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while the scope of the language developed herein is to allow the description of weak equivalences [...]

of colimits in

satisfy

Let $\Delta$ denote the simplex category whose objects are non-zero finite ordinals and whose morphisms are order-preserving maps.

1.2.1.23. Cones. Let $D$ be a small category and $\mathcal{C}$ be a category. Denote by $\Delta_D$ the functor $\mathcal{C} \to \mathcal{C}^D$ that maps every object $X$ of $\mathcal{C}$ to its associated constant functor $D \to \mathcal{C}$ mapping any object and arrow in $D$ to the object $X$ and identity on $X$, respectively. We shall speak of a cone in $\mathcal{C}$ over $D$ to refer to a natural transformation of the form $\Delta_D(X) \Rightarrow F$ where $X$ is an object in $\mathcal{C}$, which will be referred to as the peak, and $F$ is a functor from $D$ to $\mathcal{C}$.

1.2.1.24. Cocones. Let $D$ be a small category, $\mathcal{C}$ be a category and denote by $\Delta_D$ the functor $\mathcal{C} \to \mathcal{C}^D$ defined in section 1.2.1.23. We shall speak of a cocone in $\mathcal{C}$ over $D$ to refer to a natural transformation of the form $F \Rightarrow \Delta_D(X)$ where $X$ is an object in $\mathcal{C}$, which will be referred to as the peak, and $F$ is a functor from $D$ to $\mathcal{C}$.

cone / cocone

A cone / cocone such as $\alpha$ will later be said to be universal over its diagram $(D,F)$.

A colimit sketch is a small category $\mathtt{S}$ equipped with a subset $Q$ of its universal cocones (see section 1.2.1.27) such that, for every object $x$ in $\mathtt{S}$, there exists a unique cocone in $Q$ defined over a diagram of the form $(\mathbf{1},x)$. The cocones in $Q$ as well as their associated diagrams and colimits (i.e. their peaks) will be said to be chosen. A model for a colimit sketch $\mathtt{S}$ in a category $\mathcal{C}$ is a functor $\mathtt{S} \to \mathcal{C}$ that sends the chosen cocones of $\mathtt{S}$ to universal cocones in $\mathcal{C}$.

A limit sketch is a small category $\mathtt{S}$ equipped with a subset $Q$ of its universal cones (see section 1.2.1.26) such that, for every object $x$ in $\mathtt{S}$, there exists a unique cone in $Q$ defined over a diagram of the form $(\mathbf{1},x)$. The cones in $Q$ as well as their associated diagrams and limits (i.e. their peaks) will be said to be chosen. A model for a limit sketch $\mathtt{S}$ in a category $\mathcal{C}$ is a functor $\mathtt{S} \to \mathcal{C}$ that sends the chosen cones of $\mathtt{S}$ to universal cones in $\mathcal{C}$.

$\varkappa \circ \varkappa_*$

where $\Delta$ is the category of non-zero finite ordinals and order-preserving maps between them.

For every non-zero ordinal $n \in \omega$,

First, consider the canonical arrow given below, on the left, from the initial object to the terminal object in $\mathbf{sSet}$.

by replacing the canonical arrow $\emptyset \to \mathbf{1}$ with

For every non-zero $n \in \omega$,

$\mathbb{D}_1$ / $\beta_{*}$ / $\delta_2^{*}$ / $\beta$ / $\delta_1$

For every non-zero $n \in \omega$,

By definition

that results from the leftmost factorisation of (5.42), which is generated with respect to the set of arrows $J_{\mathrm{cof}}$, is [...]

for every small category $\mathtt{T}$, functor $\upvarphi$$:\mathtt{T} \to J_{\mathrm{cof}}$ that is compatible with the numbered category $(\mathcal{E}^D, \kappa)$ and pushout arrow $p:X \to Y$ of the arrow $\mathrm{col}_{\mathtt{T}}$$\upvarphi$ (see the diagram below, on the left), the image of p via the functor $\nabla_d:\mathcal{E}^D \to \mathcal{E}$ is a surtraction (resp. intraction) in $\hat{\mathcal{E}}_d$ for every object $d$ in $D$.

reducing the length of the different inductive definitions involved as [...]

