P}) \to (R, P)$ , this defines a log derivation $(d, d^\flat ) \colon (R, P) \to (R, P) \oplus J[1]$ in a natural way, as the proof of loc. cit. shows. The analog of this appears at the very end of the proof of Lemma 7.7, where we make implicit reference to a homotopy inverse of the equivalence (7.5). Indeed, in the context of ordinary rings, this is given by

$$\begin{align*}\mathrm{GL}_1(R \oplus J[1]) \xrightarrow{\cong} \mathrm{GL}_1(R) \oplus (1 + J[1]), \quad (x, j) \mapsto (x, (1, x^{-1}j)).\end{align*}$$

We now return to working towards a proof of Proposition 7.5. Lemma 7.7 implies that the logification of the square (7.3) has an underlying square of commutative ${\mathcal J}$ -space monoids of the form

(7.6)

Proof. Observe that the projection $P \oplus J[1] \to P$ is replete by Corollary 4.6. Hence, $d^\flat $ is replete by pasting, and the assumption that J is connective. It thus suffices to prove that the square is cartesian after replacing $d^\flat $ with its group completion $d^{\flat , \mathrm {gp}}$ . Consider the square

of graded ${\Bbb E}_{\infty }$ -spaces. Lemma 7.6 shows that the upper right-hand and lower left-hand corners model P, while (the proof of) Lemma 7.7 shows that the lower right-hand corner models $P \oplus J[1]$ . Hence, this square models (7.6) with $d^\flat $ replaced with $d^{\flat , \mathrm {gp}}$ , and since J is connective, Lemma 3.10 applies to conclude.

7.10 Starting with a log square-zero extension

We now aim to prove the other direction of Theorem 1.16.

Proof. Consider the commutative diagram

(7.7)

of commutative ${\mathcal J}$ -space monoids, in which both squares are homotopy cartesian. We will show that the left-hand square is homotopy cocartesian, which we claim suffices to conclude. Indeed, if the left-hand square in (7.7) is cocartesian, then both squares in the diagram

are cocartesian, where the right-hand square is the defining cocartesian square for the logification of $(R, P)$ . Since the left-hand square is cocartesian, so is the outer square. But the pushout of $\widetilde {P} \xleftarrow {} {(\alpha \circ p^\flat )^{-1}}\mathrm {GL}_1^{\mathcal J}(R) \xrightarrow {} \mathrm {GL}_1^{\mathcal J}(R)$ is the logification of $\widetilde {P} \xrightarrow {p^\flat } P \xrightarrow {\alpha } \Omega ^{\mathcal J}(R)$ , and so $(p, p^\flat )$ is strict.

We now proceed to prove that the left-hand square in (7.7) is cocartesian. Notice that the map $\overline {i}$ gives rise to a map $\overline {i} \oplus J[1] \colon \alpha ^{-1}\mathrm {GL}_1^{\mathcal J}(R) \oplus J[1] \to P \oplus J[1]$ . Consider the commutative cube

of commutative ${\mathcal J}$ -space monoids. Here, the back face is the homotopy cartesian square exhibiting $\widetilde {P} \to P$ as part of a log square-zero extension. The right-hand face arises from pulling back the monoid derivation $d^\flat $ along $\overline {\alpha }$ and is as such homotopy cartesian. The left-hand face is the left-hand homotopy cartesian square in the diagram (7.7), and this implies that the front face is homotopy cartesian. Moreover, the bottom face is homotopy cocartesian. Since $d^\flat $ is replete (cf. proof of Lemma 7.9), Proposition 3.6 applies to conclude the proof.

7.12 The log Postnikov tower

Let $(R, P)$ be a log ring spectrum with R connective. We wish to define a tower of log square-zero extensions under $(R, P)$ that is compatible with the Postnikov tower of R. As a consequence of Theorem 1.16, there is essentially no choice:

Definition 7.13. There $(R, P)$ be a log ring spectrum with R connective. We define the log Postnikov tower of $(R, P)$ to be the tower

$$\begin{align*}\cdots \to (\tau_{\le 2}(R), P_{\le 2}) \to (\tau_{\le 1}(R), P_{\le 1}) \to (\pi_0(R), P_{\le 0}),\end{align*}$$

where each $(\tau _{\le n}(R), P_{\le n}) := (\tau _{\le n}(R), P^a)$ is defined as the logification of the inverse image pre-log structure $P \to \Omega ^{\mathcal J}(R) \to \Omega ^{\mathcal J}(\tau _{\le n}(R))$ .

