2.1 Poincaré categories and Poincaré objects

A hermitian structure on a small stable $\infty $ -category ${\mathcal {C}}$ is a reduced, quadratic functor . A pair consisting of this data is called a hermitian category. These organise into an $\infty $ -category $\mathrm {Cat^h}$ whose morphisms consist of hermitian functors, that is pairs $(f,\eta )$ where $f \colon {\mathcal {C}} \rightarrow {\mathcal {C}}'$ is an exact functor and is a natural transformation, see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 1.2].

To such a hermitian category is associated its category of hermitian forms : Objects are pairs $(X,q)$ where $X \in {\mathcal {C}}$ and q is a -hermitian form on X, that is, a point in . Morphisms are maps in ${\mathcal {C}}$ preserving the hermitian forms. We shall denote the groupoid core of by . In order to impose a nondegeneracy condition on the forms in , one needs a nondegeneracy condition on the hermitian $\infty $ -category itself. To this end recall the classification of quadratic functors from Goodwillie calculus, see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 1.3]: Any reduced quadratic functor uniquely extends to a cartesian diagram

where is linear (i.e., exact) and is bilinear (i.e., exact in each variable) and symmetric (i.e., comes equipped with a refinement to an element of $\mathrm {Fun}({\mathcal {C}}^{\mathrm {op}} \times {\mathcal {C}}^{\mathrm {op}},\mathrm {Sp})^{\mathrm {hC}_2}$ , with ${\mathrm {C}_2}$ acting by flipping the input variables).

A hermitian structure is called Poincaré if there exists an equivalence ${\mathrm {D}} \colon {\mathcal {C}}^{\mathrm {op}} \rightarrow {\mathcal {C}}$ such that

naturally in $X,Y \in {\mathcal {C}}^{\mathrm {op}}$ . By Yoneda’s lemma, such a functor ${\mathrm {D}}$ is uniquely determined if it exists, so we refer to it as . By the symmetry of the functor then automatically satisfies . Any hermitian functor between Poincaré-categories (i.e., hermitian categories whose hermitian structure is Poincaré) induces a tautological map

and we say that $(F,\eta )$ is a Poincaré functor if this transformation is an equivalence. Poincaré categories together with Poincaré functors form a (nonfull) subcategory $\mathrm {Cat}^{\mathrm {p}}$ of $\mathrm {Cat^h}$ .

Now, if is Poincaré, then to any hermitian form there is tautologically associated a map

as the image of q under

and we say that $(X,q)$ is Poincaré if $q^\sharp $ is an equivalence, see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 2.1]. The full subspace of the core of spanned by the Poincaré forms is denoted by and provides a functor

$$\begin{align*}\mathrm{Pn} \colon \mathrm{Cat}^{\mathrm{p}} \rightarrow \mathrm{An}. \end{align*}$$

2.2 The hermitian $\mathcal Q$ -construction

To a Poincaré category are associated its Grothendieck-Witt space or spectrum and an algebraic cobordism category. Both are defined using the hermitian $\mathcal Q$ -construction which we now recall. For $K \in \mathrm {Cat}_\infty $ let $\operatorname {TwAr}(K)$ denote the twisted arrow category of K. We define $\mathcal Q_K({\mathcal {C}}) \subseteq \mathrm {Fun}(\operatorname {TwAr}(K),{\mathcal {C}})$ for stable ${\mathcal {C}}$ to be the full subcategory spanned by all diagrams F for which the squares

are (co)cartesian for every functor $[3] \rightarrow K$ with value $i \rightarrow j \rightarrow k \rightarrow l$ ; for example this pertains to the middle square in $\operatorname {TwAr}[2]$ :

and by pasting pullbacks it always suffices to consider chains of the form $i\to k \xrightarrow {\mathrm {id}} k\to l$ . If ${\mathcal {C}}$ is equipped with a hermitian structure then

defines a hermitian structure on $\mathrm {Fun}(\operatorname {TwAr}(K),{\mathcal {C}})$ . If is Poincaré, then so is the restriction of to $\mathcal Q_K({\mathcal {C}})$ (though is not even Poincaré for $K = [2]$ ). The construction is functorial in K yielding the desired hermitian $\mathcal Q$ -construction

$$\begin{align*}\mathcal Q \colon \mathrm{Cat}_\infty^{\mathrm{op}} \times \mathrm{Cat}^{\mathrm{p}} \longrightarrow \mathrm{Cat}^{\mathrm{p}},\end{align*}$$

see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Section 2.2]. It preserves limits in both variables, so fixing the input Poincaré category to be , its restriction to defines a complete Segal object of $\mathrm {Cat}^{\mathrm {p}}$ , and thus is a complete Segal space, since $\mathrm {Pn}\colon \mathrm {Cat}^{\mathrm {p}} \rightarrow \mathrm {An}$ also preserves limits.

Denote now by

Rezk’s nerve adjunction (i.e., $\mathrm {asscat}$ is the colimit preserving extension of the inclusion and $\mathrm N$ , the nerve, its right adjoint), whose right adjoint is fully faithful with essential image the complete Segal spaces see, for example, [Reference Joyal and TierneyJT07, Reference Hebestreit and SteinebrunnerHS25]. We note that for a simplicial space X, its realisation is canonically equivalent to the realisation $|\mathrm {asscat} X|$ of the associated category since adjoints compose and spaces have constant nerve (the realisation functor of categories being the left adjoint to the inclusion $\mathrm {An}\to \mathrm {Cat}_\infty $ ). We put

Unwinding the definitions objects of are exactly Poincaré objects in , and a morphism is an algebraic cobordism à la Ranicki, that is, given by a span

together with an equivalence $\eta \colon f^*q \sim g^*q'$ in , which satisfies Lefschetz duality in the sense that the induced map

is an equivalence. Here indexing conventions adhere to the geometric definitions where $\mathsf {Cob}_d$ has d-dimensional cobordisms as morphisms. In particular, we find given by the cobordism group of Poincaré objects in , which is isomorphic to . For the purposes of this paper we shall treat it as the definition of the $\mathrm L$ -groups on the left (and more generally of ), and refer the reader to [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Section 2.3] for a proof that this agrees with the Witt-style definition, given by taking the quotient of by metabolic forms (which we will recall below). Finally, we recall that there is a tautological equivalence

that we shall exploit throughout.

The Grothendieck-Witt space of can be defined in a number of ways. Closest in spirit to classical constructions for exact categories is

but as part of the Genauer-Bott sequence described at the end of this subsection we also established an equivalence

in [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Section 4.1] and showed that $\mathcal {GW} \colon \mathrm {Cat}^{\mathrm {p}} \rightarrow \mathrm {An}$ is the initial functor equipped with a transformation from $\mathrm {Pn} \colon \mathrm {Cat}^{\mathrm {p}} \rightarrow \mathrm {An}$ that is additive and group-like; this is arguably the cleanest definition, but as we won’t need it we refrain from expounding the meaning of these terms here.

Any of the definitions of the Grothendieck-Witt space can be extended to that of a Grothendieck-Witt spectrum with

for all $i> 0$ , where denotes the i-fold iteration of $\mathcal Q$ , see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Section 4.2]. In particular, we find

in analogy with the theorem of Galatius, Madsen, Tillmann and Weiss. While this spectrum, which in contrast to the situation for $\mathrm K({\mathcal {C}})$ is generally not bounded below, will play essentially no role in the present paper, the iterated $\mathcal Q$ -construction on the right hand side does. We note that

is an i-fold complete Segal space, and thus models an i-fold (or i-tuple) category. We call it in analogy with the higher cobordism categories in manifolds topology; of course i-fold complete Segal spaces are a perfectly good definition of i-fold categories in which case the above definition merely amounts to an abbreviation. In the double category , a morphism square is given by a diagram

in which the corners are Poincaré objects of , the outer spans are cobordisms between these, which can be glued to a single Poincaré object in , of which U is then a nullbordism.

In a different direction, restricting the Poincaré structure on to the fibre of the functor

$$\begin{align*}d_0 \colon \mathcal Q_1({\mathcal{C}}) \longrightarrow {\mathcal{C}}\end{align*}$$

gives a Poincaré structure on the arrow category $\mathrm {Ar}({\mathcal {C}})$ , which we call the metabolic Poincaré category associated to . Its Poincaré objects can be thought of either as the equivalent of unimodular forms equipped with a Lagrangian, or as Poincaré objects with boundary in , see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 2.3]. The latter point of view nicely fits with the null-composite sequence

of Poincaré categories, where $\mathrm {met}$ is the restriction of and the left hand arrow takes X to $X \rightarrow 0$ . Applying the Grothendieck-Witt functor results in the fibre sequence

of spectra, which we termed the Bott-Genauer sequence, as it is analogous to Genauer’s fibre sequence form geometric topology, see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Section 4.3].

2.3 Examples of Poincaré categories

As mentioned above the material of this somewhat lengthy subsection is not strictly speaking necessary to read most of the present paper, but we shall freely reference the examples explained here throughout the text.