convergent conjugation

the definition of $\omega$-globular and $\omega$-spinal sketches

for the generating of

by equipping coheroids with sets of canonical vertebrae and show that these give rise to vertebral categories when the coheroids in question are endowed with certain parallel arrows. We shall come across two intermediate structures, namely:
$\to$ Reflexive spinal coheroids, which will come along with reflexive vertebrae;
$\to$ Magmoidal spinal coheroids, which will come along with framing of vertebrae;

We introduce the kappa and tau constructions $\kappa_{1,k}^m$, $\kappa_{2,k}^m$ and $\tau_{1,k}^m(\beta)$, $\tau_{2,k}^m(\beta)$ linking globular sums of some dimension to others of lower dimension.

Specifically, using the analogy between the notion of cell in an $\infty$-groupoid and the notion of topological discs, the kappa and tau constructions will link a gluing of discs of the form given below, on the left, to the gluing of lower dimensional discs given by their respective borders, as shown on the right.

The kappa constructions $\kappa_{1,k}^m$, $\kappa_{2,k}^m$ are to handle gluings taking the form of whiskerings whose whiskers should be thought of as 'reversible' cells and whose middle parts should specifically be viewed as 'non-reversible' cells. This non-reversibility will concretely be charactirised by the use of seeds where we would most often use stems. For their part, tau constructions $\tau_{1,k}^m(\beta)$, $\tau_{2,k}^m(\beta)$ are to handle whiskerings whose cells should all be viewed as potentially 'reversible' cells.

$\to$ $(k,0)$-normal spinal coheroids come along with a morphism $\pi_k^0$ linking a 'non-reversible' cell of dimension $k$ to a gluing of a 'non-reversible' cell of dimension $k$ along two reversible ones of the same dimension and thus emulates a ternary composition of discs;

$\to$ $(k,1)$-normal spinal coheroids come along with a morphism $\pi_k^1$ linking a 'non-reversible' cell of dimension $k+1$ to a gluing of a 'non-reversible' cell of dimension $k+1$ along two reversible ones of dimension $k$ and thus emulates a ternary composition of discs. This notion will depend on the construction $\pi_k^0$, which will take care to compose the borders at dimension $k$.

Broadly, a $(k,m-k)$-transitive spinal coheroid is equipped with morphisms $\upsilon_k^{m-k}(\beta)$ encoding the compositions of 'reversible' cells of dimension $m$ along pairs of 'reversible' cells of dimension $k$.

The functoriality of these compositions will be ensured via the notion of closedness, which will require the compositions to be compatible with the spheres induced by the borders of the discs. Precisely, this requires the existence of gluings of spheres along any pair of 'reversible' cells.

Closedness will come along with canonical morphisms $d_{1,k}^m$, $d_{2,k}^m$, $\kappa_{k}^m$ and $\tau_{k}^m(\beta)$ that factorises the kappa and tau constructions in canonical ways as follows.

These morphisms will help define the functoriality of our framings, which will specifically follow from the commutative diagrams constructed in section 6.3.3.2, called closedness and normality, and section 6.3.3.3, called closedness and transitivity. Because the inductive developments given in these sections are quite cumbersome, I would now like to specify these commutative diagrams as well as give a short description of their roles (see below) -- the reader is therefore excepted to come back here if they lose their way while reading these sections. Now, to resume our discussion, this so-called functoriality will follow from the canonical properties of the morphisms $d_{1,k}^m$, $d_{2,k}^m$, $\kappa_{k}^m$ and $\tau_{k}^m(\beta)$ as follows:
$\triangleright$ diagrams (6.30); (6.34) and (6.39) of section 6.3.3.2 will be used to ensure the functoriality of the whiskerings of non-reversible cells along pairs of reversible cells relative to the whiskerings of the spheres, induced by the borders of the previous non-reversible cells, along the same pairs of reversible cells;
$\triangleright$ diagrams (6.32) and (6.38) of section 6.3.3.2, or, in fact, their lower parts, will be used to ensure the functoriality of the whiskerings of the spheres along pairs of reversible cells relative to the whiskerings of the non-reversible borders of these spheres along the same pairs of reversible cells;
$\triangleright$ diagrams (6.40) and (6.41) of section 6.3.3.3 will be used to ensure the functoriality of the whiskerings of reversible cells relative to the whiskerings of the spheres, induced by the borders of the middle cells, along the same pair of reversible cells;
$\triangleright$ all the other diagrams may be viewed as intermediate steps for the purpose of defining the functoriality of our framings.

