In particular, we can write $\otimes_{\mathcal{C}} = \mu_2$ and call it the tensor product associated with $\mathcal{C}^{\otimes}$. For any object $X$ in $\mathcal{C}$, one can consider the $\infty$-functor:

$$ \Sigma_X : \mathcal{C} \to \mathcal{C}, Y \to X \otimes Y = \mu_2(X, Y). $$

According to our choice of morphisms in $\mathcal{C} \otimes_{\infty}^{\text{st}}$ (Definition 6.4.9), this is a left functor (Definition 6.3.5) so that it automatically admits a right adjoint $\Omega_X = \text{Hom}_{\mathcal{C}}(X, -)$ — according to 6.3.1(2). One deduces a bifunctor:

$$ \text{Hom} : \mathcal{C} \times \mathcal{C} \to \mathcal{C} $$

which is exact.

As the associated homotopy category functor $\text{Ho}$ commutes with products, one deduces that the associated homotopy category $\text{Ho}(\mathcal{C})$ is symmetric monoidal, and moreover closed. Besides, starting from Theorem 6.4.4, $\text{Ho}(\mathcal{C})$ is even a triangulated monoidal category.

**Example 6.5.7.** An $\infty$-category $\mathcal{C}_0$ which admits finite products can be endowed with a symmetric monoidal $\infty$-category structure, with associated tensor product given by the cartesian product $\times$. See [Lur17, §2.4.1].

Using the so-called *Day convolution product*, Lurie further proved (see [NS17, after Def. 2.1], or directly [Lur17, Cor. 4.8.1.12]) that the $\infty$-category of presheaves $\mathcal{P}\text{Sh}(\mathcal{C}_0)$ admits a symmetric monoidal structure such that the Yoneda embedding

$$ \gamma : \mathcal{C}_0 \to \mathcal{P}\text{Sh}(\mathcal{C}_0) $$

is monoidal. Beware that this monoidal structure is in general different from the one coming from the cartesian product on $\mathcal{P}\text{Sh}(\mathcal{C}_0)$, except when the $\infty$-category $\mathcal{C}_0$ admits finite products.

**6.5.8. Tensor invertible objects.** Let $\mathcal{C}$ a presentable symmetric monoidal $\infty$-category and $X$ an arbitrary object. One says that $X$ is $\otimes$-invertible if the $\infty$-functor $\Sigma_X$, defined in 6.5.6, is an equivalence of $\infty$-categories.

Marco Robalo has described in [Rob15, §2.1] a universal procedure to $\otimes$-invert the object $X$. Indeed, he shows that the sub-$\infty$-category of the comma $\infty$-category $\mathcal{C}/\mathcal{C}\otimes_{\infty}^{\otimes}$ (see Example 6.2.2) spanned by the monoidal $\infty$-functors $F : \mathcal{C} \to \mathcal{D}$ such that $F(X)$ is $\otimes$-invertible admits an initial object — combine Proposition 2.1 and Proposition 2.9(1). We denote such an initial object (well-defined up to a contractible set of choices) by $\Sigma_X^{\infty} : \mathcal{C} \to \mathcal{C}[X^{-1}]$.

If, in addition, the object $X$ is symmetric (see [Rob15, Def. 2.16]), then the monoidal $\infty$-category $\mathcal{C}[X^{-1}]$ can be described as the *spectrum objects relative to* $X$; see *loc. cit.* Corollary 2.22. In other words, $\mathcal{C}[X^{-1}]$ is the homotopy colimit in the $\infty$-category $\mathcal{C}\otimes_{\infty}^{\otimes}$ of the following tower of presentable monoidal $\infty$-categories:

$$ \mathcal{C} \xrightarrow{\Sigma_X} \mathcal{C} \xrightarrow{\Sigma_X} \dots $$

where $\Sigma_X$ was defined in 6.5.6. In this construction, the assumption that $X$ is symmetric is used via *loc. cit.* Theorem 2.14.
This implies in particular that whenever $\mathcal{C}$ is stable and $X$ is symmetric, the $\infty$-category $\mathcal{C}[X^{-1}]$ is stable and $\Sigma_X^\infty$ is exact.