We start out by discussing hermitian structures given as the animations of form parameters in the sense of Schlichting [Reference SchlichtingSch19], which includes form parameters in the sense of Bak [Reference BakBak81] as a special case. Given a ring R, a form parameter $\lambda $ consists of an $R \otimes _{\mathbb {Z}} R$ -module M, equipped with an involution $\sigma $ that is linear over the flip of $R \otimes _{\mathbb {Z}} R$ , together with a factorisation

$$\begin{align*}M_{\mathrm{C}_2} \xrightarrow{\tau} Q \xrightarrow{\rho} M^{\mathrm{C}_2}\end{align*}$$

of the norm map $[m] \mapsto m + \sigma (m)$ that satisfies condition ( $*$ ) below and is quadratic over R in the following sense: Q is an abelian group equipped with an action of the multiplicative monoid of R, such that multiplication by any $r \in R$ is an additive map $Q \rightarrow Q$ , but such that in lieu of distributivity one only has that

$$\begin{align*}\mathrm B(r,s)(x) = (r+s)x - rx - sx\end{align*}$$

defines a bilinear map $R \times R \rightarrow {\mathrm {Hom}}_{\mathrm {Ab}}(Q,Q)$ . $M_{\mathrm {C}_2}$ and $M^{\mathrm {C}_2}$ also carry such actions by restricting the $R \otimes _{\mathbb {Z}} R$ -module structure on M along the diagonal

$$\begin{align*}R \longrightarrow R \otimes R, \quad r \longmapsto r \otimes r.\end{align*}$$

In addition to being additive and R-equivariant for these actions we require that $\tau $ and $\rho $ satisfy

(*) $$ \begin{align} \mathrm B(r,s)(q) = \tau[(r \otimes s)\rho(q)], \end{align} $$

which in particular prescribes the composition $\tau \circ \rho \colon Q \rightarrow Q$ . This requirement is geared to make the notion of a $\lambda $ -form work: It consists of a finitely generated projective R-module P together with a ${\mathrm {C}_2}$ -equivariant $R \otimes R$ -linear map $b \colon P \otimes P \rightarrow M$ and an R-equivariant function $q\colon P \rightarrow Q$ that satisfy

$$\begin{align*}q(p+p') -q(p)-q(p') = \tau[b(p,p')] \quad \text{and}\quad \rho q(p) = b(p,p).\end{align*}$$

For commutative R the examples of M-valued (skew-)symmetric and (skew-)quadratic forms for some R-module M correspond to the choices $\rho = \mathrm {id}_{M^{\mathrm {C}_2}}$ and $\tau = \mathrm {id}_{M_{\mathrm {C}_2}}$ , respectively, with $\sigma = \pm \mathrm {id}_M$ and the $R \otimes _{\mathbb {Z}} R$ -module structure on M induced by the multiplication of R. A ring with skew-involution similarly fits into this framework by taking $M=R$ and letting the skew-involution provide both the $R \otimes _{\mathbb {Z}} R$ -module structure as well as $\sigma $ .

Sending a projective module P to the abelian group of $\lambda $ -forms begets the functor

$$\begin{align*}\mathrm F^\lambda \colon \mathcal{P}(R)^{\mathrm{op}} \longrightarrow \mathrm{Ab}\end{align*}$$

that already appeared in the introduction. It uniquely extends to a hermitian structure on ${\mathcal {D}}^{\mathrm {p}}(R)$ that makes the diagram

commute, see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 4.2]. This hermitian structure is Poincaré if and only if M is perfect as an R-module, when pulled back along the left, say, inclusion $R \rightarrow R \otimes _{\mathbb {Z}} R$ , and in addition the canonical map

$$\begin{align*}R \longrightarrow \mathbb R{\mathrm{Hom}}_R(M,M), \quad r \longmapsto (1 \otimes r) \cdot -\end{align*}$$

is an equivalence. In this case the duality associated to is then given by

For most of our results, in particular for Theorem B to be applicable to , we will furthermore have to assume that M is not just perfect but projective as an R-module, and for notational ease will therefore refer to $\lambda $ as invertible only if it satisfies this stronger condition (we will discuss this point further in 3.2.4 below). In the simplest example of a commutative ring R and the two R-module structures on M agreeing, this exactly amounts to M being $\otimes _R$ -invertible. At any rate, to put the description of the linear part into context, let us briefly recall the classification of hermitian structures on module categories from [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 3.2].

Fix an $\mathbb E_1$ -algebra A over a base $\mathbb E_\infty $ -ring spectrum k and a subgroup $c \subseteq \mathrm K_0(A)$ . Consider then the category of compact A-module spectra $\mathrm {Mod}^\omega _{A}$ or more generally its full subcategory ${\mathrm {Mod}}^c_A$ spanned by all those $X \in \mathrm {Mod}^\omega _{A}$ with $[X] \in c \subseteq \mathrm K_0(A)$ . For example ${\mathrm {Mod}}^{\langle A \rangle }_A$ is the stable subcategory of ${\mathrm {Mod}}(A)$ spanned by A itself. For the reader mostly interested in the applications to discrete rings as discussed above, we recall that any discrete ring R gives rise to such data, via the Eilenberg-Mac Lane functor $\mathrm {H} \colon \mathrm {Ab} \longrightarrow \mathrm {Sp}$ , which is lax symmetric monoidal and therefore induces a functor

$$\begin{align*}\mathrm{Ring} \longrightarrow \mathrm{Alg}_{\mathbb E_1}({\mathrm{Mod}}_{\mathrm{H}{\mathbb{Z}}}) \end{align*}$$

In this way, any discrete ring may be regarded as an $\mathbb E_1$ -algebra over $\mathrm {H}{\mathbb {Z}}$ . There are, furthermore, equivalences

$$\begin{align*}\mathrm{Mod}^\omega_{\mathrm{H} R} \simeq {\mathcal{D}}^{\mathrm{p}}(R) \quad \text{and} \quad {\mathrm{Mod}}^{\langle \mathrm H R \rangle} _{\mathrm{H} R} \simeq {\mathcal{D}}^{\mathrm{f}}(R), \end{align*}$$

where ${\mathcal {D}}^{\mathrm {p}}(R)$ denotes the full subcategory of the derived $\infty $ -category $\mathcal D(R)$ of R spanned by the perfect complexes, that is, finite chain complexes of finitely generated projective R-modules and ${\mathcal {D}}^{\mathrm {f}}(R)$ is the full subcategory spanned by the finite chain complexes of finite free R-modules. In this regime the reader should keep in mind that terms such as $\otimes _{\mathrm {H} {\mathbb {Z}}}$ or ${\mathrm {Hom}}_{\mathrm {H} R}$ will evaluate to the functors $\otimes ^{\mathbb {L}}_{\mathbb {Z}}$ and $\mathbb R{\mathrm {Hom}}_R$ .

Hermitian structures on the categories ${\mathrm {Mod}}^c_A$ arise from k-modules with genuine involution $(M,N,\alpha )$ over A [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Definition 3.2.3]: The first entry M plays the same role as M did above; it is an $A \otimes _k A$ -module, equipped with the structure of a homotopy fixed point in the category ${\mathrm {Mod}}_{A \otimes _k A}$ under the ${\mathrm {C}_2}$ -action flipping the two factors.

The additional data of a module with genuine involution consists of an A-module spectrum N, and an A-linear map $\alpha \colon N \rightarrow M^{\mathrm {tC}_2}$ ; to make sense of the latter term, note that upon forgetting the $A \otimes _k A$ -action, the involution equips M with the structure of a (naive) ${\mathrm {C}_2}$ -k-module spectrum. The k-module spectrum $M^{\mathrm {tC}_2}$ then becomes an $(A \otimes _k A)^{\mathrm {tC}_2}$ -module via the lax monoidality of the Tate construction and from here obtains an A-module structure on $M^{\mathrm {tC}_2}$ by pullback along the Tate diagonal $A \rightarrow (A \otimes _k A)^{\mathrm {tC}_2}$ , which is a map of $\mathbb E_1$ -ring spectra, see [Reference Nikolaus and ScholzeNS18, Chapter III.1] for an exposition of the Tate diagonal in the present language. Let us immediately warn the reader that the Tate diagonal is not generally k-linear for the k-module structure on $(A \otimes _k A)^{\mathrm {tC}_2}$ arising from the unit map $k \rightarrow (k \otimes _k k)^{\mathrm {tC}_2} = k^{\mathrm {tC}_2}$ , as this map is usually different from the Tate-diagonal of k (in particular, this is the case for $k = \mathrm {H}{\mathbb {Z}}$ by [Reference Nikolaus and ScholzeNS18, Theorem III.1.10]).

Even if only interested in discrete R, one therefore has to leave not only the realm of discrete R-modules to form the Tate construction, but even the realm of derived categories, as no replacement for the Tate diagonal can exist in that regime. In fact, k really only plays an auxiliary role in the entire discussion above and for the purposes of the present paper it is best to take $k = \mathbb S$ , as in that case the following procedure captures all hermitian structures on ${\mathrm {Mod}}^c_A$ .

The hermitian structure associated to a module with genuine involution $(M,N,\alpha )$ is given by the pullback

where the ${\mathrm {C}_2}$ -action on $\mathrm {hom}_{A \otimes _k A}(X \otimes _k X,M)$ is given by flipping the factors in the source and the involution on M. It is a Poincaré structure on $\mathrm {Mod}^\omega _A$ if M restricts to an object of $\mathrm {Mod}^\omega _{A}$ under either inclusion $A \rightarrow A \otimes _k A$ , and furthermore M is invertible, that is, the natural map

$$\begin{align*}A \rightarrow \mathrm{hom}_A(M,M) \end{align*}$$

is an equivalence; again we shall tacitly assume that this condition is satisfied whenever we speak of modules with involution over A in the sequel. In this case the associated duality is given by ${X \mapsto \mathrm {hom}_A(X,M)}$ regarded as an A-module via the extraneous A-module structure on M. Given a subgroup $c \in \mathrm K_0(A)$ one obtains a Poincaré structure on ${\mathrm {Mod}}^c_A$ if in addition c is closed under the duality on $\mathrm K_0(A)$ induced by M. In case $c = \mathrm K_0(A)$ or $c=0$ this condition is of course vacuous, and for $c = \langle A \rangle $ it simply translates to $[M]\in \langle A \rangle $ .

The hermitian structure for $\lambda = (Q,\rho ,\tau ,\sigma )$ a form parameter with underlying module M then corresponds to the $\mathrm H{\mathbb {Z}}$ -module with genuine involution over $\mathrm HR$

$$\begin{align*}\mathrm{g}\lambda = (M,\mathrm{cof}(M_{\mathrm{hC}_2} \rightarrow M_{\mathrm{C}_2} \xrightarrow{\tau} Q), \mathrm{cof}(M_{\mathrm{hC}_2} \rightarrow Q) \xrightarrow{\rho} \mathrm{cof}(M_{\mathrm{hC}_2} \rightarrow M^{\mathrm{C}_2}) \rightarrow M^{\mathrm{tC}_2});\end{align*}$$

the underlying spectra are easily determined by evaluating the fibre sequence

for $X = R \in {\mathcal {D}}^{\mathrm {p}}(R)$ , but it is somewhat tricky to lift $\mathrm {cof}(M_{\mathrm {hC}_2} \xrightarrow {\tau } Q)$ and its map to $M^{\mathrm {tC}_2}$ from $\mathrm {Sp}$ to $\mathcal D(R)$ a priori as their constituents are not themselves R-linear, see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 4.2].

