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• where the object $\mathbb{S}^{n-1}/\partial$ is the quotient of the $(n-1)$-sphere by itself;
• where the object $\mathbb{D}^{n}/\partial$ is the quotient of the $n$-disc by its boundary;
• where the object $\mathbb{S}^{n}/p$ is the quotient of the $n$-sphere by its equator;
• where the object $\mathbb{D}^{n+1}/p$ is the quotient of the $(n+1)$-disc by its equator;
• where the object $\mathbb{D}^{n+1}/h$ is the quotient of the $(n+1)$-disc by one of its hemispheres;
• where the object $\mathbb{D}^{n+1}/\partial$ is the quotient of the $(n+1)$-disc by its boundary;
• where the maps between the different objects are the obvious inclusions.

\[ \xymatrix{ *+!R(.7){\mathbb{S}^{n-1}/\partial}\ar[r]^-{\gamma_n}\ar[d]_-{\gamma_n}\ar@{}[rd]|>>>{\huge{\lrcorner}} &*+!L(.7){\mathbb{D}^n/\partial}\ar[d]^{\delta_1^n}&\\ *+!R(.7){\mathbb{D}^n/\partial}\ar[r]_-{\delta_2^n}&*+!L(.7){\mathbb{S}^{n}/p}\ar[r]_-{\beta_{n}}&*+!L(1){\mathbb{D}^{n+1}/p} } \] \[ \xymatrix{ *+!R(.7){\mathbb{D}^n/\partial}\ar[r]^-{\gamma_n'}\ar[d]_-{\gamma_n'}\ar@{}[rd]|>>>{\huge{\lrcorner}} &*+!L(.7){\mathbb{D}^{n+1}/h}\ar[d]^{\delta_1^n}&\\ *+!R(.7){\mathbb{D}^{n+1}/h}\ar[r]_-{\delta_2^n}&*+!L(.7){\mathbb{D}^{n+1}/\partial}\ar@{=}[r]&*+!L(1){\mathbb{D}^{n+1}/\partial} } \]

[...] the homotopy contractions, denoted $\alpha$, define retractions with the trivial stems [...]

Remark 5.104. Because, in the present case, $J_{\mathrm{cof}}$ is a discrete small category, it is possible to simplify the definition of well-disposedness for surtractions and intractions, most specifically, the context of item 2) regarding pushouts of diagrams in $J_{\mathrm{cof}}$ (see the beginning of the section). Precisely, one may restrict one's class of small categories $\mathtt{T}$ to the singleton $\{\mathbf{1}\}$ only, that is to say that the arrows $p:X \to Y$ are now pushouts of trivial stems $\beta \circ \delta_1:\mathbb{D}_1 \to \mathbb{D}'$ in $J_{\mathrm{cof}}$. This is because the pushout of a coproduct of arrows, say of the form \[ \xymatrix{ \coprod \mathbf{A}(i) \ar[d]_{\coprod\phi(i)}\ar[r]^{\coprod u_k}\ar@{}[rd]|>>>{\huge{\lrcorner}}&X\ar[d]^{i_{|\mathtt{T}|}}\\ \coprod \mathbf{B}(i)\ar[r] &X_{|\mathtt{T}|}, } \] can be seen as a transfinite composition of composable arrows $i_1$, $i_2$, $\dots$, $i_{n+1}\dots$ over the cardinal of $\mathtt{T}$, where each $i_k$ is a pushout of the $k$-th component of $\phi:\mathbf{A} \Rightarrow \mathbf{B}$ as pictured below. \[ \xymatrix{ \mathbf{A}(0) \ar[d]_{\phi(0)}\ar[r]^{u_0}\ar@{}[rd]|>>>{\huge{\lrcorner}}&X\ar[d]^{i_1}\\ \mathbf{B}(0)\ar[r]_{}&X_1 } \xymatrix{ \mathbf{A}(1) \ar[d]_{\phi(1)}\ar[r]^{i_1u_1}\ar@{}[rd]|>>>{\huge{\lrcorner}}&X_1\ar[d]^{i_2}\\ \mathbf{B}(1)\ar[r]_{}&X_2 } \dots \xymatrix{ \mathbf{A}(n) \ar[d]_{\phi(n)}\ar[r]^{i_n\dots i_1u_n}\ar@{}[rd]|>>>{\huge{\lrcorner}}&X_n\ar[d]^{i_{n+1}}\\ \mathbf{B}(n)\ar[r]_{}&X_{n+1} } \dots \] An obvious extension of the transfinite induction of the proof of Theorem 5.102 to such transfinite compositions allows us to only check the well-disposedness condition of Theorem 5.102 for $\mathtt{T} = \mathbf{1}$.

the mappings

The indexing will then be as given above.

Below is given some conventional notation for spinal objects.

satisfying the inequalities $-1 \leq k < m$

We now set similar notations for globular objects. An $\omega$-globular object [...]

by the inclusions of the $n$-disc $\mathbb{D}_n$ into the two hemispheres of the $(n+1)$-discs $\mathbb{D}_{n+1}$.

For every $k \in \omega$, the class $\Omega_k$ is therefore the singleton containing the arrow $\gamma_{k+1}:\mathbb{S}_k \to \mathbb{D}_{k+1}$.

$f,g:\mathbb{D}_{k} \to \mathbb{B}$ in $\mathcal{A}$, [...]

$\mathbb{S}_{k}$

if $\mathcal{A}$ is the set of all pairs

the set $A_n$ is a subset of $\mathcal{A}$;

There exist at least two canonical choices

Let $(\mathbb{S},\mathbb{D},\gamma,\delta) \cdot \Omega:\mathtt{Spine} \to (\mathcal{C},\mathcal{A})$ be a spinal coheroid and $k$ be some integer in $\omega$. This section introduces a notion of spinal coheroid that admits a symmetry between its `discs'. Recall that, any arrow $\beta:\mathbb{S}_k \to \mathbb{A}$ in $\Omega_k$ is associated with a vertebra $p_k \cdot \beta$ in $\nu_k$, which gives the following commutative square.

stemming from the pair of alliances of nodes of vertebrae $(\mathrm{id},\varkappa_q,\xi)$$:\nu_q^{\mathrm{rv}} \leadsto \nu_q$ and $(\mathrm{id},\varkappa_q,\xi)$$:\nu_q^{\mathrm{rv}} \leadsto \nu_q$ given in (6.42).

are conjugable (see section 3.3.5.1) [...]

It follows that the alliance $(\mathrm{id},\varkappa_k,\xi(\beta_{\diamond})):v_{\diamond}^{\mathrm{rv}} \leadsto (v_{\flat}^{\mathrm{rv}})^{\mathrm{rv}}$ is conjugable with any identity alliance of the form $p_k \leadsto p_k$. Similarly, the alliance $(\mathrm{id},\varkappa_k,\xi(\beta_{\bullet})):v_{\bullet} \leadsto v_{\dagger}^{\mathrm{rv}}$ is conjugable with any identity alliance of the form $p_k \leadsto p_k$.

We are now going to define the notion of $(k,n)$-coherence, inductively on $n$. This type of coherence says that conjugating a spine (at some level $k$) along a pair of vertebrae gives a spine that is `$k$-homotopic' to the initial spine by providing a pair of mates between the former and the latter (at level $k$) -- see diagram (6.1), page 274 for more intuition . Note that using the terms `a spine' is a simplification of what will happen below since only the (inifite) canonical spine $P=(p_k)_k$ will be considered. We also include intermediate cases to prepare the successive inductive steps. Here is a quick description of these sections.

• $\triangleright$ Prep. $\texttt{I}$: shows that the conjugation of a `sphere' along a pair of vertebrae is compatible with the conjugation of a `discs' above that sphere for the same pair of vertebrae;
• $\triangleright$ Prep. $\texttt{II}$: shows the conjugation of the `hemispheres' of a sphere along a pair of vertebrae is compatible with the conjugation of that sphere for the same pair;
• $\triangleright$ Reminder: points out that the morphisms constructed in Prep. $\texttt{I}$ and Prep. $\texttt{II}$ can be composed and gives a detail of its components
• $\triangleright$ Mates for $\texttt{I}$: shows the mates associated with the conjugation of a 'sphere' is compatible with the mates associated with the conjugation of a 'discs' above that sphere;
• $\triangleright$ Mates for $\texttt{II}$: shows the mates associated with the conjugation of the `hemispheres' of a sphere is compatible with the mates associated with the conjugation of that sphere;

defines a conjugation of vertebrae (see section 3.3.5.2) along the following three pairs.

[...] the previous conjugation gives rise, via the functor $\mathcal{S}_{\mathrm{cor}}:\mathbf{Conj}(\mathcal{C}) \to \mathbf{Scov}(\mathcal{C})$, to a strong correspondence (see section 3.3.4.2) between two copies of the vertebra $p_k \cdot \gamma_{k+1}$. In the present case, this correspondence may be written as a commutative square [...] where $\zeta^0_{k,k}:\mathbb{G}_k^k(\phi(\beta_{\flat}),\gamma_{k+1},\phi(\beta_{\dagger})) \to \mathbb{G}_k^k(\beta_{\diamond},\gamma_{k+1},\beta_{\bullet})$ is produced by the functor $\mathcal{S}_{\mathrm{cor}}:\mathbf{Conj}(\mathcal{C}) \to \mathbf{Scov}(\mathcal{C})$ as the universal morphism making the following diagrams commute.

(see section 3.3.5.4)

$q \leq r < m$

it follows from Proposition 3.76 (and the relation between $\Gamma_k^m$ and $\Gamma_{k+1}^m$; see the end of section 6.2.1.3) that the morphism of strong correspondences

may be seen as a morphism of correspondences as follows.

we use the fact that the spinal coheroid is symmetric (see section 6.4.1.1) [...]

Now, to finish this subsection, we are going to detail the construction of a similar morphism, but between strong correspondences. First, recall that, by Proposition 6.38 [...]. This framing involves the following two framings.

see section 3.3.4.5

Recall that the conventions on correspondences (see the end of section 3.3.4.6) [...]

Now, to finish this subsection, we are going to detail the construction of a similar morphism, but between strong correspondences. Recall that the framing $(\Sigma_{\sigma_m}^k,v_{\diamond}^{0},v_{\bullet}^{0}) \triangleright \Sigma_{\sigma_m}^k$ given by Proposition 6.38 involves the following two framings.

It is also possible to work out the two-out-of-six for these weak equivalences from the technics available in [4]. However the proof would use the language of groups and groupoids, mainly due to the use of the operators $\pi_n$. This means that the homotopical logic would be hidden in the underlying quotients of the operators $\pi_n$ while the point of the proof given herein was to make this homotopical logic fully explicit.