**Example 6.5.9.** Following [Lur17, §1.4.2], one can define the stabilization $\mathcal{C}_{st}$ of a pointed $\infty$-category $\mathcal{C}$ with finite limits by considering the limit of the (left) tower of pointed $\infty$-categories

$$ \dots \xrightarrow{\Omega} \mathcal{C} \xrightarrow{\Omega} \mathcal{C} $$

where $\Omega$ is the loop space functor defined in 6.4.3. If $\mathcal{C}$ is a symmetric monoidal $\infty$-category, one obtains a symmetric monoidal $\infty$-category structure on $\mathcal{C}_{st}$ by taking the colimit in the $\infty$-category $\mathcal{C}$ at $\infty$.

Building on Example 6.5.7, $\mathcal{C}$ admits a symmetric monoidal $\infty$-categorical structure whose tensor product is given by the smash product. One can view the simplicial sphere $S^1$ as an object of $\mathcal{C}$ by the formula $S^1 = \Sigma*$. Then one obtains the identification $\Omega = \Omega_{S^1}$ and $\Sigma = \Sigma_{S^1}$ (by using adjunctions properties). Note that $S^1$ is a symmetric object.<fnref n="73" />

One can show that the stabilization $\mathcal{C}_{st}$ defined above coincides with the monoidal $\infty$-category $\mathcal{C}[(S^1)^{-1}]$ constructed previously. In other words, $\mathcal{C}_{st}$ can be identified with the colimit of the (right) tower of presentable monoidal $\infty$-categories

$$ \mathcal{C} \xrightarrow{\Sigma} \mathcal{C} \xrightarrow{\Sigma} \dots $$

computed in the $\infty$-category of presentable monoidal $\infty$-categories. This follows from [Lur18, Rem. C.1.1.6]. Note that if one computes the previous colimit in the $\infty$-category of $\infty$-categories, then one obtains a smaller $\infty$-category called the Spanier-Whitehead category. It is not stable but only *pre-stable* (see loc. cit.).

**6.5.10. Monoidal structures and localizations.** Let $\mathcal{C}$ be as in 6.5.6. We consider a set of morphisms $W$ of $\mathcal{C}$ and let us consider the canonical left functor $\pi : \mathcal{C} \to \mathcal{C}[W^{-1}]$.

Then the following conditions are equivalent (see [Lur17, Prop. 2.2.1.9]):

(1) There exists a symmetric monoidal $\infty$-category structure on $\mathcal{C}[W^{-1}]$ such that the functor $\pi$ extends to a symmetric monoidal left functor.

(2) The $W$-local equivalences in $\mathcal{C}$ are stable under tensor product: for any $W$-local equivalence $f : M \to N$, and any object $P$ of $\mathcal{C}$, $f \otimes P$ is a $W$-local equivalence.

In that case, we say that the $W$-localization is monoidal. By abuse of notation, we say that the functor $\pi$ is symmetric monoidal.

**Example 6.5.11.** Let $\mathcal{S}h_t(S)$ be the $\infty$-topos of $t$-sheaves on a Grothendieck site as in Example 6.3.7. One deduces from the description of Čech covers that $\mathcal{S}h_t(S)$ admits a symmetric monoidal structure such that $\infty$-functor $a : \mathcal{P}Sh(S) \to \mathcal{S}h(S)$

<footnote n="73">⁷³This follows from the corresponding property of the simplicial set $S^1$ in the $\infty$-category of spaces. Or one can directly deduce this from the symmetric monoidal structure on $\mathcal{C}$.</footnote>
is symmetric monoidal. Moreover, when the site S admits products, the corresponding tensor product on $\mathcal{S}h(S)$ is the cartesian product.

**6.5.12. Monoidal model categories.** It is established in [NS17, Prop. 2.3] that the $\infty$-category $\mathcal{M}[W^{-1}]$ associated to any simplicial, combinatorial, tractable, left proper and symmetric monoidal model category $\mathcal{M}$ with weak equivalences $W$ admits a canonical symmetric monoidal $\infty$-categorical structure.

Moreover, it is proved in [NS17, Prop. 2.4, Th. 2.8] that every presentable symmetric monoidal $\infty$-category $\mathcal{C}$ arise in that way: there exists an equivalence $\mathcal{C} \simeq \mathcal{M}[W^{-1}]$.

In other words, one can freely use the construction of model categories to benefit from the language of $\infty$-category theory.

## 6.6. Pro-objects in $\infty$-categories.

**6.6.1.** Our reference for pro-objects is [BHH17, §5.2].$^{74}$

Given a presentable $\infty$-category $\mathcal{C}$, there exists an $\infty$-category pro-$\mathcal{C}$ which admits limits (see 6.3.2), an $\infty$-functor $\iota: \mathcal{C} \to \text{pro-}\mathcal{C}$ which satisfy the following universal property: for any $\infty$-category $\mathcal{D}$ which admits all limits, the $\infty$-functor

$$ \iota^*: \mathcal{F}\text{un}(\text{pro-}\mathcal{C}, \mathcal{D}) \to \mathcal{F}\text{un}(\mathcal{C}, \mathcal{D}) $$

is fully faithful and induces an equivalence of $\infty$-categories with the full sub-$\infty$-category of the left-hand side spanned by those functors which commute with limits (see again 6.3.2).

We refer the reader to [BHH17, Th. 3.2.19]. A simple construction of the $\infty$-category pro-$\mathcal{C}$ is to consider the full sub-$\infty$-category of $\mathcal{F}\text{un}(\mathcal{C}, \mathcal{S})^{\text{op}}$ of functors which are accessible and commute with finite limits (see [BHH17, Prop. 3.2.18]).

**Remark 6.6.2.** In fact, as we are implicitly working within universes, one can also rely on the formula:

$$ \text{pro-}\mathcal{C} = (\text{ind}(\mathcal{C}^{\text{op}}))^{\text{op}} $$

and use the treatment of ind-objects in [Lur09, §5.3].

**Example 6.6.3.** Let $\mathcal{C}$ be an (ordinary) category. Then one obtains the following identification:

$$ \text{pro-}(\mathcal{N} \mathcal{C}) = \mathcal{N} (\text{pro-}\mathcal{C}) $$

using, on the right-hand side, the classical definition of pro-objects. Indeed, this follows from the universal properties of pro-objects in both the $\infty$-categorical framework (as above) and the ordinary categorical one (see [AGV73], §8.10, 8.13).

**6.6.4.** By the very construction, the $\infty$-category pro-$\mathcal{C}$ associated with a presentable $\infty$-category admits cofiltered projective limits. In particular, given an

<footnote>$^{74}$Which also proposes a detailed treatment of set-theoretic issues inherent to the notions of limits and colimits.</footnote>
$\infty$-functor $X : \mathcal{I} \to \mathcal{C}, i \mapsto X_i$, one can consider the associated projective limit in pro-$\mathcal{C}$ which we will denote by

$$ \lim_{i \in \mathcal{I}} X_i $$

following the classical notation (see [AGV73, I, 8.5.3.2]).

According to [BHH17, Prop. 5.1.1, 5.1.2], given two $\infty$-functors $X : \mathcal{I} \to \mathcal{C}$ and $Y : \mathcal{I} \to \mathcal{C}$, there exists a canonical equivalence:

$$ (6.6.4.a) \qquad \mathrm{Map}(\text{"lim"}_{i \in \mathcal{I}} X_i, \text{"lim"}_{j \in \mathcal{I}} Y_j) \simeq \lim_{j \in \mathcal{I}} i \in \mathcal{I} \operatorname{colim} \mathrm{Map}(X_i, Y_j). $$

One should be careful that, in general,

$$ \mathrm{Ho}(\mathrm{pro-}\mathcal{C}) \neq \mathrm{pro-Ho}\mathcal{C}. $$

But see Proposition 3.2.3 for a case of interest.

## 7. CATEGORICAL PRESENTATION OF CYCLE MODULES

One can describe Rost's cycle premodules as covariant functors from a category whose morphisms are defined by generators and relations. We reproduce below the definition from [Dég08b, §4.1] for the reader's convenience.