From this formula we for example find and associated to

$$\begin{align*}(M,\tau_{\geq 2}M^{\mathrm{tC}_2}, \mathrm{can}) \quad \text{and} \quad (M,\tau_{\geq 0}M^{\mathrm{tC}_2}, \mathrm{can}),\end{align*}$$

respectively, where $\mathrm {can}$ is the canonical map from the connective cover. These structures extend to an arbitrary connective $\mathbb E_1$ -ring A, where we define associated to $(M,\tau _{\geq i} M^{\mathrm {tC}_2},\mathrm {can})$ . This includes the cases $i \in \{\pm \infty \}$ , where we obtain the homotopy symmetric and quadratic Poincaré structures

corresponding to

$$\begin{align*}(M,0,0) \quad \text{and} \quad (M,M^{\mathrm{tC}_2},\mathrm{id}),\end{align*}$$

respectively. Their Poincaré objects (in the case of a discrete ring) where originally introduced into surgery theory by Ranicki as the base objects for higher $\mathrm L$ - or Witt groups, compare [Reference RanickiRan92], and then reworked into the present language by Lurie in [Reference LurieLur11]. Let us warn immediately, that if R is a discrete ring and $2$ is not assumed invertible in R, the functor will essentially never satisfy the connectivity conditions that we have to assume in our main results, whereas our methods give information about each of the finite stages.

Another very interesting example, that our results apply to, is the universal Poincaré structure on the category of finite spectra, corresponding to the module with genuine involution $(\mathbb S, \mathbb S, u)$ , where u is the unit of the $\mathbb E_\infty $ -ring $\mathbb S^{\mathrm {tC}_2} \simeq \mathbb S_2^\wedge $ , see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 4.1]. This Poincaré structure was originally considered by Weiss and Williams in [Reference Weiss and WilliamsWW14] (in different language of course) during their pursuit of a unification of surgery and concordance theory, and generalises to arbitrary $B \in \mathrm {An}$ , as in the case of Waldhausen’s $\mathrm A$ -spectra, as follows: Consider the categories $\mathrm {Sp}/B = \mathrm {Fun}(B,\mathrm {Sp})$ for some $B \in \mathrm {An}$ . Entirely parallel to the discussion above, one can derive hermitian structures on the compact objects of $\mathrm {Sp}/B$ from triples $(M,N, \alpha )$ with $M \in (\mathrm {Sp}/B \times B)^{\mathrm {hC}_2}$ and $\alpha \colon N \rightarrow (\Delta ^*M)^{\mathrm {tC}_2}$ a map in $\mathrm {Sp}/B$ , where $\Delta \colon B \rightarrow B \times B$ is the diagonal, compare [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 4.4]. The most important examples of such functors are the visible Poincaré structures given by the triples

$$\begin{align*}(\Delta_! \xi, \xi, u \colon \xi \rightarrow (\Delta^*\Delta_!\xi)^{\mathrm{tC}_2}), \end{align*}$$

where $\xi \colon B \rightarrow \mathrm {Pic}(\mathbb S) \subset \mathrm {Sp}$ is some stable spherical fibration over B, $\Delta _! \colon \mathrm {Sp}/B \rightarrow \mathrm {Sp}/(B \times B)$ is the left adjoint to $\Delta ^*$ and u is the unit of this adjunction (which factors through $\xi \rightarrow (\Delta ^*\Delta _!\xi )^{\mathrm {hC}_2}$ since $\Delta $ is invariant under the ${\mathrm {C}_2}$ -action on $B \times B$ ). These hermitian structures are automatically Poincaré with associated duality given by

$$\begin{align*}X \longmapsto \mathrm{hom}_B(X,\Delta_!\xi), \end{align*}$$

the Costenoble-Waner duality functor twisted by $\xi $ . The case of the universal Poincaré category is contained in this discussion as the case $B = \ast $ and $\xi = \mathbb S$ . In [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Section 4.7] we produced equivalences

extending Waldhausen’s equivalence

$$\begin{align*}\mathrm K((\mathrm{Sp}/B)^\omega) \simeq \mathrm{A}(B)\end{align*}$$

for $\lambda \in \{\mathrm {q},\mathrm {s},\mathrm {v}\}$ , where the top right hand side denotes the spectra constructed by Weiss and Williams and the lower right hand side Waldhausen’s algebraic $\mathrm K$ -spectrum of the space B.

Finally, we mention once more the hyperbolic categories $\operatorname {Hyp}({\mathcal {C}}) = {\mathcal {C}}^{\mathrm {op}} \times {\mathcal {C}}$ for ${\mathcal {C}}$ a stable $\infty $ -category, with Poincaré structure given by $\mathrm {hom}_{\mathcal {C}} \colon {\mathcal {C}}^{\mathrm {op}} \times {\mathcal {C}} \rightarrow \mathrm {Sp}$ . The duality in this case is given by $(X,Y) \mapsto (Y,X)$ and the diagonal induces an equivalence

$$\begin{align*}\mathrm{Pn}(\operatorname{Hyp}({\mathcal{C}})) \simeq \mathrm{cr}({\mathcal{C}}),\end{align*}$$

through which all of our results on Grothendieck-Witt spaces contain their K-theoretic counterparts, since essentially per construction

$$\begin{align*}\mathsf{Cob}(\operatorname{Hyp}({\mathcal{C}})) \simeq \mathrm{Span}({\mathcal{C}})\end{align*}$$

and thus $\mathrm {GW}(\operatorname {Hyp}({\mathcal {C}})) \simeq \mathrm K({\mathcal {C}})$ , compare [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 2.2].

3.1 Weight structures on stable $\infty $ -categories

Weight structures were originally introduced for triangulated categories by Bondarko in [Reference BondarkoBon12], and a detailed discussion on stable $\infty $ -categories is for example given in [Reference SosniloSos19]. We recall the definition and basic properties for the reader’s convenience.

Definition 3.1.1. A weight structure on a stable $\infty $ -category consists of two full subcategories ${\mathcal {C}}_{[0,\infty ]}$ and ${\mathcal {C}}_{[-\infty ,0]}$ of ${\mathcal {C}}$ such that the following conditions hold:

1. i) ${\mathcal {C}}_{[0,\infty ]}$ and ${\mathcal {C}}_{[-\infty ,0]}$ are both closed under retracts in ${\mathcal {C}}$ ,

2. ii) for $X \in {\mathcal {C}}_{[-\infty ,0]}$ and $Y \in {\mathcal {C}}_{[0,\infty ]}$ the mapping spectrum $\mathrm {hom}_{\mathcal {C}}(X,Y)$ is connective (i.e., has trivial negative homotopy groups) and

3. iii) for every $X \in {\mathcal {C}}$ , there is a fibre sequence

   $$\begin{align*}Y \rightarrow X \rightarrow Z\end{align*}$$
   with $Y \in {\mathcal {C}}_{[-\infty ,0]}$ and $Z^{[-1]} \in {\mathcal {C}}_{[0,\infty ]}$ .

For $a \leq b \in \mathbb {Z}$ we set ${\mathcal {C}}_{[a,\infty ]}$ and ${\mathcal {C}}_{[-\infty ,b]}$ to be the full subcategories of ${\mathcal {C}}$ spanned by all objects X with $X^{[-a]} \in {\mathcal {C}}_{[0,\infty ]}$ or $X^{[-b]} \in {\mathcal {C}}_{[-\infty ,0]}$ , respectively, and put

$$\begin{align*}{\mathcal{C}}_{[a,b]} = {\mathcal{C}}_{[a,\infty]} \cap{\mathcal{C}}_{[-\infty,b]}.\end{align*}$$

The category ${\mathcal {C}}_{[0,0]} = {\mathcal {C}}^{\mathrm {ht}}$ is the heart of the weight structure. Objects of ${\mathcal {C}}_{]-\infty ,\infty [} = \bigcup _{n \in {\mathbb {Z}}} {\mathcal {C}}_{[-n,n]}$ are said to be bounded, and the weight structure is exhaustive if ${\mathcal {C}} = {\mathcal {C}}_{]-\infty ,\infty [}$ .

We call a stable $\infty $ -category equipped with a weight structure a weighted category. An exact functor $F \colon {\mathcal {C}} \rightarrow \mathcal D$ between weighted categories we call bounded below or above by $d \in {\mathbb {Z}}$ , if

$$\begin{align*}F({\mathcal{C}}_{[0,\infty]}) \subseteq \mathcal D_{[d,\infty]} \quad \text{or} \quad F({\mathcal{C}}_{[-\infty,0]}) \subseteq \mathcal D_{[-\infty,d]},\end{align*}$$

respectively. We say F has degree d if it is both bounded below and above by d.

Note the order reversal of the terms in item iii) as compared to a t-structure. As will be visible in the examples below, this has the effect that weight decompositions as in iii) are not unique.

