2.1 Generalized traces

Let $\mathscr {C}$ be a symmetric monoidal $\infty $ -category, and let $X \in \mathscr {C}$ is a dualizable object, as defined in the introduction of this section. We will be interested in the generalized endomorphisms of X, that is, morphisms in $\mathscr {C}$ of the form

where Y and Z are objects of $\mathscr {C}$ .

Definition 2.1 (Generalized trace).

Given a generalized endomorphism $f\colon Z \otimes X \to X \otimes Y$ in $\mathscr {C}$ , its generalized trace map

$$\begin{align*}{\mathrm{tr}}(f \,\vert \, X) \colon Z \longrightarrow Y \quad\in\quad \mathscr{C} \end{align*}$$

is defined as the following composition:

If Y and Z are the monoidal unit , the map is called the trace of the endomorphism $f\colon X \to X$ . If f is the identity on X, we will call its trace the dimension Footnote ⁵ of X:

We now give several examples of dimensions and traces. For an extensive discussion of symmetric monoidal traces, including more examples, we refer the reader to [Reference Ponto and Shulman58].

Example 2.3. When , the trace of $f\colon Z \to Y$ is simply f itself:

Example 2.5 (Field trace).

Consider a finite field extension $K \to L$ . The generalized trace of the multiplication map $L \otimes _K L \to L$ is a K-linear map

$$\begin{align*}{\mathrm{tr}}_{L/K}\colon L \to K, \end{align*}$$

known as the field trace of L over K. It sends an element $\alpha \in L$ to the trace of the endomorphism $m_{\alpha }\colon L \to L$ given by multiplication by $\alpha $ .

More generally, if k is a commutative ring and R is a dualizable k-algebra, applying the generalized trace construction to the multiplication map $R \otimes _k R \to R$ gives a k-linear trace map $ {\mathrm {tr}}_{R/k}\colon R \to k$ .

Remark 2.8. The generalized trace is functorial in Y and Z in the following sense: if $a\colon Z' \to Z$ and $b\colon Y \to Y'$ are morphisms in $\mathscr {C}$ , then the generalized trace of the composite

is naturally equivalent to the composite

Observation 2.9 (Trace is symmetric monoidal).

Let $f\colon X \to X$ and $g\colon Y \to Y$ be two traceable endomorphisms in $\mathscr {C}$ , i.e. X and Y are dualizable objects of $\mathscr {C}$ . The tensor product $X \otimes Y$ is again dualizable, with evaluation and coevaluation given by the tensor product of those for X and Y. A simple computation shows that the trace of the endomorphism $f \otimes g\colon X \otimes Y \to X \otimes Y$ is equivalent to the tensor product of the traces of f and g:

$$\begin{align*}{\mathrm{tr}}(f\otimes g) \simeq {\mathrm{tr}}(f)\otimes {\mathrm{tr}}(g). \end{align*}$$

In Corollary 2.27, we will see that these equivalences can be made fully coherent.

2.2 Functoriality of traces

Let $\mathscr {C}$ be a symmetric monoidal $\infty $ -category. If X and $X'$ are dualizable objects in $\mathscr {C}$ , then any equivalence $X \simeq X'$ in $\mathscr {C}$ can be used to transfer the duality data from X to duality data on $X'$ , giving rise to a homotopy $\dim (X) \simeq \dim (X')$ in between the dimensions of X and $X'$ . It is possible to choose these homotopies in a fully coherent fashion. For a precise formulation of this claim, we need the following definitions:

As we are about to make precise, the assignment $X \mapsto \dim (X)$ can be promoted to a map of spaces

$$\begin{align*}\dim \colon \mathscr{C}^{\mathrm{dbl}} \longrightarrow \Omega \mathscr{C} \quad\in\quad \mathcal{S}, \end{align*}$$

naturally in $\mathscr {C}$ . Similarly, one can promote the assignment $(X,f) \mapsto {\mathrm {tr}}(f \,\vert \, X)$ to a map of spaces

$$\begin{align*}{\mathrm{tr}} \colon \mathscr{C}^{\mathrm{trl}} \longrightarrow \Omega \mathscr{C} \quad\in\quad \mathcal{S}, \end{align*}$$

naturally in $\mathscr {C}$ . A convenient way of obtaining these two maps, as observed by [Reference Toën and Vezzosi64], is to use the fact that the functors $\mathscr {C} \mapsto \mathscr {C}^{\mathrm {dbl}}$ and $\mathscr {C} \mapsto \mathscr {C}^{\mathrm {trl}}$ from symmetric monoidal $\infty $ -categories to spaces are corepresented by certain symmetric monoidal $\infty $ -categories $\mathrm {Fr}^{\mathrm {rig}}(\mathrm {pt})$ and $\mathrm {Fr}^{\mathrm {rig}}(B\mathbb {N})$ . The main input for this is the following observation.

Lemma 2.11. The functor $(-)^{\mathrm {rig}}\colon {\mathrm {CAlg}}(\mathrm {Cat}_\infty ) \longrightarrow \mathrm {Cat}_{\infty }$ admits a left adjoint

$$\begin{align*}\mathrm{Fr}^{\mathrm{rig}}\colon \mathrm{Cat}_{\infty} \longrightarrow {\mathrm{CAlg}}(\mathrm{Cat}_\infty). \end{align*}$$

For an $\infty $ -category I, the symmetric monoidal $\infty $ -category $\mathrm {Fr}^{\mathrm {rig}}(I)$ comes equipped with a unit map $I \to \mathrm {Fr}^{\mathrm {rig}}(I)$ which lands in dualizable objects. Restriction along this map induces for every symmetric monoidal $\infty $ -category $\mathscr {C}$ an equivalence of spaces

In particular, specializing to $I = \mathrm {pt}$ and $I = B\mathbb {N}$ gives equivalences of spaces

$$\begin{align*}{\mathrm{Map}}^\otimes(\mathrm{Fr}^{\mathrm{rig}}(\mathrm{pt}),\mathscr{C}) \simeq \mathscr{C}^{\mathrm{dbl}} \qquad \text{ and } \qquad {\mathrm{Map}}^\otimes(\mathrm{Fr}^{\mathrm{rig}}(B\mathbb{N}),\mathscr{C}) \simeq \mathscr{C}^{\mathrm{trl}}, \end{align*}$$

showing that the functors $(-)^{\mathrm {dbl}}, (-)^{\mathrm {trl}}\colon {\mathrm {CAlg}}(\mathrm {Cat}_{\infty }) \to \mathcal {S}$ are corepresentable.

In the case of $B\mathbb {N}$ , we will denote the unit map by $\gamma _{ {\mathrm {univ}}}\colon B\mathbb {N} \to \mathrm {Fr}^{\mathrm {rig}}(B\mathbb {N})$ , and refer to it as the ‘universal traceable endomorphism’. Forming its trace produces an element $ {\mathrm {tr}}(\gamma _{ {\mathrm {univ}}}) \in \Omega \mathrm {Fr}^{\mathrm {rig}}(B\mathbb {N})$ , thought of as the ‘universal trace’. By the Yoneda lemma, it determines for every $\mathscr {C} \in {\mathrm {CAlg}}(\mathrm {Cat}_{\infty })$ a natural map of spaces $ {\mathrm {tr}}\colon \mathscr {C}^{\mathrm {trl}} \to \Omega \mathscr {C}$ . More explicitly:

Definition 2.12. Given a symmetric monoidal $\infty $ -category $\mathscr {C}$ , we define the trace map $ {\mathrm {tr}}\colon \mathscr {C}^{\mathrm {trl}} \to \Omega \mathscr {C}$ as the composite

By Remark 2.7, this functor agrees on objects with the formula for the trace given in the beginning of this section.