We then finish the section by summarising the aformentioned constructions in the form of propositions expliciting the desired framings of spines and their functoriality.

These reflections give a 'reversible' structure to the stems of our vertebrae and thereby allow us to define conjugations of spines.

The goal of this section is to use the kappa and tau constructions -- which may be seen as source and target maps between different types of gluing of discs -- to define dimensionally coherent composition of cells.
On the one hand, normal spinal coheroids will permit the composition of three cells in the form of a whiskering (or cylinder of discs) where the middle cell (or base of the cylinder) should be regarded as non-reversible and where the whiskers (are sides of the cylinder) should be regarded as reversible. The term normal refers to the fact that the composition of a non-reversible cell along two others is what one would like to think as an elementary operation (in contrast to the next type of operation).
On the other hand, transitive spinal coheroids will permit the composition of three cells in the form of a whiskering (or cylinder of discs) where all the cells are considered reversible. The term transitive now refers to the fact that the composition of three reversible cells is reminiscent of a transitive property if one sees $\infty$-groupoids as generalisations of the notion of equivalence relation.

under certain pushouts

Then, by copying our previous reasoning by which every category of Grothendieck's $\infty$-groupoids has been equipped with a spinal category structure, one may show that the free cocompletion $C^{_{\triangledown}} = \mathbf{Psh}(C)$ has a spinal category structure too (see Remark 6.16). Finally, we could try to conclude by using Remark 4.96, which requires us to show that the unit of the adjunction \begin{align} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!(6.68) \quad \quad \quad \quad \quad & \int^{g \in C} \nabla_g(\_) \otimes i(g) \vdash \mathbf{Top}(i(\_),\_)\\ \end{align} (defined thereof) is a componentwise weak equivalence in $\mathbf{Mod}(C^{\mathrm{op}})$. This would imply that the functor $\mathbf{Top}(i(\_),\_):\mathbf{Top} \to \mathbf{Mod}(C^{\mathrm{op}})$ is a covertebral equivalence, which by Proposition 1.48 and Proposition 4.95, would prove the Homotopy Hypothesis. But this argument happens to fail in some cases and therefore needs to be refined. Indeed, it simply suffices to notice that the argument does not work for strict groupoids. For instance, the group with two elements $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ may be seen as an obvious one-object strict $\infty$-groupoid, let us call it $B(\mathbb{Z}/2\mathbb{Z})$, and hence a Grothendieck $\infty$-groupoid by the result of [5]. But the image of $B(\mathbb{Z}/2\mathbb{Z})$ via the left adjoint functor of adjunction (6.68) is a terminal topological space since the relation $1+1=0$ forces any path in $B(\mathbb{Z}/2\mathbb{Z})$ to contract to the topological realisation of the point 0. \[ \int^{g \in C} B(\mathbb{Z}/2\mathbb{Z})(g) \otimes i(g) = \{0\} \] We thus conclude that the unit of adjunction (6.68) at the Grothendieck $\infty$-groupoid $B(\mathbb{Z}/2\mathbb{Z})$ is the canonical map $!:B(\mathbb{Z}/2\mathbb{Z}) \to \mathbf{1}$, which is far from being a weak equivalence in $\mathbf{Mod}(C^{\mathrm{op}})$. Instead, we should rather seek to prove the following property: For every object $X:C^{\mathrm{op}} \to \mathbf{Set}$ in $\mathbf{Mod}(C^{\mathrm{op}})$ for which there exists a topological space $Y$ and a morphisms $f:X(\_) \Rightarrow \mathbf{Top}(i(\_),Y)$ in $\mathbf{Mod}(C^{\mathrm{op}})$, the unit of (6.68) at the object $X$ is a weak equivalence in $\mathbf{Mod}(C^{\mathrm{op}})$. At least, this property does not hold for the Grothendieck $\infty$-groupoid $B(\mathbb{Z}/2\mathbb{Z})$, since it is too strict for being mapped to an image of $\mathbf{Top}(i(\_),\_)$, and it does not contradict Quillen's criteria for Quillen equivalences (see Proposition 1.48).