If $X \in {\mathcal {C}}_{[-\infty ,b]}$ and $Y \in {\mathcal {C}}_{[a,\infty ]}$ , then the spectrum $\mathrm {hom}_{\mathcal {C}}(X,Y)$ is $(a-b)$ -connective. In particular,

$$\begin{align*}{\mathcal{C}}_{[-\infty, a]} \cap {\mathcal{C}}_{[b, \infty]}=0 \quad \mathrm{if}\quad a<b.\end{align*}$$

Lemma 3.1.2. We have

$$\begin{align*}\{X \in {\mathcal{C}} \mid \mathrm{hom}_{\mathcal{C}}(X,Y) \text{ is connective for all } Y \in {\mathcal{C}}_{[b,\infty]}\} = {\mathcal{C}}_{[-\infty,b]}\end{align*}$$

and dually

$$\begin{align*}\{X \in {\mathcal{C}} \mid \mathrm{hom}_{\mathcal{C}}(Y,X) \text{ is connective for all } Y \in {\mathcal{C}}_{[-\infty,a]}\} = {\mathcal{C}}_{[a,\infty]}.\end{align*}$$

In particular, ${\mathcal {C}}_{[a,\infty ]}$ is closed under pushouts and ${\mathcal {C}}_{[-\infty ,b]}$ under pullbacks, and all ${\mathcal {C}}_{[a,b]}$ are closed under extensions and thus additive.

Proof. The inclusion of the right hand sides into the left holds by axiom ii), and for the converse in the first statement pick a decomposition $Y \rightarrow X \rightarrow Z$ with $Y \in {\mathcal {C}}_{[-\infty ,b]}$ and $Z \in {\mathcal {C}}_{[b+1,\infty ]}$ using iii), and note that per assumption on X its second map is null-homotopic. Thus the rotated fibre sequence

$$\begin{align*}Z^{[-1]} \longrightarrow Y \longrightarrow X\end{align*}$$

splits, displaying X as a retract of Y, which suffices for the claim by axiom i).

We will refer to objects in ${\mathcal {C}}_{[n,\infty ]}$ as n-connective as this will fit our examples, and dually we shall call objects in ${\mathcal {C}}_{[-\infty ,n]} n$ -coconnective. We warn the reader explicitly, that in most examples n-coconnectivity is neither directly related to n-connectivity in a t-structure with the same connective part (which may or may not exist) nor to the notion of n-truncated objects in ${\mathcal {C}}$ . Furthermore, we call a map $X \rightarrow Y$ an n-connective if its fibre lies in ${\mathcal {C}}_{[n,\infty ]}$ , and n-coconnective if its cofibre lies in ${\mathcal {C}}_{[-\infty ,n]}$ .

Finally, let us mention that an exhaustive weight structure is entirely determined by its heart, which can be an arbitrary full subcategory $H \subseteq {\mathcal {C}}$ that is closed under taking finite sums and retracts, has ${\mathcal {C}}$ as its stable envelope and whose morphism spectra are connective: One inductively sets $H_{0,k+1}$ to consist of all objects X which can be written in an extension

$$\begin{align*}Y \rightarrow X \rightarrow Z\end{align*}$$

with $Y \in H_{0,k}$ and $X^{[-k-1]} \in H$ , and then

$$\begin{align*}{\mathcal{C}}_{[0,\infty]} = \bigcup_{k \in \mathbb N} H_{0,k},\end{align*}$$

and similarly for ${\mathcal {C}}_{[-\infty ,0]}$ . It is then an elementary exercise to check that these definitions give an exhaustive weight structure, that the construction really is inverse to taking hearts and that ${\mathcal {C}}_{[0,k]} = H_{0,k}$ .

In fact, even the category ${\mathcal {C}}$ is determined by the heart: Sosnilo shows in [Reference SosniloSos19, Section 3] that taking hearts sets up a fully faithful functor $(-)^{\mathrm {ht}} \colon \mathrm {Cat}^{\mathrm {st}}_{\mathrm {wt}} \rightarrow \mathrm {Cat}^{\mathrm {add}}$ from the category of stable $\infty $ -categories equipped with an exhaustive weight structure and functors of degree $0$ to the category of additive ( $\infty $ -)categories and product preserving functors. Its essential image is $\mathrm {Cat}^{\mathrm {add}}_{\flat }$ , the full subcategory spanned by the weakly idempotent complete additive categories, that is, those for which every map that admits a retraction is the inclusion of a direct summand. We shall also refer to these as $\flat $ -additive for brevity ( $\flat $ being an accidental lower than $\natural $ , the standard symbol for idempotent completion). A typical example of an additive category that is not weakly idempotent complete is given by the finitely generated free modules over certain rings, for example, the sections in the tangent bundles of $\mathrm S^2$ as a module over $\mathrm {C}(\mathrm S^2,\mathbb R)$ . Its weak idempotent closure is the category of stably free modules.

The functor $(-)^\flat $ admits a right adjoint $\mathrm {Stab} \colon \mathrm {Cat}^{\mathrm {add}} \rightarrow \mathrm {Cat}^{\mathrm {st}}_{\mathrm {wt}}$ , which takes any additive category $\mathcal A$ to its stable envelope $\mathrm {Stab}(\mathcal A)$ , that is, the smallest stable subcategory of $\mathrm {Fun}^\times (\mathcal A^{\mathrm {op}},\mathrm {Sp})$ containing the (connective) deloopings of the representable functors $\mathcal A^{\mathrm {op}} \rightarrow \mathrm {Grp}_{\mathbb E_\infty }(\mathrm {An})$ , where the superscript denotes product preserving functors. It is easily checked invariant under weak idempotent completion and necessarily gives the inverse to $(-)^{\mathrm {ht}}$ on weakly idempotent complete additive categories. The invariance of stabilisation under this operation will allow us to also treat general additive categories in many cases. For example, due to the group completion theorem forming weak idempotent completions has no effect on direct sum $\mathrm K$ -theory, that is, the association $\mathcal A \mapsto \mathrm {cr}(\mathcal A)^{\mathrm {grp}}$ , see for example [Reference Hebestreit, Lachmann and SteimleHLS24, Lemma 7.5], and an analogous statement holds in the hermitian context.

For easier reference later we record the following simple statements:

Lemma 3.1.4. If $f \colon X \rightarrow Y$ is n-connective then for any $Z \in {\mathcal {C}}_{[-\infty ,n]}$ the map

$$\begin{align*}\pi_0{\mathrm{Hom}}_{\mathcal{C}}(Z,X) \xrightarrow{f_*} \pi_0{\mathrm{Hom}}_{\mathcal{C}}(Z,Y)\end{align*}$$

is surjective, and even bijective if $Z \in {\mathcal {C}}_{[-\infty ,n-1]}$ .

Proof. There is a fibre sequence of spectra

$$\begin{align*}\mathrm{hom}_{\mathcal{C}}(Z,\mathrm{fib}(f)) \longrightarrow \mathrm{hom}_{\mathcal{C}}(Z,X) \longrightarrow \mathrm{hom}_{\mathcal{C}}(Z,Y)\end{align*}$$

and by assumption $\mathrm {fib}(f) \in {\mathcal {C}}_{[n,\infty ]}$ . The claim follows from axiom (2).

Proof. The claim follows by inspecting the rotated fibre sequences

$$\begin{align*}Z^{[-1]} \longrightarrow Y \longrightarrow X \quad \text{and} \quad X \longrightarrow Z \longrightarrow Y^{[1]}\end{align*}$$

using the fact that the categories ${\mathcal {C}}_{[a,\infty ]}$ and ${\mathcal {C}}_{[-\infty ,b]}$ are closed under extensions.

Proof. In the rotated sequence

$$\begin{align*}Y \longrightarrow Z \xrightarrow{\partial} X^{[1]}\end{align*}$$

we find $\partial $ nullhomotopic since $\pi _0{\mathrm {Hom}}_{\mathcal {C}}(Z,X^{(1)}) = \pi _{-1}\mathrm {hom}_{\mathcal {C}}(Z,X)= 0$ by assumption.

3.2 Weight structures on Poincaré categories

Definition 3.2.1. A Poincaré category of dimension at least d is a Poincaré category equipped with an exhaustive weight structure, such that the duality functor is bounded below by d, explicitly

Similarly, it is of dimension at most d if it is bounded above by d, that is,

and of dimension exactly d if both are true. In either case, we shall write for the collection of path components in spanned by those objects $(X,q)$ with $X \in {\mathcal {C}}_{[a,b]}$ .

It is easy to see from exhaustiveness that it in fact suffices to check for all $X \in {\mathcal {C}}^{\mathrm {ht}}$ to conclude that has dimension at least d, and that in this case automatically

A similar statement of course applies to upper dimension bounds. In particular, has dimension exactly d if and only if

To describe the interaction of passing to hearts with Poincaré structures, we have to introduce a bit terminology: We shall call a functor $F \colon \mathcal A \rightarrow \mathcal B$ additively quadratic if its cross-effect

$$\begin{align*}\mathrm B_F \colon \mathcal A \times \mathcal A \longrightarrow \mathcal B^\natural, \quad (a,b) \longmapsto \mathrm{ker}(F(a \oplus b) \rightarrow F(a) \oplus F(b))\end{align*}$$

preserves direct sums in each variable; here $\mathcal B^\natural $ denotes the idempotent completion of $\mathcal B$ . Let us immediately say, that for two stable $\infty $ -categories this condition is weaker than being quadratic in the sense required in the definition of a Poincaré structure (for example every product preserving functor between stable $\infty $ -categories is additively quadratic, but quadratic if and only if it is exact). The distinction is discussed in detail in [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Section 4.2] and [Reference Barwick, Glasman, Mathew and NikolausBGMN21, Section 2]. One particular upshot is that additively quadratic functors $\mathcal A^{\mathrm {op}} \rightarrow \mathrm {Sp}$ uniquely extend to quadratic functors $\mathrm {Stab}(\mathcal A)^{\mathrm {op}} \rightarrow \mathrm {Sp}$ , see [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, Propostion 4.2.15] (for module categories, but the proof carries over verbatim) or [Reference Barwick, Glasman, Mathew and NikolausBGMN21, Theorem 2.19]. Combined with Sosnilo’s description of exhaustive weight structures in terms of their heart (as discussed in the previous section), we therefore deduce a characterisation of Poincaré categories of dimension $0$ : Let $\mathrm {Cat}^{\mathrm {p}}_{\mathrm {wt}}$ denote the category spanned by these with degree $0$ Poincaré functors as morphisms, and let $\mathrm {Cat}^{\mathrm {aq}}$ denote the category of additive $\infty $ -categories equipped with additively quadratic functors, that is the cartesian unstraightening of the functor

$$\begin{align*}(\mathrm{Cat}^{\mathrm{add}})^{\mathrm{op}} \longrightarrow \mathrm{Cat}_\infty, \quad \mathcal A \longmapsto \mathrm{Fun}^{\mathrm{aq}}(\mathcal A^{\mathrm{op}}, \mathrm{Sp})\end{align*}$$

where the superscript denotes the full subcategory of additively quadratic functors. We obtain:

Proposition 3.2.5. The functor $(-)^{\mathrm {ht}} \colon \mathrm {Cat}^{\mathrm {p}}_{\mathrm {wt}} \rightarrow \mathrm {Cat}^{\mathrm {aq}}$ is an equivalence onto the following subcategory of the target: Its objects are those pairs such that

1. (1) $\mathcal A$ is weakly idempotent complete,

2. (2) the cross-effect of takes values in connective spectra, and

3. (3) naturally in $a,b \in \mathcal A$ for an equivalence ${\mathrm {D}} \colon \mathcal A^{\mathrm {op}} \rightarrow \mathcal A$ .