**Definition 7.0.1.** We let $\tilde{\mathcal{E}}_k$ be the category whose objects are given by pairs $(E, n)$ where $E/k$ is a function field, and $n \in \mathbb{Z}$ an integer. The morphisms of $\tilde{\mathcal{E}}_k$ are abelian groups defined by the following generators (D\*) and relations (R\*) as displayed below, where maps are always assumed to be $k$-algebra morphisms, and the valuations are always understood as (geometric) valuations of function fields:
**Generators:**

D1: $\varphi_* : (E, n) \to (L, n)$ for $\varphi : E \to L, n \in \mathbb{Z}$.

D2: $\varphi' : (L, n) \to (E, n)$, for $\varphi : E \to L$ finite, $n \in \mathbb{Z}$.

D3: $\gamma_x : (E, n) \to (E, n + r)$, for $x \in K_r^M(E), n \in \mathbb{Z}$.

D4: $\partial_v : (E, n) \to (\kappa(v), n - 1)$, for $(E, v)$ valued function field over $k, n \in \mathbb{Z}$.

**Relations:**

R0: For all $x, y \in K_*^M(E)$, $\gamma_x \circ \gamma_y = \gamma_{x,y}$.

R1a: $(\psi \circ \varphi)_* = \psi_* \circ \varphi_*$.

R1b: $(\psi \circ \varphi)' = \varphi' \circ \psi'$.

R1c: Let $\varphi : K \to E$, $\psi : K \to L$ be finite. For any $z \in \mathrm{Spec}(E \otimes_K L)$, let $\bar{\varphi}_z : L \to E \otimes_K L/z$ and $\bar{\psi}_z : E \to E \otimes_K L/z$ be the induced morphisms:
$$ \psi_* \varphi' = \sum_{z \in \mathrm{Spec}(E \otimes_K L)} \lg(E \otimes_K L_z) \cdot (\bar{\varphi}_z)' (\bar{\psi}_z)_*, $$
where $\lg(A)$ is the length of an Artin ring $A$.

R2a: For all $\varphi : E \to L, x \in K_*^M(E)$, $\varphi_* \circ \gamma_x = \gamma_{\varphi_*(x)} \circ \varphi_*$.

R2b: For $\varphi : E \to L$ finite and all $x \in K_*^M(E)$, $\varphi' \circ \gamma_{\varphi_*(x)} = \gamma_x \circ \varphi'$.

R2c: For $\varphi : E \to L$ finite and all $y \in K_*^M(L)$, $\varphi' \circ \gamma_y \circ \varphi_* = \gamma_{\varphi'(y)}$.
**R3a:** Let $\varphi : E \to L$ be a morphism, $v$ and $w$ be valuations $L/k$ and $E/k$ respectively, and $e > 0$ an integer such that $v|_{E^\times} = e.w$. Let $\bar{\varphi} : \kappa(w) \to \kappa(v)$ be the induced morphism:

$$ \partial_v \circ \varphi_* = e.\bar{\varphi}_* \circ \partial_w. $$

**R3b:** Let $\varphi : E \to L$ be finite, and $v$ a valuation on $E/k$. Given a valuation $w$ on $L$ which extends $v$, we let $\bar{\varphi}_w : \kappa(w) \to \kappa(v)$ be the induced morphism on the respective residue fields:

$$ \partial_v \circ \varphi^! = \sum_{w/v} \bar{\varphi}_w^! \circ \partial_w. $$

**R3c:** Let $\varphi : E \to L$, $v$ a valuation on $L/k$ which is zero on $E^\times$: $\partial_v \circ \varphi_* = 0$.

**R3d:** Let $(E, v)$ be a valued function field over $k$ and $\pi$ uniformizer of $v$: $\partial_v \circ \gamma_{\{-\pi\}} \circ \varphi_* = \bar{\varphi}_*.$

**R3e:** Let $(E, v)$ be a valued function field over $k$, $u \in E^\times$ such that $v(u) = 0$: $\partial_v \circ \gamma_{\{u\}} = -\gamma_{\{\bar{u}\}} \circ \partial_v.$

According to [Ros96, Def. 1.1], a cycle premodule over $k$ is simply a (covariant) functor $M : \tilde{\mathcal{E}} \to Ab$.

**7.0.2. Higher order valuations.**— The preceding category admits a more elegant description due to Rost. We introduce some notation to state it.

Let $F$ be field with an arbitrary valuation (not necessarily discrete)

$$ v : F^\times \to \Gamma $$

where $\Gamma$ is a totally ordered abelian group. We refer the reader to [Vaq06] about arbitrary valuation rings.

As usual, we denote by:

$$ \mathcal{O}_v = \{x \in F \mid v(x) \geq 0\} $$

$$ \mathcal{M}_v = \{x \in F \mid v(x) > 0\} $$

the associated valuation ring, and the maximal ideal of the latter, where we extend $v$ as usual by setting $v(0) = \infty$, for $\Gamma_\infty = \Gamma \cup \{\infty\}$, obtained by adding a maximal element $\infty$. We also let $\kappa_v = \mathcal{O}_v/\mathcal{M}_v$ be the residue (class) field.

Recall that the rank of $v$ is the rank of the abelian group $\text{Im}(v)$ (loc. cit. Definition after Th. 1.7), or equivalently the (Krull) dimension of the valuation ring $\mathcal{O}_v$ (loc. cit. Corollary after Th. 1.7). If $v$ has finite rank $r$, then $\Gamma \simeq \mathbb{Z}^r$ and $v$ can be defined as a composition of $r$ valuations $(v_1, \cdots, v_r)$, $v_{i+1}$ being a valuation on the residue field $\kappa_{v_i}$ (see loc. cit. Remark 1.7).

**7.0.3. Milnor K-theory of valuation rings.**— We consider an arbitrary valuation $v : F \to \Gamma$. Using the above notation, we have an exact sequence of abelian groups:

$$ (7.0.3.a) \qquad 1 \to (1 + \mathcal{M}_v) \to \mathcal{O}_v^\times \to \kappa_v^\times \to 1 $$

Following [Ros96, Rem. 1.6], we define a $\mathbb{Z}$-graded ring associated to $v$ by the formula:

$$ K_*^M(v) = K_*^M(F)/(1 + \mathcal{M}_v). $$
Considering the obvious projection map $p : K_*^M(F) \to K_*^M(v)$, there exists a morphism $i : K_*^M(\kappa_v) \to K_*^M(v)$ of graded rings uniquely defined by the universal property$^{75}$:

$$ \begin{array}{ccc} K_*^M(\mathcal{O}_v) & \xrightarrow{p} & K_*^M(v) \\ \bigdownarrow & & \bigdownarrow \\ K_*^M(\kappa_v) & & \kappa_v \end{array} $$

where the vertical map is induced by the obvious surjection. The following proposition was stated in [Ros96, Rem. 1.10].

**Proposition 7.0.4.** Let $k$ be an arbitrary field, and $E, F$ be function fields over $k$. Given an integer $r \ge 0$, we consider the following set:

$$ \mathcal{I}_r(F, E) = \left\{ (v_*, L) \;\middle|\; \begin{array}{l} v_1, \dots, v_r \text{ geom. valuations on } F/k, v = v_1 \circ \dots \circ v_r \\ L = \kappa_x, x \in \text{Spec}(E \otimes_k \kappa_v), E \xrightarrow{\varphi} L \xleftarrow{\psi} \kappa_v, \varphi \text{ finite} \end{array} \right\} $$

Then the following map:

$$ \bigoplus_{r \ge 0, (v_*, L) \in \mathcal{I}_r(F, E)} K_*^M(L) \otimes_{K_*^M(\kappa_v)} K_*^M(v) \to \operatorname{Hom}_{\tilde{\mathcal{E}}_k}((F, 0), (L, *)) \\ \sigma \otimes \bar{\tau} \mapsto \varphi' \circ \gamma_\sigma \circ \psi_* \circ \partial_{v_1} \circ \dots \circ \partial_{v_r} \circ \gamma_\tau $$

where $\tau \in K_*^M(\mathcal{O}_v)$ is an arbitrary element such that $p(\tau) = \bar{\tau}$, is well-defined and induces an isomorphism of graded abelian groups.

Obviously, given that the functor $(E, 0) \mapsto (E, 1)$ is an auto-equivalence of the category $\tilde{\mathcal{E}}_k$, this completely describes the morphism in the latter category. The proof follows by applying the relations $(R*)$ to put morphisms in $\tilde{\mathcal{E}}_k$ in the above normalized form. A key point is Lemma 1.9 of [Ros96].

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