Proof. This is true by [Reference Lurie25, Proposition 7.1.3.9] at the level of ${\Bbb E}_{\infty }$ -rings. By Theorem 1.16, it suffices to prove that the map is strict. Each truncation morphism $(R, P) \to (\tau _{\le n}(R), P_{\le n})$ is strict by definition. Lemma 5.11 thus implies that

$$\begin{align*}(\tau_{\le (n - 1)}(R), P_{\le (n - 1)}) \simeq (\tau_{\le (n - 1)}(R), \mathrm{GL}_1^{\mathcal J}(\tau_{\le (n - 1)}(R))) \otimes_{(R, \mathrm{GL}_1^{\mathcal J}(R))} (R, P).\end{align*}$$

Using that the map $(R, \mathrm {GL}_1^{\mathcal J}(R)) \to (\tau _{\le (n - 1)}(R), \mathrm {GL}_1^{\mathcal J}(\tau _{\le (n - 1)}(R)))$ factors over $(\tau _{\le n}(R), P_{\le n})$ and applying Lemma 5.11 again to the map $(R, P) \to (\tau _{\le n}(R), P_{\le n})$ , we identify the above relative coproduct with

$$\begin{align*}(\tau_{\le (n - 1)}(R), \mathrm{ GL}_1^{\mathcal J}(\tau_{\le (n - 1)}(R))) \otimes_{(\tau_{\le n}(R), \mathrm{GL}_1^{\mathcal J}(\tau_{\le n}(R)))} (\tau_{\le n}(R), P_{\le n}),\end{align*}$$

which concludes the proof by a third application of Lemma 5.11.

Remark 7.15. By Lemma 7.6, each of the graded ${\Bbb E}_{\infty }$ -spaces $P_{\le n}$ admit the explicit description $P \boxtimes _{\mathrm {GL}_1^{\mathcal J}(R)} \mathrm {GL}_1^{\mathcal J}(\tau _{\le n}(R))$ . In particular, the underlying tower of graded ${\Bbb E}_{\infty }$ -spaces is cobase-changed from the Postnikov tower of R in the sense that it is of the form

$$\begin{align*}\cdots \! \to \!P \boxtimes_{\mathrm{GL}_1^{\mathcal J}\!\!(R)} \mathrm{GL}_1^{\mathcal J}(\tau_{\le 2}(R)) \to P \boxtimes_{\mathrm{GL}_1^{\mathcal J}\!\!(R)} \mathrm{GL}_1^{\mathcal J}(\tau_{\le 1}(R)) \to P \boxtimes_{\mathrm{GL}_1^{\mathcal J}\!\!(R)} \mathrm{GL}_1^{\mathcal J}(\pi_0(R)).\end{align*}$$

We thus see that the bottom-most stage of the tower is not necessarily discrete, in contrast to the Postnikov tower of a connective ${\Bbb E}_{\infty }$ -ring.

Proof. As Postnikov towers of ring spectra converge, it suffices to show that the limit of the diagram of underlying commutative ${\mathcal J}$ -space monoids is P. Lemma 3.10 shows that both squares in the diagram

are cartesian. As limits commute with limits, the limit in question is equivalent to

$$\begin{align*}(P \boxtimes_{\mathrm{GL}_1^{\mathcal J}(R)} \mathrm{GL}_1^{\mathcal J}(\pi_0(R))) \times_{(P^{\mathrm{gp}} \boxtimes_{\mathrm{GL}_1^{\mathcal J}(R)} \mathrm{GL}_1^{\mathcal J}(\pi_0(R)))} \mathrm{lim}(P^{\mathrm{gp}} \boxtimes_{\mathrm{GL}_1^{\mathcal J}(R)} \mathrm{GL}_1^{\mathcal J}(\tau_{\le n}(R))).\end{align*}$$

It thus suffices to prove that $\mathrm {lim}(P^{\mathrm {gp}} \boxtimes _{\mathrm {GL}_1^{\mathcal J}(R)} \mathrm {GL}_1^{\mathcal J}(\tau _{\le n}(R))) \simeq P^{\mathrm {gp}}$ , as the outer rectangle above is cartesian. The result now follows from Corollary 7.18 below.

In the part of the proof of Proposition 7.16 provided above, we reduced to checking that the tower

$$\begin{align*}\cdots \to P^{\mathrm{gp}} \boxtimes_{\mathrm{GL}_1^{\mathcal J}\!\!(R)} \mathrm{GL}_1^{\mathcal J}(\tau_{\le 1}(R)) \to P^{\mathrm{gp}} \boxtimes_{\mathrm{GL}_1^{\mathcal J}\!\!(R)} \mathrm{GL}_1^{\mathcal J}(\pi_0(R))\end{align*}$$

converges to $P^{\mathrm {gp}}$ . For this, we shall exploit the relationship between grouplike commutative ${\mathcal J}$ -space monoids and the category of connective spectra over the sphere of [Reference Sagave39, Theorems 1.6, 5.10], therein modelled by an appropriate slice of the stable model structure on Segal $\Gamma $ -spaces. The slice is over a certain Segal $\Gamma $ -space $b{\mathcal J}$ , which models the sphere spectrum. These are connected by a chain of Quillen equivalences, and the resulting derived functor is denoted $\gamma $ , which gives an equivalence relating their underlying $\infty $ -categories. As a first step, we need the following:

We warn the reader that this is not as obvious as it might seem at first sight. While the analogous statement is clear for the spectrum of units $\mathrm {gl}_1(R)$ , the spectrum of graded units $\gamma (\mathrm { GL}_1^{\mathcal J}(R))$ exhibits different behaviour, even when the ring spectrum R is connective.

Proof of Lemma 7.17.

Consider first the case where $R = \tau _{\le (n + 1)}(R)$ , so that the truncation map $\tau _{\le (n + 1)}(R) \to \tau _{\le n}(R)$ is a square-zero extension. The cartesian square exhibiting this map as such gives a cartesian square of commutative ${\mathcal J}$ -space monoids upon applying $\mathrm {GL}_1^{\mathcal J}(-)$ . Hence, there is a cartesian square

of Segal $\Gamma $ -spaces over $b{\mathcal J}$ . By [Reference Sagave38, Lemma 5.4, Proposition 5.5, Lemma 7.2], the lower right-hand corner splits as $\gamma (\mathrm {GL}_1^{\mathcal J}(\tau _{\le n}(R))) \times \gamma ((1 \oplus \pi _{n + 1}(R)[n + 2]))$ , and the term $\gamma ((1 \oplus \pi _{n + 1}(R)[n + 2])$ models the connective spectrum $\pi _{n + 1}(R)[n + 2]$ . Hence, the fiber of the map of spectra $\gamma (\mathrm {GL}_1^{\mathcal J}(\tau _{\le (n + 1)}(R))) \to \gamma (\mathrm {GL}_1^{\mathcal J}(\tau _{\le n}(R)))$ is n-connected, as desired.

The result follows as $\gamma (\mathrm {GL}_1^{\mathcal J}(R))$ is the limit of the tower

$$\begin{align*}\cdots \to \gamma(\mathrm{GL}_1^{\mathcal J}(\tau_{\le (n + 1)}(R))) \to \gamma(\mathrm{GL}_1^{\mathcal J}(\tau_{\le n}(R)))\end{align*}$$

where each map from $\gamma (\mathrm {GL}_1^{\mathcal J}(\tau _{\le (n + k)}(R)))$ is n-connected; cf. [Reference Bhatt, Morrow and Scholze8, Proof of Lemma 3.3].

Corollary 7.18. Each map $\gamma (P^{\mathrm {gp}}) \to \gamma (P^{\mathrm {gp}} \boxtimes _{\mathrm {GL}_1^{\mathcal J}(R)} \mathrm {GL}_1^{\mathcal J}(\tau _{\le n}(R)))$ is n-connected. In particular, the canonical map

$$\begin{align*}\gamma(P^{\mathrm{gp}}) \to \mathrm{lim}(\gamma(P^{\mathrm{gp}} \boxtimes_{\mathrm{GL}_1^{\mathcal J}(R)} \mathrm{GL}_1^{\mathcal J}(\tau_{\le n}(R))))\end{align*}$$

is an equivalence. Since the functor $\gamma $ is an equivalence, the canonical map

$$\begin{align*}P^{\mathrm{gp}} \to \mathrm{lim}(P^{\mathrm{gp}} \boxtimes_{\mathrm{GL}_1^{\mathcal J}(R)} \mathrm{GL}_1^{\mathcal J}(\tau_{\le n}(R)))\end{align*}$$

is also an equivalence.

Proof. This follows from Lemma 7.17 and the cocartesian squares

of connective spectra over the sphere. It is, in particular, a cocartesian square of spectra, so the lower map is n-connected because the upper one is.