The functor ${\mathrm {D}}$ is then uniquely determined by , and we denote it by , and the morphisms in the image of $(-)^{\mathrm {ht}}$ are given by those pairs such that the natural map

is an equivalence for every $a \in \mathcal A$ .

The inverse process takes to $\mathrm {Stab}(\mathcal A)$ equipped with the unique weight structure with $\mathrm {Stab}(\mathcal A)^{\mathrm {ht}} \simeq \mathcal A$ and the unique quadratic extension of .

We shall denote the subcategory of $\mathrm {Cat}^{\mathrm {aq}}$ described in the Proposition by $\mathrm {Cat}^{\mathrm {ap}}_{\flat }$ .

Finally, we record two elementary statements for easier reference. Note that for a Poincaré category of dimension at least d, we have , whenever $X \in {\mathcal {C}}_{[-\infty ,b]}$ , but that no statement about the duals of connective objects can be inferred. The dual statement, however, applies to Poincaré categories of dimension bounded above, so if is of exact dimension d then if $X \in {\mathcal {C}}_{[a,b]}$ . The individual statements imply:

Here, and also below, we shall state connectivity estimates for Poincaré objects in , since the objects of the cobordism category are given by .

4.1 Review of algebraic surgery

Just as in geometric topology we will use algebraic surgery to modify objects up to cobordism. Given a Poincaré object , a surgery datum on $(X,q)$ consists of a map $f \colon T \rightarrow X$ in ${\mathcal {C}}$ , together with a null homotopy r of . Associated to such a datum is a cobordism

$$\begin{align*}(X,q) \longleftarrow \chi(f) \longrightarrow X_f,\end{align*}$$

the trace of the surgery, whose right end we call the result of surgery; the underlying span of this cobordism is given as follows: $\chi (f)$ is the fibre of the composite

It receives a canonical map from T, since its precomposition with f is canonically identified with $f^*(q)_\sharp $ , which r provide a null-homotopy of. The object underlying $X_f$ is then the cofibre of this map $T \rightarrow \chi (f)$ .

Let us also recall that surgery is a functorial equivalence in the following sense: A surgery datum on $(X,q)$ amounts to the same thing as a hermitian form in restricting to $(X,q)$ along and we define the category of surgery data in by

If we denote for $\mathcal A \in \mathrm {Cat}_\infty $ by $\operatorname {dec}(\mathcal A)$ the pull-back $\operatorname {Ar}(\mathcal A)\times _{\mathcal A} \mathrm {cr} \mathcal A$ along the source-projection, the results of [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Section 2.3] upgrade the extraction of the trace as above to a commutative diagram

natural in the Poincaré category , whose top horizontal map is an equivalence, the surgery equivalence, and whose bottom horizontal map is the equivalence taking a map $b \colon X \rightarrow Y$ in to the span

$$\begin{align*}X \xleftarrow{\mathrm{id}} X \xrightarrow{b} Y\end{align*}$$

with its canonical structure as a cobordism . The symbol $\operatorname {dec}$ originates from the equivalence $\mathrm N_n\operatorname {dec}(\mathcal A) \simeq \mathrm N_{1+n}(\mathcal A)$ which identifies the Rezk nerve of $\operatorname {dec}\mathcal A$ with the décalage of $\mathrm N(A)$ .

The surgery equivalence takes a morphism in from f to g with underlying square

and b an equivalence, to a commutative square in

The lower cobordism $\chi (a)$ arises (necessarily) by surgery along some surgery datum on $X_g$ ; its underlying map is of the form $\mathrm {cof}(a) \to X_g$ , and since this will suffice for our purposes let us refrain from spelling out the hermitian refinement.

The inverse of $\chi $ takes a cobordism with underlying span $X\leftarrow Z \rightarrow Y$ to the map $\mathrm {fib}(Z\to Y) \to X$ , along which the Poincaré form on X canonically inherits a null-homotopy.

Now let K be a small $\infty $ -category. From the surgery equivalence, we obtain an induced equivalence

where $\mathrm {lpc} K$ , the left pre-cone, is defined by the pushout

in $\mathrm {Cat}_\infty $ , which makes it adjoint to the décalage construction. We are more interested in describing functors out of the actual left cone $^\triangleleft {K}$ , rather than the pre-cone, so we make the following definition:

Definition 4.1.2. The space of K-fold surgery data in a Poincaré category is the space defined by the pull-back

For $K=*$ , in which case we abbreviate the above to , one of course simply obtains the core of , that is, the space of surgery data, and more generally, if $|K|\simeq *$ one simply has

From the considerations above we immediately conclude:

Proposition 4.1.3. The surgery equivalence induces an equivalence

such that the extraction of the underlying Poincaré object on the left corresponds to evaluation at the cone point on the right.

Besides the map

forgetting the multiple surgery datum, there is also a map

extracting the diagram underlying the surgery datum: Per construction we have a map forgetting hermitian structures. It induces a map

which (by definition of ) restricts as desired. Somewhat informally, we will often write this underlying diagram as $f \colon T\to X$ , where X is the value at the cone point, and T is the restriction $K\to {\mathcal {C}}$ . In generalisation of the case $K=*$ the result of surgery in this generality is the diagram , or equivalently an object of .

Lemma 4.1.5. If $K\to L$ is cofinal, then the square

is cartesian.

Proof. The map identifies, via the equivalence of [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺23, 6.3.15 Corollary], with the right fibration

classified by the infinite loop space of the hermitian structure on ${\mathcal {C}}^K = \mathrm {Fun}(K,{\mathcal {C}})$ . From the explicit formula

we conclude that the square

is cartesian if $K\to L$ is cofinal.

By using the description of from 4.1.4 (1), we therefore find given as the limit of

computing vertical limits first. Computing horizontal pullbacks first gives the claim.

We conclude this section by recording some easy general facts about surgery for later reference; most of them are of course well-known at least in the case of module categories. We start with the general connectivity estimate for the trace.

Lemma 4.1.6. Let be a Poincaré category of dimension at least d, $f \colon T\to X$ a surgery datum on and consider the associated cobordism

$$\begin{align*}X \longleftarrow \chi(f) \longrightarrow X_f.\end{align*}$$

If T is k-connective, then $\chi (f) \rightarrow X_f$ is k-connective; and if T is k-coconnective then $\chi (f) \rightarrow X$ is $d-k$ -connective. Furthermore, if $T\in {\mathcal {C}}_{[k,k]}$ and is of exact dimension d, then the fibres of these two maps are concentrated in degrees k and $d-k$ , respectively.

Proof. This is immediate from the fibre sequences

in the definition of $\chi (f)$ and $X_f$ .

To improve connectivity of objects and morphisms, we will use the following two estimates.

In particular, if $T \in {\mathcal {C}}_{[p,p]}$ we find $X_f$ is $p+1$ -connective if f is p-connective, as long as $2p<d$ , that is, as long as T is a surgery datum below the middle dimension (note that in this case $X \in {\mathcal {C}}_{[p,\infty ]}$ is a definite requirement for $X/T \in {\mathcal {C}}_{[p+1,\infty ]}$ ).

Proof. Checking vertical fibres one finds the square

cartesian. Considering horizontal fibres one then finds a fibre sequence

and by assumption the left hand term is in ${\mathcal {C}}_{[d-l,\infty ]}$ , so closure of ${\mathcal {C}}_{[p+1,\infty ]}$ under extensions gives the claim once $d-l \geq p+1$ . The converse is similar.

Lemma 4.1.8. Given a surgery datum in with underlying diagram

such that the total fibre of the left square is m-connective, then the result of surgery has left pointing map $m+1$ -connective.

Proof. We need to show that in the result of surgery

the lower left arrow is $m+1$ -connective. But its fibre is the same as that of $W/T \rightarrow X/S$ , and thus the suspension of the totel fibre of the square in question.

To take care that the connectivity of morphisms is not destroyed by surgery, we use the following estimate.

Lemma 4.1.9. Let be a Poincaré category of dimension at least d and let

be a surgery datum on an object with $W \rightarrow X$ is m-connective. Then if $S \rightarrow T$ is l-coconnective and $S \rightarrow R k$ -connective, also the left pointing map in the result of the surgery is m-connective as long as $m \leq k+1, d-l$ .

Proof. Set $M = R \cup _ST$ for legibility. Then the trace of the surgery yields a commutative diagram

whose bottom row is the result of surgery. Let $F_b$ , $F_m$ and $F_u$ denote the fibres of the bottom, middle and upper left pointing arrows, respectively. Then computing the total fibre of the lower left square first vertically and then horizontally results in a (rotation of the) fibre sequence

$$\begin{align*}F_m \longrightarrow F_b \longrightarrow R/S\end{align*}$$

and doing the same for the upper left square gives

But the source in the second term is equivalent to . Thus if $T/S \in {\mathcal {C}}_{[-\infty ,l]}$ this terms lies in ${\mathcal {C}}_{[d-l,\infty ]}$ and the result follows.