The map $\dim \colon \mathscr {C}^{\mathrm {dbl}} \to \Omega \mathscr {C}$ may be produced in a similar way from the Yoneda lemma by using instead the ‘universal dimension’ $\dim (\mathrm {pt}_{ {\mathrm {univ}}}) \in \Omega \mathrm {Fr}^{\mathrm {rig}}(\mathrm {pt})$ , where $\mathrm {pt}_{ {\mathrm {univ}}} \in \mathrm {Fr}^{\mathrm {rig}}(\mathrm {pt})$ is the ‘universal dualizable object’. Alternatively, one can obtain the map $\dim \colon \mathscr {C}^{\mathrm {dbl}} \to \Omega \mathscr {C}$ from the map $ {\mathrm {tr}}\colon \mathscr {C}^{\mathrm {trl}} \to \Omega \mathscr {C}$ by precomposing with the map $\mathscr {C}^{\mathrm {dbl}} \to \mathscr {C}^{\mathrm {trl}}\colon X \mapsto (X,\mathrm {id})$ .

2.3 Lax and oplax transformations

When $\mathscr {C} \in {\mathrm {CMon}}(\mathrm {Cat}_{(\infty ,n)})$ is a symmetric monoidal $(\infty ,n)$ -category, then the traceable endomorphisms $f\colon X \to X$ in $\mathscr {C}$ naturally form an $(\infty ,n-1)$ -category $\mathscr {C}^{\mathrm {trl}}$ rather than just a space. The construction of this $(\infty ,n-1)$ -category proceeds by realizing it as a subcategory of the $(\infty ,n)$ -category of functors $B\mathbb {N} \to \mathscr {C}$ and symmetric monoidal oplax natural transformations between them, satisfying certain dualizability and adjointability constraints. In this subsection, we will recall the necessary background on lax and oplax transformations.

Following [Reference Hoyois, Scherotzke and Sibilla37], we will model $(\infty , n)$ -categories by Barwick’s n-fold complete Segal spaces $\mathscr {C}\colon (\Delta ^{\mathrm {op}})^{\times n} \to \mathcal {S}$ [Reference Barwick and Schommer-Pries6, §Reference Ben-Zvi and Nadler14]. For an $(\infty ,n)$ -category $\mathscr {C}$ and a vector $\vec {k} = (k_1,\dots ,k_n)\in \mathbb {N}^n$ , we denote by $\mathscr {C}_{\vec {k}}$ the value of $\mathscr {C}$ , considered as an n-fold complete Segal space, at $([k_1],\dots ,[k_n])\in \Delta ^n$ . The collection of $(\infty ,n)$ -categories assembles into an $\infty $ -category that we denote by $\mathrm {Cat}_{(\infty ,n)}$ . By a symmetric monoidal $(\infty ,n)$ -category we mean a commutative monoid in the $\infty $ -category $\mathrm {Cat}_{(\infty ,n)}$ .

For each vector $\vec {k}\in \mathbb {N}^n$ , the functor $\mathrm {Cat}_{(\infty ,n)} \to \mathcal {S}$ given by $\mathscr {C}\mapsto \mathscr {C}_{\vec {k}}$ is corepresentable by the so called ‘walking $\vec {k}$ -tuple’ $\theta ^{\vec {k}}\in \mathrm {Cat}_{(\infty ,n)}$ (see [Reference Johnson-Freyd and Scheimbauer39, Definition 5.1, Remark 5.4]). Namely, we have a natural equivalence of spaces

$$\begin{align*}{\mathrm{Map}}(\theta^{\vec{k}},\mathscr{C}) \simeq \mathscr{C}_{\vec{k}}. \end{align*}$$

The $(\infty ,n)$ -categories $\theta ^{\vec {k}}$ generate $\mathrm {Cat}_{(\infty ,n)}$ under colimits. For $n = 1$ , the $\infty $ -category $\theta ^{(k)}$ is just the k-simplex $[k] \in \mathrm {Cat}_{(\infty ,1)}$ . The following are some examples for $n = 2$ :

Given two $(\infty ,n)$ -categories $\mathscr {C}$ and $\mathscr {D}$ , one can form two new $(\infty ,n)$ -categories

$$\begin{align*}{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{C},\mathscr{D}) \qquad \text{ and } \qquad {\mathrm{Fun}}_{\mathrm{oplax}}(\mathscr{C},\mathscr{D}), \end{align*}$$

whose objects are the n-functors $\mathscr {C} \to \mathscr {D}$ and whose higher morphisms are given by certain (op)lax transformations (see [Reference Hoyois, Scherotzke and Sibilla37, §2.2]). We emphasize that there is no laxness in the objects of $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {C},\mathscr {D})$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}(\mathscr {C},\mathscr {D})$ , only in the higher morphisms. These two $(\infty ,n)$ -categories are most easily described in terms of a certain conjectural monoidal structure $\times ^{\mathrm {lax}}$ on $\mathrm {Cat}_{(\infty ,n)}$ , called the Gray product: for $(\infty ,n)$ -categories $\mathscr {C}$ and $\mathscr {D}$ , we would like to define the $(\infty ,n)$ -categories $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {C},\mathscr {D})$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}(\mathscr {C},\mathscr {D})$ via the natural isomorphisms

$$ \begin{align*} \begin{split} {\mathrm{Map}}(\mathscr{E}, {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{C},\mathscr{D})) &\simeq {\mathrm{Map}}(\mathscr{E} \times^{\mathrm{lax}} \mathscr{C}, \mathscr{D}) \\ {\mathrm{Map}}(\mathscr{E}, {\mathrm{Fun}}_{\mathrm{oplax}}(\mathscr{C},\mathscr{D})) &\simeq {\mathrm{Map}}(\mathscr{C} \times^{\mathrm{lax}} \mathscr{E}, \mathscr{D}) \end{split} \end{align*} $$

for $\mathscr {E} \in \mathrm {Cat}_{(\infty ,n)}$ . Unfortunately, as far as we know the construction of the Gray product of $(\infty ,n)$ -categories with all of its expected properties is not yet fully furnished in the literature.

To circumvent this technical difficulty, [Reference Johnson-Freyd and Scheimbauer39] observed that in order to define $ {\mathrm {Fun}}_{\mathrm {lax}}(-,-)$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}(-,-)$ one only needs the Gray products of the walking $\vec {k}$ -tuples $\theta ^{\vec {k}}$ . In [Reference Johnson-Freyd and Scheimbauer39, Definition 5.7], a combinatorial model is given for a Gray product $\theta ^{\vec {k}}\times ^{\mathrm {lax}} \theta ^{\vec {\ell }}$ , which we denote by $\theta ^{\vec {k},\vec {\ell }}$ . We have the following low-dimensional examples, cf. [Reference Johnson-Freyd and Scheimbauer39, Example 4.2,4.3]:

The construction of $\theta ^{\vec {k},\vec {l}}$ is functorial in $\vec {k}, \vec {l} \in \Delta ^{n}$ , and satisfies $\theta ^{\vec {k},\vec {0}} \simeq \theta ^{\vec {k}} \simeq \theta ^{\vec {0},\vec {k}}$ . In particular, it comes equipped with a natural map $\theta ^{\vec {k},\vec {l}} \to \theta ^{\vec {k}} \times \theta ^{\vec {l}}$ to the cartesian product. Using the $(\infty ,n)$ -categories $\theta ^{\vec {k},\vec {l}}$ , one now defines $ {\mathrm {Fun}}_{\mathrm {lax}}(-,-)$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}(-,-)$ by the following two-step procedure:

Definition 2.13. For $\mathscr {C} \in \mathrm {Cat}_{(\infty ,n)}$ and $\vec {k}\in \mathbb {N}^n$ , we define the following n-fold simplicial spaces:

$$\begin{align*}{\mathrm{Fun}}_{\mathrm{lax}}(\theta^{\vec{k}},\mathscr{C})_{\vec{\ell}} := {\mathrm{Map}}(\theta^{\vec{k},\vec{\ell}},\mathscr{C}) \qquad \text{ and } \qquad {\mathrm{Fun}}_{\mathrm{oplax}}(\theta^{\vec{k}},\mathscr{C})_{\vec{\ell}} := {\mathrm{Map}}(\theta^{\vec{\ell},\vec{k}},\mathscr{C}). \end{align*}$$

We then define n-fold simplicial spaces $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E},\mathscr {C})$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}(\mathscr {E},\mathscr {D})$ by:

$$ \begin{align*} {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C})_{\vec{k}}&:= {\mathrm{Map}}(\mathscr{E},{\mathrm{Fun}}_{\mathrm{oplax}}(\theta^{\vec{k}},\mathscr{C})),\\ {\mathrm{Fun}}_{\mathrm{oplax}}(\mathscr{E},\mathscr{C})_{\vec{k}}&:= {\mathrm{Map}}(\mathscr{E},{\mathrm{Fun}}_{\mathrm{lax}}(\theta^{\vec{k}},\mathscr{C})). \end{align*} $$

These are n-fold complete Segal spaces by [Reference Johnson-Freyd and Scheimbauer39, Corollary 5.19] and these constructions assemble into functors

$$\begin{align*}{\mathrm{Fun}}_{\mathrm{lax}}(-,-), \; {\mathrm{Fun}}_{\mathrm{oplax}}(-,-)\colon \mathrm{Cat}_{(\infty,n)}^{\mathrm{op}} \times \mathrm{Cat}_{(\infty,n)} \to \mathrm{Cat}_{(\infty,n)}. \end{align*}$$

It is immediate from the construction that these functors preserve limits in both variables. Furthermore, they come equipped with a natural map from the ordinary functor category: by precomposition with the maps $\theta ^{\vec {k},\vec {l}} \to \theta ^{\vec {k}} \times \theta ^{\vec {l}}$ one obtains maps

$$\begin{align*}{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{C},\mathscr{D}) \longleftarrow {\mathrm{Fun}}(\mathscr{C},\mathscr{D}) \longrightarrow {\mathrm{Fun}}_{\mathrm{oplax}}(\mathscr{C},\mathscr{D}). \end{align*}$$

Remark 2.15. In the case $n = 2$ , a full definition of the Gray product is given by [Reference Gagna, Harpaz and Lanari28] in terms of scaled simplicial sets: by [Reference Gagna, Harpaz and Lanari28, Corollary 2.15], their construction determines a presentably (non-symmetric) monoidal structure

$$\begin{align*}-\times^{\mathrm{lax}} - \colon \mathrm{Cat}_{(\infty,2)} \times \mathrm{Cat}_{(\infty,2)} \longrightarrow \mathrm{Cat}_{(\infty,2)} \end{align*}$$

on the $\infty $ -category $\mathrm {Cat}_{(\infty ,2)}$ . In [Reference Haugseng, Hebestreit, Linskens and Nuiten34, Proposition 5.1.9], it was computed that this lax product takes the expected values on pairs of simplices $[k] \in \mathrm {Cat}_{(\infty ,1)}$ (as in the example $\theta ^{(1),(1)}$ above). Consequently, for $\mathscr {C},\mathscr {D} \in \mathrm {Cat}_{(\infty ,1)}$ and $\mathscr {E} \in \mathrm {Cat}_{(\infty ,2)}$ , we have

$$\begin{align*}{\mathrm{Map}}(\mathscr{C} , {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{D},\mathscr{E})) \simeq {\mathrm{Map}}(\mathscr{C} \times^{\mathrm{lax}} \mathscr{D}, \mathscr{E}). \end{align*}$$

In particular, the underlying $(\infty ,1)$ -category of $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {D},\mathscr {E})$ can be described in terms of $\times ^{\mathrm {lax}}$ .

Finally, we discuss the interaction of the constructions $ {\mathrm {Fun}}_{\mathrm {lax}}(-,-)$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}(-,-)$ with symmetric monoidal structures.

If both $\mathscr {C}$ and $\mathscr {D}$ have symmetric monoidal structures, one can define the following variants of $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {C},\mathscr {D})$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}(\mathscr {C},\mathscr {D})$ in which the functors and (op)lax natural transformations are symmetric monoidal:

Definition 2.17. Assume $\mathscr {C}$ and $\mathscr {D}$ are symmetric monoidal $(\infty ,n)$ -categories. We define $(\infty ,n)$ -categories $ {\mathrm {Fun}}_{\mathrm {lax}}^{\otimes }(\mathscr {C},\mathscr {D})$ and $ {\mathrm {Fun}}_{\mathrm {oplax}}^{\otimes }(\mathscr {C},\mathscr {D})$ by the following representability conditions: for $\mathscr {E} \in \mathrm {Cat}_{(\infty ,n)}$ we have natural equivalences

$$ \begin{align*} {\mathrm{Map}}(\mathscr{E}, {\mathrm{Fun}}_{\mathrm{lax}}^{\otimes}(\mathscr{C},\mathscr{D})) &\simeq {\mathrm{Map}}^{\otimes}(\mathscr{C}, {\mathrm{Fun}}_{\mathrm{oplax}}(\mathscr{E},\mathscr{D})) \\ {\mathrm{Map}}(\mathscr{E}, {\mathrm{Fun}}_{\mathrm{oplax}}^{\otimes}(\mathscr{C},\mathscr{D})) &\simeq {\mathrm{Map}}^{\otimes}(\mathscr{C}, {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{D})). \end{align*} $$

2.4 Higher categorical traces

In this section, we will recall from [Reference Hoyois, Scherotzke and Sibilla37] the definition of the symmetric monoidal $(n-1)$ -functor $ {\mathrm {tr}}\colon \mathscr {C}^{\mathrm {trl}} \to \Omega \mathscr {C}$ for a symmetric monoidal $(\infty ,n)$ -category $\mathscr {C}$ . We will start by formally defining its source and target.

Definition 2.18. Let $\mathscr {C} \in {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,n)})$ be a symmetric monoidal $(\infty ,n)$ -category. We define the $(\infty ,n-1)$ -category $\mathscr {C}^{\mathrm {dbl}}$ of dualizable objects in $\mathscr {C}$ as

$$\begin{align*}\mathscr{C}^{\mathrm{dbl}} := \iota_{n-1}{\mathrm{Fun}}^{\otimes}_{\mathrm{oplax}}(\mathrm{Fr}^{\mathrm{rig}}(\mathrm{pt}),\mathscr{C}). \end{align*}$$

Similarly, we define the $(\infty ,n-1)$ -category $\mathscr {C}^{\mathrm {trl}}$ of traceable endomorphisms in $\mathscr {C}$ as

$$\begin{align*}\mathscr{C}^{\mathrm{trl}} := \iota_{n-1}{\mathrm{Fun}}^{\otimes}_{\mathrm{oplax}}(\mathrm{Fr}^{\mathrm{rig}}(B\mathbb{N}),\mathscr{C}). \end{align*}$$

Precomposition with the symmetric monoidal functor $\mathrm {Fr}^{\mathrm {rig}}(B\mathbb {N}) \to \mathrm {Fr}^{\mathrm {rig}}(\mathrm {pt})$ induced by the map $B\mathbb {N} \to \mathrm {pt}$ gives an $(n-1)$ -functor $\mathscr {C}^{\mathrm {dbl}} \to \mathscr {C}^{\mathrm {trl}}$ .

For $n = 1$ , there is an equivalence of $\infty $ -groupoids $\iota _0 {\mathrm {Fun}}^{\otimes }_{\mathrm {oplax}}(-,-) \simeq {\mathrm {Map}}^{\otimes }(-,-)$ , and thus by corepresentability the definitions of the $\infty $ -groupoids $\mathscr {C}^{\mathrm {dbl}}$ and $\mathscr {C}^{\mathrm {trl}}$ agree with the ones from the previous subsection. For larger n, the constructions of $\mathscr {C}^{\mathrm {dbl}}$ and $\mathscr {C}^{\mathrm {trl}}$ are compatible with the inclusions $\mathrm {Cat}_{(\infty ,n)} \hookrightarrow \mathrm {Cat}_{(\infty ,n+1)}$ , in the sense that the following squares commute:

Moreover, these squares are vertically right adjointable: for $1 \leq k \leq n$ there are equivalences

$$\begin{align*}(\iota_k \mathscr{C})^{\mathrm{dbl}} \simeq \iota_{k-1}\mathscr{C}^{\mathrm{dbl}} \qquad \qquad \text{ and } \qquad \qquad (\iota_k \mathscr{C})^{\mathrm{trl}} \simeq \iota_{k-1}\mathscr{C}^{\mathrm{trl}}. \end{align*}$$

It follows that the space of objects of the $(\infty ,n-1)$ -category $\mathscr {C}^{\mathrm {dbl}}$ is the space of dualizable objects in $\mathscr {C}$ , while the space of objects of $\mathscr {C}^{\mathrm {trl}}$ is the space of traceable endomorphisms in $\mathscr {C}$ .

We will next show that also the morphisms in $\mathscr {C}^{\mathrm {dbl}}$ and $\mathscr {C}^{\mathrm {trl}}$ are as claimed in the introduction of this section.

Lemma 2.20. Let $\mathscr {C}$ be a symmetric monoidal $(\infty ,n)$ -category. For any $(\infty ,n-1)$ -category $\mathscr {E}$ there are equivalences of spaces

$$ \begin{align*} {\mathrm{Map}}(\mathscr{E},\mathscr{C}^{\mathrm{dbl}}) &\simeq (\iota_1{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}))^{\mathrm{dbl}},\\ {\mathrm{Map}}(\mathscr{E},\mathscr{C}^{\mathrm{trl}}) &\simeq (\iota_1{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}))^{\mathrm{trl}}. \end{align*} $$

Proof. For the second equation, there are equivalences

$$ \begin{align*} {\mathrm{Map}}(\mathscr{E},\mathscr{C}^{\mathrm{trl}}) &\simeq {\mathrm{Map}}^{\otimes}(\mathrm{Fr}^{\mathrm{rig}}(B\mathbb{N}),{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C})) \\ &\simeq {\mathrm{Map}}^{\otimes}(\mathrm{Fr}^{\mathrm{rig}}(B\mathbb{N}),\iota_1{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}))\\ &\simeq (\iota_1{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}))^{\mathrm{trl}}, \end{align*} $$

where the first equivalence is immediate from the definitions, the second equivalence holds as $\mathrm {Fr}^{\mathrm {rig}}(B\mathbb {N})$ is an $(\infty ,1)$ -category, and the third is the defining property of $\mathrm {Fr}^{\mathrm {rig}}(B\mathbb {N})$ . The computation for the first equation is analogous, using $\mathrm {Fr}^{\mathrm {rig}}(\mathrm {pt})$ instead.

Corollary 2.21 (cf. [Reference Hoyois, Scherotzke and Sibilla37, Lemma 2.4]).

Let $\mathscr {C}$ be an $(\infty ,n)$ -category. There are natural monomorphisms of $(\infty ,n)$ -categories

$$\begin{align*}\mathscr{C}^{\mathrm{dbl}} \hookrightarrow \mathscr{C} \qquad \text{ and } \qquad \mathscr{C}^{\mathrm{trl}} \hookrightarrow {\mathrm{Fun}}_{\mathrm{oplax}}(B\mathbb{N},\mathscr{C}). \end{align*}$$

An object $X \in \mathscr {C}$ is in $\mathscr {C}^{\mathrm {dbl}}$ if and only if it is dualizable, and a morphism $f\colon X \to Y$ between dualizable objects is in $\mathscr {C}^{\mathrm {dbl}}$ if and only if it is a left adjoint morphism. Similarly, an endomorphism $f\colon X \to X$ is an object in $\mathscr {C}^{\mathrm {trl}}$ if and only if X is dualizable, and a lax square

is a morphism in $\mathscr {C}^{\mathrm {trl}}$ if and only if X and Y are dualizable and $\varphi \colon X \to Y$ is a left adjoint in $\mathscr {C}$ .

Proof. We start by producing the two monomorphisms. We will do this for $\mathscr {C}^{\mathrm {trl}}$ , and leave the case for $\mathscr {C}^{\mathrm {dbl}}$ to the reader. For every $(\infty ,n-1)$ -category $\mathscr {E}$ , we consider the following natural composite map of spaces:

$$\begin{align*}{\mathrm{Map}}(\mathscr{E},\mathscr{C}^{\mathrm{trl}}) \overset{2.20}{\simeq} (\iota_1 {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}))^{\mathrm{trl}} \hookrightarrow {\mathrm{Map}}(B\mathbb{N},{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C})) \simeq {\mathrm{Map}}(\mathscr{E},{\mathrm{Fun}}_{\mathrm{oplax}}(B\mathbb{N},\mathscr{C})). \end{align*}$$

Note that the middle map is an inclusion of path components by definition of $(-)^{\mathrm {trl}}$ . By the Yoneda lemma, we obtain the desired monomorphism $\mathscr {C}^{\mathrm {trl}} \hookrightarrow {\mathrm {Fun}}_{\mathrm {oplax}}(B\mathbb {N},\mathscr {C})$ .

We now prove the descriptions of the objects and morphisms of $\mathscr {C}^{\mathrm {dbl}}$ and $\mathscr {C}^{\mathrm {trl}}$ . As mentioned earlier, the statement on objects is clear from the equivalences $\iota _0\mathscr {C}^{\mathrm {dbl}} \simeq (\iota _1\mathscr {C})^{\mathrm {dbl}}$ and $\iota _0\mathscr {C}^{\mathrm {trl}} \simeq (\iota _1\mathscr {C})^{\mathrm {trl}}$ . For the statement on morphisms, we apply Lemma 2.20 to $\mathscr {E} = [1]$ to reduce to a statement about the dualizable objects of $ {\mathrm {Fun}}_{\mathrm {lax}}([1],\mathscr {C})$ . This then becomes an instance of Lemma 2.4 in [Reference Hoyois, Scherotzke and Sibilla37], which says that an object $(\varphi \colon X \to Y)$ of $ {\mathrm {Fun}}_{\mathrm {lax}}([1],\mathscr {C})$ is dualizable if and only if X and Y are dualizable and $\varphi \colon X \to Y$ is a left adjoint in $\mathscr {C}$ .

The target $\Omega \mathscr {C}$ of the higher trace functor is the $(\infty ,n-1)$ -category of endomorphisms of the monoidal unit, defined formally as follows:

Definition 2.22. The $(\infty ,n-1)$ -category $\Omega \mathscr {C}$ is defined as the following pullbackFootnote ⁶ in $\mathrm {Cat}_{(\infty ,n-1)}$ :

That is, the objects of $\Omega \mathscr {C}$ are given by 1-morphisms in $\mathscr {C}$ , the morphisms are given by 2-morphisms

and so forth. If $\mathscr {C}$ is symmetric monoidal, then so is $\Omega \mathscr {C}$ , as all the functors in the diagram of which $\Omega \mathscr {C}$ is defined to be the limit are manifestly symmetric monoidal.

Proposition 2.23. For every $(\infty ,n-1)$ -category $\mathscr {E}$ the canonical map

$$\begin{align*}{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\Omega \mathscr{C}) \to \Omega {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}) \end{align*}$$

is an equivalence. In particular, there is an equivalence

$$\begin{align*}\Omega (\iota_1 {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C})) \simeq {\mathrm{Map}}(\mathscr{E},\Omega \mathscr{C}). \end{align*}$$

Proof. In the special case $\mathscr {E} = \theta ^{\vec {k}}$ , the first statement is [Reference Hoyois, Scherotzke and Sibilla37, Proposition 2.6]. The general case follows from the fact that the $(\infty ,n-1)$ -categories $\theta ^{\vec {k}}$ generate $\mathrm {Cat}_{(\infty ,n-1)}$ under colimits and both sides take colimits in the variable $\mathscr {E}$ to limits.

For the second statement, we observe that $\Omega \iota _{k} \mathscr {C} \simeq \iota _{k-1} \Omega \mathscr {C}$ for $1 \leq k \leq n-1$ , so that

$$\begin{align*}\Omega \iota_1 {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}) \simeq \iota_0 \Omega {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C}) \simeq \iota_0 {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\Omega \mathscr{C}) \simeq {\mathrm{Map}}(\mathscr{E},\Omega \mathscr{C}), \end{align*}$$

proving the claim.

We can now bootstrap the $(\infty ,1)$ -categorical trace map from Definition 2.12 to the $(\infty ,n)$ -categorical setting.

Definition 2.24. Let $\mathscr {C}$ be a symmetric monoidal $(\infty ,n)$ -category. We define the trace functor

$$\begin{align*}{\mathrm{tr}}\colon \mathscr{C}^{\mathrm{trl}} \to \Omega \mathscr{C} \end{align*}$$

as the $(n-1)$ -functor inducing the following natural map of spaces for every $\mathscr {E} \in \mathrm {Cat}_{(\infty ,n-1)}$ :

Here the first and last equivalences are Lemma 2.20 and Proposition 2.23 respectively and the middle map is the trace map of spaces for the symmetric monoidal $(\infty ,1)$ -category $\iota _1 {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E},\mathscr {C})$ , as defined in Definition 2.12.

We define the dimension functor $\dim \colon \mathscr {C}^{\mathrm {dbl}} \to \Omega \mathscr {C}$ as the $(n-1)$ -functor obtained by precomposing $ {\mathrm {tr}}\colon \mathscr {C}^{\mathrm {trl}} \to \Omega \mathscr {C}$ with the map $\mathscr {C}^{\mathrm {dbl}} \to \mathscr {C}^{\mathrm {trl}}$ from Definition 2.18.

By definition, the $(n-1)$ -functor $ {\mathrm {tr}}\colon \mathscr {C}^{\mathrm {trl}} \to \Omega \mathscr {C}$ is on groupoid cores simply given by the trace map of the $(\infty ,1)$ -category $\iota _1 \mathscr {C}$ , and thus it has the correct behavior on objects. The following lemma shows that also its behavior on morphisms is as described in the introduction of this section:

Proof. Under the identification $ {\mathrm {Map}}([1],\mathscr {C}^{\mathrm {trl}}) \simeq (\iota _1 {\mathrm {Fun}}_{\mathrm {lax}}([1],\mathscr {C}))^{\mathrm {trl}}$ , the morphism $(\varphi ,\alpha )\colon (X,f) \to (Y,g)$ in $\mathscr {C}^{\mathrm {trl}}$ corresponds to an endomorphism $(f,g,\alpha )$ of the dualizable object $(\varphi \colon X \to Y)$ in the symmetric monoidal $\infty $ -category $ {\mathrm {Fun}}_{\mathrm {lax}}([1],\mathscr {C})$ , and the map $ {\mathrm {tr}}(\alpha )$ is by definition the trace of this endomorphism:

In the proof of [Reference Hoyois, Scherotzke and Sibilla37, Lemma 2.4], the authors write down explicit duality data of $\varphi $ as an object of $ {\mathrm {Fun}}_{\mathrm {lax}}([1],\mathscr {C})$ , and plugging this in gives the explicit description of the map $ {\mathrm {tr}}(\alpha )\colon {\mathrm {tr}}(f) \to {\mathrm {tr}}(g)$ as given in (2).

We finish the section by showing that the trace functor $ {\mathrm {tr}}\colon \mathscr {C}^{\mathrm {trl}} \to \Omega \mathscr {C}$ admits a canonical enhancement to a symmetric monoidal $(n-1)$ -functor.

Lemma 2.26. The functors

$$ \begin{align*} (-)^{\mathrm{trl}}\colon {\mathrm{CAlg}}(\mathrm{Cat}_{(\infty,n)}) &\to \mathrm{Cat}_{(\infty,n-1)} \\ \Omega\colon {\mathrm{CAlg}}(\mathrm{Cat}_{(\infty,n)}) &\to \mathrm{Cat}_{(\infty,n-1)} \end{align*} $$

preserve limits.

Proof. The functor $\Omega (-)$ is the pullback of the limit-preserving functors sending $\mathscr {C}$ to , $\mathscr {C}_0 \times \mathscr {C}_0$ and $\mathscr {C}_1$ , and thus also $\Omega $ preserves limits. For $(-)^{\mathrm {trl}}$ , this follows from the fact that for every $\mathscr {E} \in \mathrm {Cat}_{(\infty ,n-1)}$ and $\mathscr {C} \in {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,n)})$ there is an equivalence

$$\begin{align*}{\mathrm{Map}}(\mathscr{E},\mathscr{C}^{\mathrm{trl}}) \simeq {\mathrm{Map}}^{\otimes}(\mathrm{Fr}^{\mathrm{rig}}(B\mathbb{N}),{\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},\mathscr{C})), \end{align*}$$

where the right-hand side preserves limits in the variable $\mathscr {C} \in {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,n)})$ .

Since the functors $(-)^{\mathrm {trl}}$ and $\Omega (-)$ in particular preserve finite products, they induce functors on commutative algebra objects. Employing the equivalence $ {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,n)}) \simeq {\mathrm {CAlg}}( {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,n)}))$ , we may regard every symmetric monoidal $(\infty ,n)$ -category as a commutative algebra in the $\infty $ -category $ {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,n)})$ . This leads to a symmetric monoidal enhancement of the trace functor.

3.1 The $(\infty ,2)$ -category of spans

We will start by introducing the $(\infty ,2)$ -category $ {\mathrm {Span}}(\mathscr {D})$ of spans associated with an $\infty $ -category $\mathscr {D}$ which admits finite limits. There are various ways to define $ {\mathrm {Span}}(\mathscr {D})$ : explicitly as a 2-fold complete Segal space [Reference Haugseng32, Reference Gaitsgory and Rozenblyum29] generalizing Barwick’s $(\infty ,1)$ -categorical construction [Reference Barwick and Rognes5, Reference Barwick4], via bivariant fibrations [Reference Stefanich63], or via bivariant theories [Reference Macpherson53]. All these $(\infty ,2)$ -categories are equivalent because they satisfy a universal property: they come equipped with a functor $\mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ which is universal among bivariant theories from $\mathscr {D}$ into some $(\infty ,2)$ -category $\mathscr {E}$ , cf. Definition 3.2. For the purposes of this article, it is convenient to simply define $ {\mathrm {Span}}(\mathscr {D})$ via its universal property.

Recall that a commutative square

in an $(\infty ,2)$ -category $\mathscr {E}$ is called vertically right adjointable if the morphisms b and c admit right adjoints $b^r$ and $c^r$ in $\mathscr {E}$ , and the Beck-Chevalley map

is an equivalence in $\mathscr {E}$ .

Definition 3.2 [Reference Macpherson53, Section 3.2].

Let $\mathscr {D}$ be a left exact $\infty $ -category and let $\mathscr {E}$ be an $(\infty ,2)$ -category. A functor $F\colon \mathscr {D} \to \mathscr {E}$ is called a bivariant theory if the following two conditions are satisfied:

1. (1) For every morphism $f\colon A \to B$ in $\mathscr {D}$ , the morphism $F(f)\colon F(A) \to F(B)$ admits a right adjoint $F(f)^r$ in $\mathscr {E}$ ;

2. (2) For every pullback square

   
   in $\mathscr {D}$ , the induced square
   
   in $\mathscr {E}$ is vertically right adjointable.

Given another bivariant theory $F'\colon \mathscr {C} \to \mathscr {E}$ , a transformation $\alpha \colon F \Rightarrow F'$ is called bivariant if it commutes with the right adjoints of $F(f)$ : for every morphism $f\colon A \to B$ in $\mathscr {D}$ , the resulting naturality square

is vertically right adjointable.

We let $ {\mathrm {Biv}}(\mathscr {D}, \mathscr {E}) \subseteq {\mathrm {Fun}}(\mathscr {D},\mathscr {E})$ denote the $(\infty ,2)$ -category of bivariant theories $F\colon \mathscr {D} \to \mathscr {E}$ , bivariant transformations and arbitrary 3-transformations.

We may now introduce $ {\mathrm {Span}}(\mathscr {D})$ as the $(\infty ,2)$ -category equipped with the universal bivariant theory $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ .

Proposition 3.3 [Reference Macpherson53, Theorem 4.2.6].

Let $\mathscr {D}$ be a left exact $\infty $ -category. There exists an $(\infty ,2)$ -category $ {\mathrm {Span}}(\mathscr {D})$ equipped with a bivariant theory $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ satisfying the following property: for any other $(\infty ,2)$ -category $\mathscr {E}$ , composition with $h_{\mathscr {D}}$ induces an equivalence of $(\infty ,2)$ -categories

The construction $\mathscr {D} \mapsto {\mathrm {Span}}(\mathscr {D})$ and the maps $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ uniquely assemble into a functor $ {\mathrm {Span}}\colon \mathrm {Cat}^{\mathrm {lex}}_{\infty } \to \mathrm {Cat}_{(\infty ,2)}$ equipped with a natural transformation h from the forgetful functor $\mathrm {Cat}^{\mathrm {lex}}_{\infty } \to \mathrm {Cat}_{\infty } \hookrightarrow \mathrm {Cat}_{(\infty ,2)}$ .

The above universal property of $ {\mathrm {Span}}(\mathscr {D})$ admits a symmetric monoidal variant:

Proposition 3.6 [Reference Macpherson53, 4.4.6 Theorem].

Let $\mathscr {D}$ be a left exact $\infty $ -category. There is a unique enhancement of $ {\mathrm {Span}}(\mathscr {D})$ to a symmetric monoidal $(\infty ,2)$ -category together with an enhancement of the bivariant theory $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ to a symmetric monoidal bivariant theory. It satisfies the following property: for any other symmetric monoidal $(\infty ,2)$ -category $\mathscr {E}$ , composition with $h_{\mathscr {D}}$ induces an equivalence of $(\infty ,2)$ -categories

The construction $\mathscr {D} \mapsto {\mathrm {Span}}(\mathscr {D})$ and the maps $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ uniquely assemble into a functor $ {\mathrm {Span}}\colon \mathrm {Cat}^{\mathrm {lex}}_{\infty } \to {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,2)})$ equipped with a natural transformation h from the forgetful functor $\mathrm {Cat}^{\mathrm {lex}}_{\infty } \to {\mathrm {CAlg}}(\mathrm {Cat}_{\infty }) \hookrightarrow {\mathrm {CAlg}}(\mathrm {Cat}_{(\infty ,2)})$ .

In [Reference Stefanich63, Section 3.1], Stefanich introduces for every left exact $\infty $ -category $\mathscr {D}$ an explicit 2-fold complete Segal space $2 {\mathrm {Corr}}(\mathscr {D})$ equipped with a functor $\mathscr {D} \to 2 {\mathrm {Corr}}(\mathscr {D})$ satisfying the universal property of $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ , cf. [Reference Stefanich63, Theorem 3.4.18]. In [Reference Stefanich63, Remark 3.1.10], the mapping spaces in $2 {\mathrm {Corr}}(\mathscr {D})$ are computed to be

$$\begin{align*}{\mathrm{Map}}_{2{\mathrm{Corr}}(\mathscr{D})}(d,d') \simeq \mathscr{D}_{/d \times d'} \end{align*}$$

for $d,d'\in \mathscr {D}$ . One observes from his computation (cf. [Reference Stefanich63, Notation 3.1.4, Notation 3.1.9, Remark 3.1.8]) that this identification is natural in the triple $(\mathscr {D},d,d')$ . In particular, letting both d and $d'$ be the final object of $\mathscr {D}$ , we obtain the following result:

Proposition 3.7 [Reference Stefanich63].

Let $\mathscr {D}$ be a left exact $\infty $ -category. Then there is a functorial equivalence

We finish this subsection by recalling from [Reference Haugseng32, Reference Stefanich63] that all objects in $ {\mathrm {Span}}(\mathscr {D})$ are dualizable and that every morphism f in $\mathscr {D}$ gives rise to an adjunction in $ {\mathrm {Span}}(\mathscr {D})$ between the associated left- and right-pointing morphisms.

Lemma 3.8 [Reference Haugseng32, Theorem 1.4], [Reference Stefanich63, Proposition 3.3.3].

For a left exact $\infty $ -category $\mathscr {D}$ and an object $X \in \mathscr {D}$ , the span

is part of a duality datum exhibiting X as self-dual in $ {\mathrm {Span}}(\mathscr {D})$ . The coevaluation is given by the span . In particular, every object in $ {\mathrm {Span}}(\mathscr {D})$ is dualizable.

Lemma 3.9 [Reference Stefanich63, Proposition 3.3.1], see also [Reference Haugseng32, Lemma 12.3].

For a morphism $f\colon X \to Y$ in $\mathscr {D}$ , the morphisms of spans

are the counit resp. unit of an adjunction in $ {\mathrm {Span}}(\mathscr {D})$ between the span and the span .

3.2 Traces and dimensions in the $(\infty ,2)$ -category of spans

In this subsection, we recall the folklore fact that the trace of an endomorphism in $ {\mathrm {Span}}(\mathscr {D})$ is given by the equalizer $ {\mathrm {Eq}}(f,g)$ of f and g. An instance of this appears for example in [Reference Ben-Zvi and Nadler14, Section 4]. For completeness, we provide a proof of this fact. Furthermore, we will show that there is a natural identification $\dim _{ {\mathrm {Span}}(\mathscr {D})}(X) \simeq LX$ for $X \in \mathscr {D}$ .

Definition 3.10. Let $f,g\colon Z \to X$ be two maps in $\mathscr {D}$ . The equalizer $ {\mathrm {Eq}}(f,g)$ of f and g is defined via the pullback square

When $Z = X$ and $f = g = \mathrm {id}_X$ is the identity on X, we write the equalizer as

$$\begin{align*}LX := {\mathrm{Eq}}(\mathrm{id}_X,\mathrm{id}_X) \end{align*}$$

and call it the free loop space, or the free loop object of X. Observe that it is equivalent to the cotensor $X^{S^1}$ of X by $S^1$ .

The proof that traces in $ {\mathrm {Span}}(\mathscr {D})$ are given by equalizers is a straightforward computation.

Proof. Spelling out the definition of trace and plugging in the explicit duality data from Lemma 3.8, the trace is given by the following composite of spans:

Observe that we have the following pullback diagram:

It follows that the above composite span is equivalent to the span , as desired.

The above computation of dimensions in $ {\mathrm {Span}}(\mathscr {D})$ gives rise to explicit descriptions of dimensions in $(\infty ,2)$ -categories $\mathscr {E}$ which receive a bivariant theory from $\mathscr {D}$ :

3.2.1 Coherence

We have seen above that for an object X of a left exact $\infty $ -category $\mathscr {D}$ , the dimension of X in $ {\mathrm {Span}}(\mathscr {D})$ is given by its free loop space: $\dim _{ {\mathrm {Span}}(\mathscr {D})}(X) \simeq LX$ . The goal of the remainder of subsection is to make this calculation functorial in X, see Theorem 3.19 below. This is more subtle than it looks: to get at the functoriality of the trace functor, we need to produce all the higher duality data for the objects in $ {\mathrm {Span}}(\mathscr {D})$ in a coherent fashion.

We will work around this problem by observing that the computation of dimensions in $ {\mathrm {Span}}(\mathscr {D})$ can be done uniformly in $\mathscr {D} \in \mathrm {Cat}_{\infty }^{\mathrm {lex}}$ , allowing us to reduce to the ‘universal’ left exact $\infty $ -category. More precisely, we will consider the left exact $\infty $ -category $\mathscr {D} = (\mathcal {S}^{\mathrm {fin}})^{\mathrm {op}}$ , the opposite of the $\infty $ -category of finite spaces. By using the fact that $(\mathcal {S}^{\mathrm {fin}})^{\mathrm {op}}$ is freely generated under finite limits by the point $\mathrm {pt} \in (\mathcal {S}^{\mathrm {fin}})^{\mathrm {op}}$ , it is possible to reduce the coherence problem to a non-coherent statement, which was solved in Lemma 3.11. In the remainder of this subsection, we will fill in the details of this proof strategy.

Lemma 3.13. The functor $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ factors through the subcategory $ {\mathrm {Span}}(\mathscr {D})^{\mathrm {dbl}}$ .

Proof. Since $ {\mathrm {Span}}(\mathscr {D})^{\mathrm {dbl}} \hookrightarrow {\mathrm {Span}}(\mathscr {D})$ is a non-full subcategory by Corollary 2.21, it suffices to check this on the level of objects and morphisms. By Lemma 3.8, $h_{\mathscr {D}}$ carries every object in $\mathscr {D}$ to a dualizable object of $ {\mathrm {Span}}(\mathscr {D})$ , and since $h_{\mathscr {D}}$ is a bivariant theory, it carries morphisms in $\mathscr {D}$ to left adjoint morphisms in $ {\mathrm {Span}}(\mathscr {D})$ . The statement thus follows from the description of objects and morphisms of $ {\mathrm {Span}}(\mathscr {D})^{\mathrm {dbl}}$ given in Corollary 2.21.

Remark 3.14. The functor $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})^{\mathrm {dbl}}$ is in fact an equivalence of $\infty $ -categories. As we will not need this statement, we leave the proof to the reader.

It is relatively straightforward to show that the equivalence $\dim _{ {\mathrm {Span}}(\mathscr {D})}(X) \simeq LX$ is natural with respect to equivalences in the $\infty $ -category $\mathscr {D}$ .

Lemma 3.15. For every left exact $\infty $ -category $\mathscr {D}$ , the following diagram naturally commutes:

Proof. The four corners of the square are natural in $\mathscr {D} \in \mathrm {Cat}^{\mathrm {lex}}_{\infty }$ and the four edges form natural transformations. The assignment $\mathscr {D} \mapsto \mathscr {D}^{\simeq }$ is corepresented by the left exact $\infty $ -category $(\mathcal {S}^{\mathrm {fin}})^{\mathrm {op}}$ . Indeed, the $\infty $ -category $\mathcal {S}^{\mathrm {fin}}$ is freely generated under finite colimits by the point, see (the proof of) [Reference Lurie49, Proposition 5.3.6.2]. It thus suffices, by the Yoneda lemma, to show that the two composites agree for $\mathscr {D} = (\mathcal {S}^{\mathrm {fin}})^{\mathrm {op}}$ when evaluated at the point $\mathrm {pt} \in (\mathcal {S}^{\mathrm {fin}})^{\mathrm {op}}$ . This is an instance of Lemma 3.11.

In order to enhance the above commutative diagram to a diagram defined on all of $\mathscr {D}$ rather than just its groupoid core $\mathscr {D}^{\simeq }$ , we will need some understanding of the interaction between span-categories and functor categories.

Lemma 3.16. Let $\mathscr {E}$ and $\mathscr {D}$ be $\infty $ -categories such that $\mathscr {D}$ is left exact. The inclusion $ {\mathrm {Fun}}(\mathscr {E},\mathscr {D}) \hookrightarrow {\mathrm {Fun}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D})) \hookrightarrow {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D}))$ is a symmetric monoidal bivariant theory, i.e. it is symmetric monoidal, sends all morphisms to left adjoint morphisms and sends pullback squares to right adjointable squares.

Proof. The inclusion $ {\mathrm {Fun}}(\mathscr {E},\mathscr {D}) \hookrightarrow {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D}))$ inherits a symmetric monoidal structure from applying the finite-product-preserving functor $ {\mathrm {Fun}}(\mathscr {E},-)$ to the symmetric monoidal functor $\mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ .

Next, given a morphism $\alpha \colon F \to G$ in $ {\mathrm {Fun}}(\mathscr {E},\mathscr {D})$ , we have to show its image in $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D}))$ is a left adjoint. By [Reference Haugseng33, Theorem 4.6], it suffices to show that this image in $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D}))$ comes from a morphism in the functor category $ {\mathrm {Fun}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D}))$ , and that for every $e \in \mathscr {E}$ the map $\alpha (e)\colon F(e) \to G(e)$ is a left adjoint morphism in $ {\mathrm {Span}}(\mathscr {D})$ . But the former is automatic, and the latter follows from the fact that $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ sends morphisms to left adjoints.

Finally, given a pullback square

in $ {\mathrm {Fun}}(\mathscr {E},\mathscr {D})$ , we have to show that the resulting Beck-Chevalley map (see e.g. [Reference Carmeli, Schlank and Yanovski18, Section 2.2])

in $ {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D}))$ is an equivalence. For every $e \in \mathscr {E}$ , evaluation of the pullback square at $e \in \mathscr {E}$ gives a pullback square in $\mathscr {D}$ . Hence, since $h_{\mathscr {D}}\colon \mathscr {D} \to {\mathrm {Span}}(\mathscr {D})$ is a bivariant theory, the above Beck-Chevalley map is an equivalence in $ {\mathrm {Span}}(\mathscr {D})$ after evaluating at e:

Since the 2-functors $\mathrm {ev}_e\colon {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D})) \to {\mathrm {Span}}(\mathscr {D})$ for $e \in \mathscr {E}$ are jointly conservative on morphism $\infty $ -categories, this gives the claim.

By the previous lemma, the symmetric monoidal inclusion $ {\mathrm {Fun}}(\mathscr {E},\mathscr {D}) \hookrightarrow {\mathrm {Fun}}_{\mathrm {lax}}(\mathscr {E}, {\mathrm {Span}}(\mathscr {D}))$ universally extends to a symmetric monoidal functor

$$\begin{align*}\Phi_{\mathscr{E},\mathscr{D}}\colon {\mathrm{Span}}({\mathrm{Fun}}(\mathscr{E},\mathscr{D})) \to {\mathrm{Fun}}_{\mathrm{lax}}(\mathscr{E},{\mathrm{Span}}(\mathscr{D})). \end{align*}$$

The following lemma shows that this functor interacts well with the equivalence :

Lemma 3.17. For every $\infty $ -category $\mathscr {E}$ , there is a natural homotopy making the following diagram of $\infty $ -groupoids commute:

(3)

Proof. Under the identification $ {\mathrm {Fun}}(\mathscr {E},\mathscr {E}')^{\simeq } \simeq {\mathrm {Map}}_{\mathrm {Cat}_{\infty }}(\mathscr {E},\mathscr {E}')$ , the composite along the left and bottom of the diagram (3) constitutes a map from $ {\mathrm {Map}}_{\mathrm {Cat}_{\infty }}(\mathscr {E},\mathscr {D})$ to $ {\mathrm {Map}}_{\mathrm {Cat}_{\infty }}(\mathscr {E},\Omega {\mathrm {Span}}(\mathscr {D}))$ which is natural in $\mathscr {E} \in \mathrm {Cat}_{\infty }$ . By the Yoneda lemma, it is thus induced by a functor $j_{\mathscr {D}}\colon \mathscr {D} \to \Omega {\mathrm {Span}}(\mathscr {D})$ . From the definition it is immediate that the functors $i_{\mathscr {D}}$ and $j_{\mathscr {D}}$ agree on groupoid cores and that the diagram (3) commutes if we replace $i_{\mathscr {D}}$ by $j_{\mathscr {D}}$ . It thus remains to show that the maps $i_{\mathscr {D}}$ and $j_{\mathscr {D}}$ are equivalent as functors $\mathscr {D} \to \Omega {\mathrm {Span}}(\mathscr {D})$ .

Note that the construction of $j_{\mathscr {D}}$ is natural in $\mathscr {D} \in \mathrm {Cat}^{\mathrm {lex}}_{\infty }$ . We may thus consider the composite $k_{\mathscr {D}}:= i^{-1}_{\mathscr {D}}j_{\mathscr {D}}\colon \mathscr {D} \to \mathscr {D}$ and regard it as an endomorphism $k\colon U \to U$ of the forgetful functor $U\colon \mathrm {Cat}^{\mathrm {lex}}_{\infty } \to \mathrm {Cat}_{\infty }$ . We need to show that k is the identity of U. Since $i_{\mathscr {D}}$ and $j_{\mathscr {D}}$ agree on groupoid cores, the induced map $k_{\mathscr {D}}\colon \mathscr {D}^{\simeq } \to \mathscr {D}^{\simeq }$ on groupoid cores is naturally equivalent to the identity. It thus remains to prove the following result, Lemma 3.18, which we record separately because of its independent interest.

Lemma 3.18. Let $k\colon U \to U$ be an endomorphism of the forgetful functor $U\colon \mathrm {Cat}^{\mathrm {lex}}_{\infty } \to \mathrm {Cat}_{\infty }$ . Assume that the induced endomorphism $k^{\simeq }\colon U^{\simeq } \to U^{\simeq }$ of the functor $U^{\simeq }\colon \mathrm {Cat}^{\mathrm {lex}}_{\infty } \to \mathcal {S}$ is equivalent to the identity of $U^{\simeq }$ . Then k is equivalent to the identity of U.

Proof. The forgetful functor U admits a left adjoint $F\colon \mathrm {Cat}_{\infty } \to \mathrm {Cat}_{\infty }^{\mathrm {lex}}$ , sending an $\infty $ -category $\mathscr {E}$ to the subcategory of the presheaf category $\mathrm {PSh}(\mathscr {E}^{\mathrm {op}})^{\mathrm {op}}$ generated under finite limits by the representable objects. In particular, the transformation $k\colon U \to U$ corresponds to a transformation $k'\colon \mathrm {id} \to U \circ F$ , and the problem translates to showing that $k'$ is equivalent to the unit transformation $u\colon \mathrm {id} \to U \circ F$ .

First we will show that $k'$ factors through u via some transformation $k"\colon \mathrm {id} \to \mathrm {id}$ . To see this, observe that for an $\infty $ -category $\mathscr {E}$ the unit map $u_{\mathscr {E}}\colon \mathscr {E} \to U(F(\mathscr {E}))$ is fully faithful, both sides being full subcategories of $\mathrm {PSh}(\mathscr {E}^{\mathrm {op}})^{\mathrm {op}}$ . It follows in particular that $u\colon \mathrm {id} \to U \circ F$ is a monomorphism in $ {\mathrm {Fun}}(\mathrm {Cat}_{\infty },\mathrm {Cat}_{\infty })$ , and thus it is a property for $k'$ to factor through u. Moreover, as the property of factoring through a monomorphism in a functor category can be checked pointwise, it suffices to show that for each $\mathscr {E} \in \mathrm {Cat}_{\infty }$ the functor $k^{\prime }_{\mathscr {E}}\colon \mathscr {E} \to U(F(\mathscr {E}))$ factors through $u_{\mathscr {E}}\colon \mathscr {E} \to U(F(\mathscr {E}))$ via some map $k^{\prime \prime }_{\mathscr {E}}\colon \mathscr {E} \to \mathscr {E}$ . By construction, $k^{\prime }_{\mathscr {E}}\colon \mathscr {E} \to U(F(\mathscr {E}))$ is given by the composite

By assumption on k, the second map induces the identity on groupoid cores, proving that $k^{\prime }_{\mathscr {E}}$ factors through $u_{\mathscr {E}}$ on objects. By fully faithfulness of $u_{\mathscr {E}}\colon \mathscr {E} \to U(F(\mathscr {E}))$ this means $k^{\prime }_{\mathscr {E}}$ factors through $u_{\mathscr {E}}$ as desired.

We are thus left with showing that the resulting endomorphism $k"\colon \mathrm {id}_{\mathrm {Cat}_{\infty }} \to \mathrm {id}_{\mathrm {Cat}_{\infty }}$ of the identity on $\mathrm {Cat}_{\infty }$ is equivalent to the identity. This is immediate: the space $ {\mathrm {End}}(\mathrm {id}_{\mathrm {Cat}_\infty })$ of endomorphisms of $\mathrm {id}_{\mathrm {Cat}_{\infty }}$ is contractible. This follows from a theorem by Toën, which says that the full subcategory of $ {\mathrm {Fun}}(\mathrm {Cat}_{\infty },\mathrm {Cat}_{\infty })$ spanned by the equivalences is equivalent to the discrete category $\{\mathrm {id}, (-)^{\mathrm {op}}\}$ , see [Reference Lurie50, Theorem 4.4.1].

We are now ready for the main theorem of this section.

Theorem 3.19. For every left exact $\infty $ -category $\mathscr {D}$ , the following diagram naturally commutes:

Proof. By the Yoneda lemma, it suffices to prove that the diagram commutes after applying the functor $ {\mathrm {Map}}_{\mathrm {Cat}_{\infty }}(\mathscr {E},-) = {\mathrm {Fun}}(\mathscr {E},-)^{\simeq }$ for every $\infty $ -category $\mathscr {E}$ , naturally in $\mathscr {E}$ . Here we may use the following naturally commutative diagram:

The left square commutes by Lemma 3.15 applied to $ {\mathrm {Fun}}(\mathscr {E},\mathscr {D})$ . The top square commutes by definition of $\Phi _{\mathscr {E},\mathscr {D}}$ . The right square commutes by definition of the higher categorical dimension functor (Definition 2.24). The bottom square commutes by Lemma 3.17. Finally, the middle square commutes by naturality of the dimension functor applied to the symmetric monoidal 2-functor $\Phi _{\mathscr {E},\mathscr {D}}$ .

4.1 Free and cofree $ \mathscr {C}$ -linear $\infty $ -categories

There are two classes of $\mathscr {C}$ -linear $\infty $ -categories which will play an important role throughout this article: for every space A we have the free $\mathscr {C}$ -linear $\infty $ -category $\mathscr {C}[A]$ and the cofree $\mathscr {C}$ -linear $\infty $ -category $\mathscr {C}^A$ . The goal of this subsection is to recall the basic properties these $\infty $ -categories have.

Definition 4.7. Let A be a space and let $\mathscr {D}$ be a $\mathscr {C}$ -linear $\infty $ -category. We define the $\mathscr {C}$ -linear $\infty $ -categories $\mathscr {D}[A]$ and $\mathscr {D}^A$ by

$$\begin{align*}\mathscr{D}[A] := {\mathrm{colim}}_A \mathscr{D} \quad\in\quad {\mathrm{Mod}}_{\mathscr{C}} \qquad\qquad \text{ and } \qquad\qquad \mathscr{D}^A := \lim_A \mathscr{D} \quad\in\quad {\mathrm{Mod}}_{\mathscr{C}}. \end{align*}$$

For a map $f\colon A \to B$ of spaces, we denote by

$$\begin{align*}f_!\colon \mathscr{D}[A] \to \mathscr{D}[B] \qquad\qquad \text{ and } \qquad\qquad f^*\colon \mathscr{D}^B \to \mathscr{D}^A \end{align*}$$

the induced $\mathscr {C}$ -linear functors. Their right adjoints in $\widehat {\mathrm {Cat}}_{\infty }$ are denoted by

$$\begin{align*}f^*\colon \mathscr{D}[B] \to \mathscr{D}[A] \qquad\qquad \text{ and } \qquad\qquad f_*\colon \mathscr{D}^A \to \mathscr{D}^B. \end{align*}$$

We will show in Corollary 4.15 that there is an equivalence $\mathscr {D}[A] \simeq \mathscr {D}^A$ . Nevertheless, we choose to distinguish them in the notation to emphasize the different roles they play. Under this identification, the two functors called $f^*$ get identified, justifying the notation.

The $\mathscr {C}$ -linear $\infty $ -category $\mathscr {C}[A]$ is called the free $\mathscr {C}$ -linear $\infty $ -category on A, as the $\mathscr {C}$ -linear functors out of it into some $\mathscr {C}$ -linear $\infty $ -category $\mathscr {D}$ correspond to functors $A \to \mathscr {D}$ . In fact, we have the following stronger statement:

Lemma 4.9. Let $\mathscr {D}$ be a $\mathscr {C}$ -linear $\infty $ -category and let A be a space. There are natural equivalences of $\mathscr {C}$ -linear $\infty $ -categories

The assignments $A \mapsto \mathscr {C}[A]$ and $A \mapsto \mathscr {D}^A$ satisfy various adjointability properties, making them bivariant theories in the sense of Definition 3.2.