We shall also need the following automatic disjointness, which reflects the geometric fact that any two embedded submanifolds of less than complementary dimension can be isotoped apart:

Proof. We have to show that nullhomotopies of the restrictions to S and T of the Poincaré form extend to a nullhomotopy of the restriction of q in . To see this simply recall that

so it will suffice to show that is connective, but this is guaranteed by the assumption on k and l together with Lemma 3.2.3.

4.2 Forwards and backwards surgery

Recall that the cobordism category for a Poincaré category is the $\infty $ -category associated to the complete Segal space . Adding the space of surgery data to the mix yields another simplicial space , whose associated category we will denote , the cobordism category with surgery data. Since the functor $\mathrm {Surg}\colon \mathrm {Cat}^{\mathrm {p}}\to \mathrm {Cat}_\infty $ preserves limits (each of its constituents $\mathrm {Pn}$ , $\mathrm {He}$ , and $\operatorname {Met}$ does), and therefore $\mathrm {Pn}_{\mathrm {sd}} \colon \mathrm {Cat}^{\mathrm {p}} \rightarrow \mathrm {An}$ also preserves limits, [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Lemmas 2.2.4] implies that is also a complete Segal space, giving us good control over .

There is an evident forgetful functor induced by the natural projection $\mathrm {Pn}_{\mathrm {sd}}\to \mathrm {Pn}$ , which may be rewritten as

$$\begin{align*}\mathrm{Pn}_{\mathrm{sd}} \xrightarrow{\chi} \mathrm{Pn}\mathcal Q_1 \xrightarrow{d_1} \mathrm{Pn}\mathcal Q_0 = \mathrm{Pn}.\end{align*}$$

But similarly the transformation

$$\begin{align*}\mathrm{Pn}_{\mathrm{sd}} \xrightarrow{\chi} \mathrm{Pn}\mathcal Q_1 \xrightarrow{d_0} \mathrm{Pn}\end{align*}$$

induces another functor that extracts the result of surgery.

The key insight allowing for parametrised surgery arguments is that these two functors should be related by a natural transformation, and hence become homotopic upon realisation, once restricted appropriately.

To explain this, we note that by construction the functors are related by a cobordism, namely by the trace of the surgery, that is, . In more categorical language this can be regarded as a functor into the category of vertical morphisms in the double category represented by the double Segal space . Such a cobordism, however, is not the same thing as a natural transformation between its source and target, and generally does not induce a homotopy at the level of classifying spaces.

It turns out, however, that the double category contains the double category of commutative squares in : A square in the double category is by definition a $(1,1)$ -simplex in , that is, a cobordism between cobordisms, which may be written as a diagram

(discarding the hermitian structures from the notation) and we will argue that such a datum can be regarded as a commutative square

in just in case its top right and lower left corner are cartesian. Restricting the domains of $\mathrm {fgt}$ and $\mathrm {srg}$ to the wide subcategory of consisting of surgery data that produce such diagrams one then obtains the desired transformation $\mathrm {fw} \colon \mathrm {fgt} \Rightarrow \mathrm {srg}$ , since functors into the category of vertical morphisms in $\operatorname {Sq}(\mathcal D)$ are the same thing as natural transformations. We shall also need a version of this transformation for multiple surgery, and dual statements obtained from the symmetry of the cobordism category.

To start the analysis recall that the double category of commutative squares $\operatorname {Sq}(\mathcal D)$ in a given category $\mathcal D$ is associated to the two-fold complete Segal space

$$\begin{align*}(i,j) \mapsto {\mathrm{Hom}}_{\mathrm{Cat}_\infty}([i] \times [j], \mathcal D) \simeq {\mathrm{Hom}}_{{\mathrm{s}\mathrm{An}}}(\Delta^i \times \Delta^j, \mathrm N(\mathcal D)),\end{align*}$$

(the two terms are equivalent because the Rezk nerve $\mathrm N$ is fully faithful). Our first order of business will be to verify that the sub-Segal-space of spanned by the squares described above indeed agrees with the nerve of .

From the natural equivalence

we obtain:

Observation 4.2.2. The double category is associated to the two-fold complete Segal space

for every Poincaré category .

This makes it very easy to construct the embedding of into .

Construction 4.2.3. Both $\mathcal Q_{[i] \times [j]}({\mathcal {C}})$ and $\mathcal Q^{(2)}_{i,j}({\mathcal {C}})=\mathcal Q_i\mathcal Q_j({\mathcal {C}})$ are (essentially by definition) full subcategories of

$$\begin{align*}\mathrm{Fun}(\operatorname{TwAr}([i] \times [j]),{\mathcal{C}}) \simeq \mathrm{Fun}(\operatorname{TwAr}([i]) \times \operatorname{TwAr}([j]), {\mathcal{C}})\end{align*}$$

and their hermitian structure are the restricted ones. In the former case all squares associated to maps $[3] \rightarrow [i] \times [j]$ are required to be pullbacks, whereas in the latter case one only requires this for maps $[3] \rightarrow [i] \times [j]$ that are of the form $k=k \rightarrow l=l$ in one of the two factors (since limits in functor categories are computed pointwise). Therefore, we clearly find a fully faithful functor

$$\begin{align*}\mathcal Q_{[i] \times [j]}({\mathcal{C}}) \longrightarrow \mathcal Q^{(2)}_{i,j}({\mathcal{C}})\end{align*}$$

natural in .

One easily decodes that it is an equivalence for $i=0$ or $j=0$ , but not in general as the following example shows:

Observation 4.2.4. The category $Q^{(2)}_{1,1}({\mathcal {C}})$ is the full functor category $\mathrm {Fun}(\operatorname {TwAr}([1]\times [1]), {\mathcal {C}})$ , whereas directly decoding definitions shows that a (solid) diagram

defines an element in $\mathcal Q_{[1] \times [1]}({\mathcal {C}})$ if and only if the lower left and the upper right squares are cartesian. Alternative to a direct check, one can use the decomposition

$$\begin{align*}\mathcal Q_{[1]\times[1]}({\mathcal{C}})\simeq \mathcal Q_2({\mathcal{C}})\times_{\mathcal Q_1({\mathcal{C}})} \mathcal Q_2({\mathcal{C}})\end{align*}$$

arising from $[1]\times [1]=[2]\cup _{[1]} [2]$ , indicated by the dashed line in the diagram, to reduce this to the characterisation of $\mathcal Q_2({\mathcal {C}})$ inside $\mathrm {Fun}(\operatorname {TwAr}[2], {\mathcal {C}})$ . We further note that if the lower left square is cartesian, then so is the upper right square in the diagram obtained by the duality of , and vice versa: To see this, apply the commutativity of

both to and to . It follows $\mathcal Q_{[1]\times [1]}({\mathcal {C}})$ is closed under the duality of , and therefore the inclusion constructed above is Poincaré. Furthermore, it follows that for a Poincaré object of cartesianness of one of the two squares implies cartesianness of the other.

From this example and the Segal conditions on both sides, we see that is a Poincaré subcategory for all values of $i,j$ . Upon applying Poincaré objects we thus find an inclusion of path components

and investing 4.2.2, we obtain the desired faithful inclusion

From $\operatorname {TwAr}({\mathcal {D}}^{\mathrm {op}}) \simeq \operatorname {TwAr}(\mathcal D)$ one furthermore obtains an equivalence of simplicial Poincaré categories, the reflection, where the superscript on the right denotes the order reversal of the simplicial object. Postcomposing the reverse of $\mathrm {fw}$ in, say, the first simplicial direction, with this reflection gives the other embedding

as promised, which is characterised by the upper left and lower right square in 4.2.4 being cartesian.

Let us note immediately that both

exchange the essential images of the (appropriate opposites of the) embeddings $\mathrm {fw}$ and $\mathrm {bw}$ ; this holds essentially by definition for $\mathrm {refl}_1$ and for $\mathrm {refl}_2$ it follows from the characterisation of the image in the observation above.

Now for a category K and a double category $\mathcal D$ let us denote by ${\mathrm {Hom}}_{\mathrm {Cat}_\infty }^1(K,\mathcal D)$ the category with objects functors from K into the first direction of $\mathcal D$ and morphisms given by the second direction; at the level of nerves this corresponds to forming the simplicial space

$$\begin{align*}[n] \longmapsto {\mathrm{Hom}}_{{\mathrm{s}\mathrm{An}}}(\mathrm N K,\mathrm N_{-, n}\mathcal D).\end{align*}$$

Per construction then, the transformation is the same as a functor

Furthermore, there is a tautological equivalence

$$\begin{align*}{\mathrm{Hom}}_{\mathrm{Cat}_\infty}^1(K,\operatorname{Sq}({\mathcal{C}})) \simeq \mathrm{Fun}(K,{\mathcal{C}})\end{align*}$$

for any category $\mathcal D$ .

Definition 4.2.5. We call a surgery datum on a Poincaré cobordism (i.e., a morphism in ) forwards if $\chi $ takes it into the image of the subcategory embedding

The wide subcategory of spanned by these pieces of surgery data we denote , and there is an entirely analogous definition of . We call these the cobordism categories with forwards (or backwards) surgery data.

Per construction $\chi $ restricts to a functor

that adjoins to the desired transformation

$$\begin{align*}\chi_{\mathrm{fw}} \colon \mathrm{fgt} \Longrightarrow \mathrm{srg}\end{align*}$$

of functors ; applying the same analysis to the backwards inclusion we obtain a transformation

$$\begin{align*}\chi_{\mathrm{bw}} \colon\mathrm{srg} \Longrightarrow \mathrm{fgt}\end{align*}$$

of functors .

Forwards surgery data can be explicitly characterised as follows:

Proposition 4.2.6. A surgery datum

in is forwards if and only if the map $S\to R$ is an equivalence and backwards if the map $S\to T$ is an equivalence.

Recall our convention that the diagram in the proposition is to be read as a morphism from f to h in .

Proof. Writing the associated cobordism as

the lower line is obtained from the middle one by taking the quotient of $R\leftarrow S \to T$ , so the left lower square is cartesian if and only if $S\to R$ is an equivalence. The claim in the forwards case then follows from 4.2.4 and the fact that the cobordism is Poincaré. The backwards case follows by reflection.

Remark 4.2.7. This proposition makes our forwards and backwards surgery moves correspond to those of Galatius and Randal-Williams in a very close manner: In their implementation one first replaces $|\mathsf {Cob}_d|$ by what has become known as the space of long manifolds, that is, the space of d-dimensional submanifolds in $\mathbb R \times \mathbb R^{(\infty )}$ , such that the projection to the first coordinate is proper and eventually displays M as a cylinder of some slice in both the positive and negative directions. Roughly speaking, the equivalence from $|\mathsf {Cob}_d|$ takes an object N to $\mathbb R \times N$ (for some embedding $N \rightarrow \mathbb R^{(\infty )}$ of which there is a contractible choice), and a cobordism W represents the path given by sliding the cylindrical extension of a similarly embedded W from positive to negative infinity; such a path indeed starts at $\mathbb R \times \partial _0 W$ and ends at $\mathbb R \times \partial _1 W$ .

In this language Galatius and Randal-Williams perform a surgery move on some geometric surgery datum $\varphi \colon \mathrm S^k \times {\mathrm {D}}^{d-1-k} \rightarrow N$ , with N an object of $\mathsf {Cob}_d$ , by extending it to an embedding $\mathrm {id} \times \varphi \colon \mathbb R_{\geq 0} \times \mathrm S^k \times {\mathrm {D}}^{d-1-k} \rightarrow \mathbb R \times N$ , and then sliding a handle (i.e., the trace of the surgery) in from positive infinity; compare [Reference Galatius and Randal-WilliamsGRW14, Figure 8]. In particular, the slide connects the result of surgery to the original manifold by a canonical path in the space of long manifolds. If more generally N is a vertex of some higher simplex, whose centre in the geometric realisation of $\mathsf {Cob}_d$ gives rise to a long manifold M, say, which contains N as a slice, a similar move requires the extension of $\varphi $ to an embedding $\mathbb R_{\geq 0} \times \mathrm S^k \times {\mathrm {D}}^{d-1-k} \rightarrow M$ compatible with the projections to the first coordinate. This is roughly the data Galatius and Randal-Williams start their surgery moves with. Note that this process performs a surgery parallel to $\varphi $ on all slices of M to the right of N (i.e., at higher values of the first coordinate) as well, and that this seems necessary to produce a canonical path to the original situation.

The proposition above can be read as saying that in order for our transformation $\chi _{\mathrm {fw}}$ to exist, our surgery data needs to satisfy a similar condition: We can perform surgery on some datum $\varphi \colon T \rightarrow C_{i \leq i}$ that is part of a higher simplex of in a forwards manner once we have extended it to a diagram of the form

which clearly has the same behaviour as the kind of data used by Galatius and Randal-Williams. In fact, unwinding definitions one finds that $\chi _{\mathrm {fw}}$ is simply induced by regarding the trace of the surgery datum $\varphi $ as a morphism in , which in turn seems to correspond exactly to the slide performed by Galatius and Randal-Williams.

We thus, in particular, expect our analysis to be portable to the geometric set-up, where it would lead to a variant of parametrised surgery that dispenses with the need for the space of long manifolds.

Let us now upgrade the discussion to multiple surgery. Applying the category of surgery data to each entry in the $\mathcal Q$ -construction yields a two-fold complete Segal space

whose associated double category we shall denote . The surgery equivalence then induces an equivalence of double categories

where $\operatorname {dec}_1$ indicates the décalage construction in the first direction. For ease of notation we shall simply refer to the first direction of as the surgery direction, and to the second one as the cobordism direction. We denote by

the subcategories of spanned in the cobordism direction by the forwards and backwards morphisms, respectively, and all morphisms in the surgery direction. In other words, is the full subspace on those elements such that for each vertex in the first direction and each edge in the second direction, the restricted element of is a forwards surgery datum.

Straight from the definitions we then find the following forwards and backwards surgery equivalences:

Proposition 4.2.8. The equivalence $\chi $ restricts to an equivalence

along the forwards embedding, and similarly to an equivalence

To extract the case of K-fold surgery data let denote the category associated to the complete Segal space ; in categorical language this is described by the cartesian square

Using either description we obtain an equivalence

from 4.1.3. Inside we find subcategories

consisting of those morphisms that lie in and , respectively. We thus find:

Corollary 4.2.9. The surgery equivalence induces equivalences

natural in $K \in \mathrm {Cat}_\infty $ and .

From the case $K=\emptyset $ we learn that the extraction of underlying objects on the left hand side corresponds to the evaluation at the cone point on the right hand side, and for $K=\ast $ we recover the transformation

$$\begin{align*}\chi_{\mathrm{fw}} \colon \mathrm{fgt} \Longrightarrow \mathrm{srg}\end{align*}$$

on constructed above, along with its backwards companion.

Proof. From the forwards surgery equivalence we obtain an equivalence

the backwards case is analogous.

4.3 Reflection of cubes in the cobordism category

In the present section we study the process of reflecting commutative squares in the cobordism category. More specifically, we analyse under what circumstances a commutative square in as in the middle of

induces commutative squares as on the left and the right, and how this can be read off from surgery data defining the original square. To this end view the middle square as an element in . Then the outer two diagrams canonically upgrade to squares in the double category as well and we are asking that they lie in . Whenever this is correct the squares above combine into a larger commutative square

which we call the double of the original square. This doubling construction will be important while doing surgery in the middle dimension; roughly, it will allow us to undo the effect of certain surgeries while keeping other surgeries done. We will also need the variants for higher dimensional and longer cubes later, so treat diagrams of the form generally for natural numbers $a,r>0$ .

To get started recall the two embeddings of Poincaré categories

where the first one is bisimplicial and the second one simplicial in the second and reverse simplicial in the first variable. The reflection equivalence in the first or the second variable exchanges their images.

Definition 4.3.1. We denote by

the Poincaré category given by the intersection of the essential images of $\mathrm {fw}$ and $\mathrm {bw}$ . A diagram in the underlying stable $\infty $ -category $\mathcal Q^\square _{1,1}({\mathcal {C}})$ will be called independent square.

Explicitly, a diagram $X\colon \operatorname {TwAr}([1]\times [1])\to {\mathcal {C}}$ is an independent square if in

all four displayed squares are bicartesian.

Varying m and n we obtain a two-fold complete Segal object of $\mathrm {Cat}^{\mathrm {p}}$ , and applying Poincaré objects we obtain a two-fold complete Segal space which is a bisimplicial subspace of the nerve of and, through the forwards embedding, also of ; as we will have to iterate this construction in a moment, we refrain from translating this definition to the multi-categorical perspective. The commutative squares in which are independent are easily recognised by means of their surgery data as follows:

The proof is essentially by inspection. It is made most transparent by the following observation:

Lemma 4.3.3. If K has a terminal object t then the composite

extracting the underlying surgery data is given by

where $\widetilde F \colon \operatorname {TwAr}(^\triangleleft {K}) \rightarrow {\mathcal {C}}$ is the diagram adjoint to F and $*$ denotes the cone point.

Let us unwind the statement for a moment: Per definition the image of F on the right hand side takes some $k \in K$ to the surgery datum associated to the cobordism and per construction this is given by $\mathrm {fib}(\widetilde F_{* \rightarrow k} \rightarrow \widetilde F_{k \rightarrow k})$ . But the square

in $\operatorname {TwAr}(^\triangleleft {K})$ is cartesian by definition of the $\mathcal Q$ -construction and identifies the fibre above with the fibre in the statement of the lemma. The trouble is now that the displayed square in ${\mathcal {C}}$ is not natural in k in any sense. But for 4.3.2 we shall require a natural identification and the point of the lemma that this can indeed be achieved.

Proof. As explained in Section 4.1 the extraction of surgery data gives a map

for an arbitrary category K. According to the constructions in [Reference Calmès, Dotto, Harpaz, Hebestreit, Land, Moi, Nardin, Nikolaus and SteimleCDH⁺25, Proposition 2.4.8 & Corollary 2.4.9] it is given for $K=[n]$ by

$$\begin{align*}\xrightarrow{\mathrm{fgt}} \mathrm{cr}(\operatorname{Ar}({\mathcal{C}}^{[n]})) = {\mathrm{Hom}}_{\mathrm{Cat}_\infty}([n] \times [1],{\mathcal{C}}) \xrightarrow{\mathrm{ev}_0} {\mathrm{Hom}}_{\mathrm{Cat}_\infty}([n],{\mathcal{C}});\end{align*}$$

the evaluation at $1 \in [1]$ in the last step extracts the constant diagram on the underlying object (i.e., the evaluation at the cone point of $^\triangleleft {K}$ ). Unwinding the definition of $s_n$ the above composite is also given (still for $K=[n]$ ) by the composite

with $\iota \colon \operatorname {TwAr}(K) \times [1] \rightarrow \operatorname {TwAr}(K) \times \operatorname {TwAr}([1]) = \operatorname {TwAr}(K \times [1])$ induced by the identity in the first coordinate and the arrow $(0 \leq 1 ) \rightarrow (1 \leq 1)$ in the second, and $s \colon \operatorname {TwAr}(K) \rightarrow K$ is the source projection. But the above composite constitutes a natural transformation in arbitrary $K\in \mathrm {Cat}_\infty ^{\mathrm {op}}$ : This is evident for all steps except the left Kan extension in the very last one, which is only lax natural via the Beck-Chevalley transformations of the adjunction between $s_!$ and the (evidently natural) $s^*$ ; see [Reference Haugseng, Hebestreit, Linskens and NuitenHHLN23a]. But this Kan extension is an optical illusion: Since $s \colon \operatorname {TwAr}(K) \rightarrow K$ is a cartesian fibration with contractible fibres it is a localisation [Reference Haugseng, Hebestreit, Linskens and NuitenHHLN23b, Lemma 5.5], so that $s^* \colon {\mathrm {Hom}}_{\mathrm {Cat}_\infty }(K,{\mathcal {C}}) \rightarrow {\mathrm {Hom}}_{\mathrm {Cat}_\infty }(\operatorname {TwAr}(K),{\mathcal {C}})$ is an equivalence onto those components ${\mathrm {Hom}}_{\mathrm {Cat}_\infty }^s(\operatorname {TwAr}(K),{\mathcal {C}})$ in the target consisting of all functors $\operatorname {TwAr}(K) \rightarrow {\mathcal {C}}$ inverting all maps in $\operatorname {TwAr}(K)$ whose source component is an equivalence. It follows that on this part the requisite Beck-Chevalley transformations are invertible, and $s_! \colon {\mathrm {Hom}}_{\mathrm {Cat}_\infty }^s(\operatorname {TwAr}(K),{\mathcal {C}}) \rightarrow {\mathrm {Hom}}_{\mathrm {Cat}_\infty }(\operatorname {TwAr}(K),{\mathcal {C}})$ is natural in K. It remains to check that composite of the first six steps takes values in ${\mathrm {Hom}}_{\mathrm {Cat}_\infty }^s(\operatorname {TwAr}(K),{\mathcal {C}})$ . But some and a map $(k \rightarrow l) \rightarrow (k' \rightarrow l')$ in $\operatorname {TwAr}(K)$ are indeed taken to the induced map on horizontal fibres in

and in case the map $k \rightarrow k'$ is an equivalence, this diagram is in the image of under the map

$$\begin{align*}[3] \longrightarrow K \times[1], \quad (k,0) \rightarrow (k,1) \rightarrow (l',1) \rightarrow (l,1).\end{align*}$$

It is therefore cartesian by definition of the $\mathcal Q$ -construction.

The proof of the actual statement is now short: If K has a terminal object t, then $|K| \simeq \ast $ , so $\mathrm {lpc}(K) = {^\triangleleft {K}}$ and the source functor $s \colon \operatorname {TwAr}(K) \rightarrow K$ admits $k \mapsto (k \rightarrow t)$ as a fully faithful adjoint, so that the resulting functor ${\mathrm {Hom}}_{\mathrm {Cat}_\infty }(\operatorname {TwAr}(K),\operatorname {Ar}({\mathcal {C}})) \rightarrow {\mathrm {Hom}}_{\mathrm {Cat}_\infty }(K,\operatorname {Ar}({\mathcal {C}}))$ agrees with $s_!$ and can thus be used to extract the diagram $K \rightarrow {\mathcal {C}}$ in the last step. This results in the functor from the statement being induced by the functoriality of

$$\begin{align*}k \longmapsto \mathrm{fib}(\widetilde F((k,0) \rightarrow (t,1)) \rightarrow \widetilde F((k,1) \rightarrow (t,1))\end{align*}$$

and for functors pulled back along $K \times [1] \rightarrow {^\triangleleft {K}}$ this is equivalent to the formula we want to show.

Proof of Proposition 4.3.2

Let

be the underlying diagram in ${\mathcal {C}}$ of a $[1] \times [1]$ -fold surgery datum on a Poincaré object C of ${\mathcal {C}}$ . The result of surgery is then given by:

with maps induced by the given diagram in ${\mathcal {C}}$ . The claim now essentially follows by inspection, once we have also identified the homotopies in the squares above with the ones induced by the original square. In fact, by the final comment in 4.2.4 it suffices to do this in the lower right square, where it is a consequence of 4.3.3: As part of the diagram on $^\triangleleft {(}[1] \times [1])$ this square extends to

and unwinding definitions 4.3.3 precisely identifies the diagram formed by the fibres of the slanted maps as the original surgery datum, which implies the claim.

We will also need a version of these considerations for higher cubes. We therefore put:

Since the forwards embedding and the reflections are Poincaré functors, this subcategory is closed under the duality of and thus Poincaré itself. Similarly, one checks that it really defines an r-fold complete Segal object of $\mathrm {Cat}^{\mathrm {p}}$ . The r-fold category associated to its Poincaré objects is thus a subcategory of , the same objects and morphisms (in all directions), but severely restricted higher squares. It can be regarded as the largest subcategory of the r-fold category of commutative r-cubes in that is closed under all r reflections.

Our next goal is to characterise the pieces of $[1]^r$ -fold surgery data that give rise to objects in in generalisation of 4.3.2 above. It will be convenient to do this directly for $[a]^r$ -fold cubes (and of course the case of an arbitrary product is in the end only notationally more involved).

We start out by characterising the image of . To this end abbreviate $\operatorname {TwAr}[1]$ by H (for horn). Then the underlying category of the target consists of diagrams $H^r \rightarrow {\mathcal {C}}$ . Now, the category $H^r$ is glued from $2^r$ many r-dimensional cubes $[1]^r$ , that we index as follows: For some $v \in [1]^r$ set $H^r_v$ to be the full subcategories spanned by all $u \leq w \in \mathrm {H}^r$ with $u \leq v \leq w$ . We will refer to the to r-cubes $H_0^r$ and $H_1^r$ as the initial and terminal cube of $H^r$ , respectively.

In particular, we see that a Poincaré object in lies in if its terminal cube is strongly cocartesian.

Proof. We begin with the first statement. So let X satisfy the condition on its subcubes. Given then a chain $u \leq v \leq w$ in $[1]^r$ we have to check that the square

goes to a cartesian square. If one of the inequalities is actually an equality this is trivial and otherwise, this square can be pasted from $2$ -dimensional faces of $H_v^r$ (which are assumed cocartesian): Namely, choose a sequence

$$\begin{align*}u = u_0 < u_1 < \dots < u_n < v =v_0 < v_1 \dots < v_m = w\end{align*}$$

such that the difference in each step is only a single coordinate and consider the squares

We will prove the converse by induction on r. The cases $r=0,1$ are vacuous and $r=2$ is precisely 4.2.4. Now per construction

so we learn from the induction hypothesis that for any of the r-cube $H^r_v$ in $H^r$ , that is neither the initial nor terminal one, the three r-cubes

$$\begin{align*}H^r_v \times \{0 \leq 0\}, \quad H^r_v \times \{0 \leq 1 \}, \quad \text{and} \quad H^r_v \times \{1 \leq 1\}\end{align*}$$

of $H^{r+1}$ go to strongly cocartesian diagrams under X. Since an $r+1$ -cube is strongly cocartesian if two of its opposing faces are, this implies that all but potentially the four cubes

$$\begin{align*}H_0^{r+1} = H^r_0 \times \{(0 \leq 1) \rightarrow (0 \leq 0)\}, \quad H_0^r \times \{(0 \leq 1) \rightarrow (1 \leq 1)\}, \quad H_1^r \times \{(0 \leq 1) \rightarrow (0\leq 0)\},\end{align*}$$

$$\begin{align*}\text{and} \quad H_1^{r+1} = H_1^r \times \{(0 \leq 1) \rightarrow (1 \leq 1)\}\end{align*}$$

are cocartesian. But splitting off the first instead of the last coordinate of $[1]^{r+1}$ covers the middle two.

To obtain the second statement simply observe that any reflection exchanges the initial and terminal cubes for one of the others.

From here it is easy to characterise the associated surgery data. To formulate the result we need:

Proof of Lemma 4.3.6

For a map $g\colon S\to [a]$ , we consider the poset P of all $h\colon S\to [a]$ with $h\leq g$ (pointwise) and h supported in at most a single element, and the subposet $P'\subset P$ where $h\in P$ is discarded if there exists some s such that $0<h(s)<g(s)$ . Then we have maps

$$\begin{align*}\operatorname{colim}_{h\in P'} X(h)\longrightarrow \operatorname{colim}_{h\in P} X(h) \longrightarrow X(g)\end{align*}$$

where the first is an equivalence since $P'\subset P$ is cofinal. By the pointwise formula for Kan extensions, condition (1) is equivalent to the right map being an equivalence, for all g, while condition (2) is equivalent to the composite map being an equivalence.

This proves the equivalence of (1) and (2). Clearly (3) implies (2), and the converse is proven by induction in the number n of elements s such that $f_s(0)>0$ : The induction beginning $n=0$ is precisely condition (2), and the inductive step is proven using the pasting law for strongly cocartesian cubes (which easily follows from the pasting law for cocartesian cubes, using [Reference LurieLur17, 6.1.1.15]).

We now proceed to construct the doubles of cubes in . This is facilitated by the following reflection lemma. For its formulation recall that a complete Segal object uniquely extends to a limit preserving functor on $\mathrm {Cat}_\infty ^{\mathrm {op}}$ . In particular, we obtain

$$\begin{align*}\mathcal Q^{\square} \colon \mathrm{Cat}_\infty^r \longrightarrow \mathrm{Cat}^{\mathrm{p}}\end{align*}$$

as such an extension for each r. Since each is a full subcategory, the same is true for , and the inclusion functor is Poincaré, $\mathrm {Cat}^{\mathrm {p}}\subset \mathrm {Cat^h}$ being closed under limits.

Lemma 4.3.9. For any Poincaré category , reflection in the second summand and the identity in the first summand of each entry combine to a canonical equivalence of Poincaré categories for any $a\in \mathbb {N}$

Both sides in fact exhibit semi-simplicial functoriality in the number of their indices: The values at $[r]$ are given by the expressions with $|[r]| = r+1$ indices. Informally, the face map $d_i$ is given by evaluation at $0$ in $[2a]$ or the first summand of $[a]\cup _{\{a\}} [a]$ in the coordinate labelled by $i\in [r]$ , and more formally, arises on the right from the inclusion

which one checks closed under the face map of the co-semi-simplicial structure of $r \mapsto \prod _{[r]} [2a]$ given by the prescription above ( in fact canonically extends to a simplicial object by doubling entries, but this does not persist to the left hand side). The left hand side in 4.3.9 is entirely analogous. The proof of 4.3.9 will also show: