Dependencies

Suppose there is a 2-simplex in a 2-truncated quasi-category with spine formed by the paths \(f\) and \(g\) and diagonal \(h\). Then if \(f \sim f'\), \(g \sim g'\), and \(h \sim h'\), there is a 2-simplex with spine formed by \(f'\) and \(g'\) and diagonal \(h'\).

Equivalences of \(\infty \)-categories \(A' \rightsquigarrow A\) and \(B \rightsquigarrow B'\) induce an equivalence of functor spaces \({\mathord {\mathsf{Fun}}}(A,B) \rightsquigarrow {\mathord {\mathsf{Fun}}}(A',B')\).

For any cartesian closed category \({\mathord {\mathcal{V}}}\) with coproducts, the underlying category construction and free category construction define adjoint 2-functors

The category of simplicial sets is cocomplete.

The category of simplicial sets is complete.

A parallel pair of 1-simplices \(f,g\) in a simplicial set \(X\) are homotopic if there exists a 2-simplex whose boundary takes either of the following forms ³

or if \(f\) and \(g\) are in the same equivalence class generated by this relation.

A parallel pair of 1-simplices \(f,g\) in a 2-truncated simplicial set \(X\) are left homotopic if there exists a 2-simplex whose boundary takes the form below-left and right homotopic if there exists a 2-simplex whose boundary takes the form below-right:

An equivalence in a 2-category is given by

• a pair of objects \(A\) and \(B\);

• a pair of 1-cells \(f \colon A \to B\) and \(g \colon B \to A\); and

• a pair of invertible 2-cells

  

When \(A\) and \(B\) are equivalent, we write \(A \simeq B\) and refer to the 1-cells \(f\) and \(g\) as equivalences, denoted by “\(\rightsquigarrow \).”

A 2-truncated simplicial set \(A\) is a 2-truncated quasi-category if it admits the following three operations:

• (2,1)-filling: any path \(f_\bullet \) of length 2 in \(A\) may be filled to a \(2\)-simplex whose spine equals the given path.

• (3,1)-filling: given any path \(f_\bullet \) of length 3 in \(A\), 2-simplices \(\sigma _3\) and \(\sigma _0\) filling the restricted paths \(f_{012}\) and \(f_{123}\) respectively, and 2-simplex \(\sigma _2\) filling the path formed by \(f_{01}\) and the diagonal of \(\sigma _0\), there is a 2-simplex \(\sigma _1\) filling the path formed by the diagonal of \(\sigma _3\) and \(f_{23}\) and whose diagonal is the diagonal of \(\sigma _2\).

• (3,2)-filling: given any path \(f_\bullet \) of length 3 in \(A\), 2-simplices \(\sigma _3\) and \(\sigma _0\) filling the restricted paths \(f_{012}\) and \(f_{123}\) respectively, and 2-simplex \(\sigma _1\) filling the path formed by the diagonal of \(\sigma _3\) and \(f_{23}\), there is a 2-simplex \(\sigma _2\) filling the path formed by \(f_{01}\) and the diagonal of \(\sigma _0\) and whose diagonal is the diagonal of \(\sigma _1\).

If \(A\) is a 2-truncated quasi-category then its homotopy category \({\mathord {\mathsf{h}}}{A}\) has

• the set of 0-simplices \(A_0\) as its objects

• the set of homotopy classes of 1-simplices \(A_1\) as its arrows

• the identity arrow at \(a \in A_0\) represented by the degenerate 1-simplex \(a \cdot \sigma ^0 \in A_1\)

• a composition relation \(h = g \circ f\) in \({\mathord {\mathsf{h}}}{A}\) between the homotopy classes of arrows represented by any given 1-simplices \(f,g,h \in A_1\) if and only if there exists a 2-simplex with boundary

  

An isofibration between categories is a functor \(f \colon A \twoheadrightarrow B\) satisfying the displayed right lifting property for the inclusion of both endpoints of the free-living isomorphism:

An \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) is cartesian closed if the product bifunctor \(- \times -\colon {\mathord {\mathcal{K}}}\times {\mathord {\mathcal{K}}}\to {\mathord {\mathcal{K}}}\) extends to a simplicially enriched two-variable adjunction

in which the right adjoints \((-)^A \colon {\mathord {\mathcal{K}}}\to {\mathord {\mathcal{K}}}\) preserve isofibrations for all \(A \in {\mathord {\mathcal{K}}}\).

The homotopy coherent isomorphism \(I\), is the nerve of the free-living isomorphism.

An \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) is a category that is enriched over quasi-categories, ² meaning in particular that

• its morphisms \(f \colon A \to B\) define the vertices of a quasi-category denoted \({\mathord {\mathsf{Fun}}}(A,B)\) and referred to as a functor space,

that is also equipped with a specified collection of maps that we call isofibrations and denote by “\(\twoheadrightarrow \)” satisfying the following two axioms:

1. (completeness) The quasi-categorically enriched category \({\mathord {\mathcal{K}}}\) possesses a terminal object, small products, pullbacks of isofibrations, limits of countable towers of isofibrations, and cotensors with simplicial sets, each of these limit notions satisfying a universal property that is enriched over simplicial sets. ³

2. (isofibrations) The isofibrations contain all isomorphisms and any map whose codomain is the terminal object; are closed under composition, product, pullback, forming inverse limits of towers, and Leibniz cotensors with monomorphisms of simplicial sets; and have the property that if \(f \colon A \twoheadrightarrow B\) is an isofibration and \(X\) is any object then \({\mathord {\mathsf{Fun}}}(X,A) \twoheadrightarrow {\mathord {\mathsf{Fun}}}(X,B)\) is an isofibration of quasi-categories.

The elementary degeneracy operators are the maps

whose images double up on the element \(i \in [n]\).

In an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\), a morphism \(f\colon A \to B\) is an equivalence just when the induced map \(f_* \colon {\mathord {\mathsf{Fun}}}(X,A) \rightsquigarrow {\mathord {\mathsf{Fun}}}(X,B)\) on functor spaces is an equivalence of quasi-categories for all \(X \in {\mathord {\mathcal{K}}}\).

The elementary face operators are the maps

whose images omit the element \(i \in [n]\).

Let \({\mathord {\mathcal{K}}}\) be an \(\infty \)-cosmos. Its homotopy 2-category is the 2-category \(\mathfrak {h}{\mathord {\mathcal{K}}}\) whose

• objects are the objects \(A, B\) of \({\mathord {\mathcal{K}}}\), i.e., the \(\infty \)-categories;

• 1-cells \(f \colon A \to B\) are the 0-arrows in the functor space \({\mathord {\mathsf{Fun}}}(A,B)\), i.e., the \(\infty \)-functors; and

• 2-cells

  

  are homotopy classes of 1-simplices in \({\mathord {\mathsf{Fun}}}(A,B)\), which we call \(\infty \)-natural transformations.

Put another way \(\mathfrak {h}{\mathord {\mathcal{K}}}\) is the 2-category with the same objects as \({\mathord {\mathcal{K}}}\) and with hom-categories defined by

that is, \({\mathord {\mathsf{hFun}}}(A,B)\) is the homotopy category of the quasi-category \({\mathord {\mathsf{Fun}}}(A,B)\).

The free category on this reflexive directed graph has \(X_0\) as its object set, degenerate 1-simplices serving as identity morphisms, and nonidentity morphisms defined to be finite directed paths of nondegenerate 1-simplices. The homotopy category \({\mathord {\mathsf{h}}}{X}\) of \(X\) is the quotient of the free category on its underlying reflexive directed graph by the congruence ⁴ generated by imposing a composition relation \(h = g \circ f\) witnessed by 2-simplices

The homotopy category of an \(\infty \)-category \(A\) in an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) is defined to be the homotopy category of its underlying quasi-category, that is:

A 1-simplex in a quasi-category is an isomorphism ¹ just when it represents an isomorphism in the homotopy category. By Lemma 1.3.8 this means that \(f \colon a \to b\) is an isomorphism if and only if there exists a 1-simplex \(f^{-1} \colon b \to a\) together with a pair of 2-simplices

A Kan complex is a simplicial set admitting extensions as in 1.2.13 along all horn inclusions \(n \geq 1, 0 \leq k \leq n\).

A (lax) monoidal functor between cartesian closed categories \({\mathord {\mathcal{V}}}\) and \({\mathord {\mathcal{W}}}\) is a functor \(T \colon {\mathord {\mathcal{V}}}\to {\mathord {\mathcal{W}}}\) equipped with natural transformations

so that the evident associativity and unit diagrams commute.

For each \(n \geq 0\), an \(n\)-simplex in \({\mathord {\mathcal{A}}}(x,y)\) is referred to as an \(n\)-arrow from \(x\) to \(y\).

The category \({\mathord {\mathcal{Cat}}}\) of 1-categories embeds fully faithfully into the category of simplicial sets via the nerve functor. An \(n\)-simplex in the nerve of a 1-category \(C\) is a sequence of \(n\) composable arrows in \(C\), or equally a functor \([n]\to C\) from the ordinal category \([n]\) with objects \(0,\ldots , n\) and a unique arrow \(i \to j\) just when \(i \leq j\).

The map \([n] \mapsto [n]\) defines a fully faithful embedding \(\Delta \hookrightarrow {\mathord {\mathcal{Cat}}}\). From this point of view, the nerve functor can be described as a “restricted Yoneda embedding” which carries a category \(C\) to the restriction of the representable functor \(\hom (-,C)\) to the image of this inclusion.

By 1-truncating, any simplicial set \(X\) has an underlying reflexive quiver or reflexive directed graph with the 0-simplices of \(X\) defining the objects and the 1-simplices defining the arrows:

By convention, the source of an arrow \(f \in X_1\) is its 0th face \(f \cdot \delta ^1\) (the face opposite 1) while the target is its 1st face \(f \cdot \delta ^0\) (the face opposite 0).

A map \(f \colon A \to B\) between quasi-categories is an equivalence if it extends to the data of a “homotopy equivalence” with the free-living isomorphism \(I\) serving as the interval: that is, if there exist maps \(g \colon B \to A\),

We write “\(\rightsquigarrow \)” to decorate equivalences and \(A \simeq B\) to indicate the presence of an equivalence \(A \rightsquigarrow B\).

A simplicial map \(f \colon A \to B\) between quasi-categories is an isofibration if it lifts against the inner horn inclusions, as displayed below-left, and also against the inclusion of either vertex into the free-living isomorphism \(I\).

To notationally distinguish the isofibrations, we depict them as arrows “\(\twoheadrightarrow \)” with two heads.

A map \(f \colon X \to Y\) between simplicial sets is a trivial fibration if it admits lifts against the boundary inclusions for all simplices

  for n ≥0

We write “\(\twoheadrightarrow \)” to decorate trivial fibrations. ³

We write \(\partial \Delta [n] \subset \Delta [n]\) for the boundary sphere of the \(n\)-simplex. The sphere \(\partial \Delta [n]\) is the simplicial subset generated by the codimension-one faces of the \(n\)-simplex.

Let \(\Delta \) denote the simplex category of finite nonempty ordinals \([n] = \{ 0 {\lt}1 {\lt}\cdots {\lt} n\} \) and order-preserving maps.

The data of a simplicial category is a simplicially enriched category with a set of objects and a simplicial set \({\mathord {\mathcal{A}}}(x,y)\) of morphisms between each ordered pair of objects. Each endo-hom space contains a distinguished 0-simplex \(\textup{id}_x \in {\mathord {\mathcal{A}}}(x,y)_0\), and composition is required to define a simplicial map

The composition is required to be associative and unital, in a sense expressed by the commutative diagrams of simplicial sets

Consider a limit cone \((\lim _{j \in J}A_j \to A_j)_{j \in J}\) in the underlying category \({\mathord {\mathcal{A}}}_0\) of a simplicially-enriched category \({\mathord {\mathcal{A}}}\). By applying the covariant representable functor \({\mathord {\mathcal{A}}}(X,-) \colon {\mathord {\mathcal{A}}}_0 \to {\mathord {\mathcal{sSet}}}\) to a limit cone \((\lim _{j \in J}A_j \to A_j)_{j \in J}\) in \({\mathord {\mathcal{A}}}_0\), we obtain a natural comparison map

We say that \(\lim _{j\in J}A_j\) defines a simplicially enriched limit if and only if 1.4.2 is an isomorphism of simplicial sets for all \(X \in {\mathord {\mathcal{A}}}\).

Let \({\mathord {\mathcal{A}}}\) be a simplicial category. The cotensor of an object \(A \in {\mathord {\mathcal{A}}}\) by a simplicial set \(U\) is given by the data of an object \(A^U \in {\mathord {\mathcal{A}}}\) together with a cone \(U \to {\mathord {\mathcal{A}}}(A^U,A)\) so that the induced map defines an isomorphism of simplicial sets:

A simplicial category \({\mathord {\mathcal{A}}}\) has cotensors when all cotensors exist.

We write \(\Lambda ^k[n] \subset \Delta [n]\) for the \(k\)-horn in the \(n\)-simplex. The horn \(\Lambda ^k[n]\) is the further simplicial subset of \(\partial \Delta [n]\) that omits the face opposite the vertex \(k\), but it is defined as a subset of \(\Delta [n]\).

The category of simplicial sets is the category \({\mathord {\mathcal{sSet}}}:={\mathord {\mathcal{Set}}}^{\Delta ^{\mathord {\textup{op}}}}\) of presheaves on the simplex category.

We write \(\Delta [n]\) for the standard \(n\)-simplex the simplicial set represented by \([n] \in \Delta \).

In an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\), a morphism \(f\colon A \to B\) is a trivial fibration just when \(f\) is both an isofibration and an equivalence.

The category \({\mathord {\mathcal{A}}}_0\) of 0-arrows is the underlying category of the simplicial category \({\mathord {\mathcal{A}}}\), which forgets the higher dimensional simplicial structure.

The underlying category of a 2-category is defined by simply forgetting its 2-cells. Note that an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) and its homotopy 2-category \(\mathfrak {h}{\mathord {\mathcal{K}}}\) share the same underlying category \({\mathord {\mathcal{K}}}_0\) of \(\infty \)-categories and \(\infty \)-functors in \({\mathord {\mathcal{K}}}\).

The underlying category \({\mathord {\mathcal{K}}}_0\) of an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) is the category whose objects are the \(\infty \)-categories in \({\mathord {\mathcal{K}}}\) and whose morphisms are the \(0\)-arrows, i.e., the vertices in the functor spaces.

A quasi-category is a simplicial set \(A\) in which any inner horn can be extended to a simplex, solving the displayed lifting problem:

  for n ≥2, 0 < k < n.

The 2-truncation of a quasi-category is a 2-truncated quasi-category.

If \(A\) is a 2-truncated quasi-category then:

1. The left and right homotopy relations are reflexive.

2. The left and right homotopy relations are symmetric.

3. The left and right homotopy relations are transitive.

4. The left homotopy relation coincides with the right homotopy relation.

\(\quad \)

1. If \(\sigma \) and \(\tau \) are 2-simplices in a 2-truncated quasi-category filling the same path, their diagonal edges are homotopic.

2. If \(h\) is the diagonal edge of a 2-simplex filling the path formed by \(f\) and \(g\) and \(g\) is homotopic to \(g'\), then \(h\) is the diagonal edge of a 2-simplex filling the path formed by \(f\) and \(g'\).

3. If \(h\) is the diagonal edge of a 2-simplex filling the path formed by \(f\) and \(g\) and \(f\) is homotopic to \(f'\), then \(h\) is the diagonal edge of a 2-simplex filling the path formed by \(f'\) and \(g\).

Any functor \(f\colon A \to B\) in an \(\infty \)-cosmos may be factored as an equivalence followed by an isofibration, where this equivalence is constructed as a section of a trivial fibration.

Moreover, \(f\) is an equivalence if and only if the isofibration \(p\) is a trivial fibration.

For any simplicial category \({\mathord {\mathcal{A}}}\) and \(n \geq 0\), the \(n\)-arrows assemble into the arrows of an ordinary category \({\mathord {\mathcal{A}}}_n\) with the same set of objects as \({\mathord {\mathcal{A}}}\).

When a simplicial category has cotensors, cotensors are associative: given \(A \in {\mathord {\mathcal{A}}}\) and simplicial sets \(U\) and \(V\) there are canonical isomorphisms

Assuming such objects exist, the simplicial cotensor defines a bifunctor

in a unique way making the isomorphism 1.4.1 natural in \(U\) and \(A\) as well.

Any adjunction comprised of finite-product-preserving functors between cartesian closed categories

defines a \({\mathord {\mathcal{V}}}\)-enriched adjunction between the \({\mathord {\mathcal{V}}}\)-categories \({\mathord {\mathcal{V}}}\) and \(U_*{\mathord {\mathcal{W}}}\); i.e., there exists a \({\mathord {\mathcal{V}}}\)-natural isomorphism \(U{\mathord {\mathcal{W}}}(Fv,w) \cong {\mathord {\mathcal{V}}}(v,Uw)\).

A map \(f \colon A \to B\) between \(\infty \)-categories in an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) is an equivalence if and only if it extends to the data of a “homotopy equivalence” with the free-living isomorphism \(I\) serving as the interval: that is, if there exist maps \(g \colon B \to A\)

in the \(\infty \)-cosmos.

The equivalences in an \(\infty \)-cosmos are closed under retracts and satisfy the 2-of-3 property: given a composable pair of functors and their composite, if any two of these are equivalences so is the third.

\(\quad \)

1. The functor \({\mathord {\mathsf{h}}}\colon {\mathord {\mathcal{sSet}}}\to {\mathord {\mathcal{Cat}}}\) preserves finite products.

2. The functor \({\mathord {\mathsf{h}}}\colon {\mathord {\mathcal{QCat}}}\to {\mathord {\mathcal{Cat}}}\) preserves small products.

If \(A\) is a quasi-category then its homotopy category \({\mathord {\mathsf{h}}}{A}\) is isomorphic to the homotopy category of its underlying 2-truncated quasi-category, as just described.

\(\quad \)

1. Every 2-cell

   

   in the homotopy 2-category of an \(\infty \)-cosmos is represented by a map of quasi-categories as below-left or equivalently by a functor as below-right

   

   and two such maps represent the same 2-cell if and only if they are homotopic as 1-simplices in \({\mathord {\mathsf{Fun}}}(A,B)\).

2. Every invertible 2-cell

   

   in the homotopy 2-category of an \(\infty \)-cosmos is represented by a map of quasi-categories as below-left or equivalently by a functor as below-right

   

   and two such maps represent the same invertible 2-cell if and only if their common restrictions along \([1]\hookrightarrow I\) are homotopic as 1-simplices in \({\mathord {\mathsf{Fun}}}(A,B)\).

The change-of-base 2-functor induced by a finite-product-preserving functor \(T \colon {\mathord {\mathcal{V}}}\to {\mathord {\mathcal{W}}}\) between cartesian closed categories preserves underlying categories, if and only if, for each \(v \in {\mathord {\mathcal{V}}}\) the composite function on hom-sets

is a bijection.

Parallel 1-simplices \(f\) and \(g\) in a quasi-category are homotopic if and only if there exists a 2-simplex of any or equivalently all of the forms displayed in Definition 1.2.19.

If \(f \colon A \to B\) is an equivalence of quasi-categories, then the functor \({\mathord {\mathsf{h}}}{f} \colon {\mathord {\mathsf{h}}}{A} \to {\mathord {\mathsf{h}}}{B}\) is an equivalence of categories, where the data displayed above defines an equivalence inverse \({\mathord {\mathsf{h}}}{g} \colon {\mathord {\mathsf{h}}}{B} \to {\mathord {\mathsf{h}}}{A}\) and natural isomorphisms encoded by the composite ² functors

Consider a finite-product-preserving adjunction between cartesian closed categories:

Then change of base along the right adjoint respects the underlying categories:

Each \(n\)-simplex \(x \in X_n\) corresponds to a map of simplicial sets \(x \colon \Delta [n] \to X\). Accordingly, we write \(x \cdot \delta ^i\) for the \(i\)th face of the \(n\)-simplex, an \((n-1)\)-simplex classified by the composite map

The trivial fibrations in an \(\infty \)-cosmos define a subcategory containing the isomorphisms and are stable under product, pullback, and forming inverse limits of towers.

In an \(\infty \)-cosmos, the Leibniz cotensors of any trivial fibration with a monomorphism of simplicial sets is a trivial fibration as is the Leibniz cotensor of an isofibration with a map in the class cellularly generated by the inner horn inclusions and the map \([0]\hookrightarrow I\).

If \(E \twoheadrightarrow B\) is a trivial fibration in an \(\infty \)-cosmos, then is \({\mathord {\mathsf{Fun}}}(X,E)\twoheadrightarrow {\mathord {\mathsf{Fun}}}(X,B)\) is a trivial fibration of quasi-categories.

The underlying category of the homotopy 2-category of an \(\infty \)-cosmos is isomorphic to the underlying category of the \(\infty \)-cosmos.

There is an equivalence between categories enriched in categories and strict bicategories. In particular, each can be converted into the other.

The category \({\mathord {\mathcal{Cat}}}\) of 1-categories defines an \(\infty \)-cosmos whose isofibrations are the isofibrations. The equivalences are the equivalences of categories and the trivial fibrations are surjective equivalences: equivalences of categories that are also surjective on objects.

A finite-product-preserving functor \(T \colon {\mathord {\mathcal{V}}}\to {\mathord {\mathcal{W}}}\) between cartesian closed categories induces a change-of-base 2-functor

Any adjunction between cartesian closed categories whose left adjoint preserves finite products induces a change-of-base 2-adjunction

Given an adjunction between cartesian closed categories

whose left adjoint preserves finite products then if \({\mathord {\mathcal{C}}}\) is co/tensored as a \({\mathord {\mathcal{W}}}\)-category, \(U_*{\mathord {\mathcal{C}}}\) is co/tensored as \({\mathord {\mathcal{V}}}\)-category with the co/tensor of \(c \in {\mathord {\mathcal{C}}}\) by \(v \in {\mathord {\mathcal{V}}}\) defined by

An arrow \(f\) in a quasi-category \(A\) is an isomorphism if and only if it extends to a homotopy coherent isomorphism

The functor that carries a category to its underlying reflexive quiver has a left adjoint, defining the free category on a reflexive quiver:

The nerve embedding admits a left adjoint, namely the functor which sends a simplicial set to its homotopy category:

\(\quad \)

1. The homotopy 2-category of any \(\infty \)-cosmos has 2-categorical products.

2. The homotopy 2-category of a cartesian closed \(\infty \)-cosmos is cartesian closed as a 2-category.

An isofibration \(p \colon E \twoheadrightarrow B\) in an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) also defines an isofibration in the homotopy 2-category \(\mathfrak {h}{\mathord {\mathcal{K}}}\): given any invertible 2-cell as displayed below-left abutting to \(B\) with a specified lift of one of its boundary 1-cells through \(p\), there exists an invertible 2-cell abutting to \(E\) with this boundary 1-cell as displayed below-right that whiskers with \(p\) to the original 2-cell.

The category \({\mathord {\mathcal{Kan}}}\) of Kan complexes defines an \(\infty \)-cosmos whose isofibrations are the Kan fibrations: maps that lift against all horn inclusions \(\Lambda ^k[n] \hookrightarrow \Delta [n]\) for \(n \geq 1\) and \(0 \leq k \leq n\).

The nerve of a category \(C\) is 2-coskeletal as a simplicial set, meaning that every sphere \(\partial \Delta [n] \to C\) with \(n \geq 3\) is filled uniquely by an \(n\)-simplex in \(C\), or equivalently that the nerve is canonically isomorphic to the right Kan extension of its restriction to 2-truncated simplicial sets. ²

The nerve functor is fully faithful.

Nerves of categories are quasi-categories.

The homotopy category of the nerve of a 1-category is isomorphic to the original category, as the 2-simplices in the nerve witness all of the composition relations satisfied by the arrows in the underlying reflexive directed graph.

The full subcategory \({\mathord {\mathcal{QCat}}}\subset {\mathord {\mathcal{sSet}}}\) of quasi-categories defines an \(\infty \)-cosmos in which the isofibrations, equivalences, and trivial fibrations coincide with the classes already bearing these names.

Every morphism in \(\Delta \) factors uniquely as an epimorphism followed by a monomorphism; these epimorphisms, the degeneracy operators, decompose as composites of elementary degeneracy operators, while the monomorphisms, the face operators, decompose as composites of elementary face operators.

For any \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) and any \(\infty \)-category \(B \in {\mathord {\mathcal{K}}}\) there is an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}_{/B}\) of isofibrations over \(B\) whose

1. objects are isofibrations \(p \colon E \twoheadrightarrow B\) with codomain \(B\)

2. functor spaces, say from \(p \colon E \twoheadrightarrow B\) to \(q \colon F \twoheadrightarrow B\), are defined by pullback

   

   and abbreviated to \({\mathord {\mathsf{Fun}}}_B(E,F)\) when the specified isofibrations are clear from context

3. isofibrations are commutative triangles of isofibrations over \(B\)

   

4. terminal object is \(\textup{id}\colon B \twoheadrightarrow B\) and products are defined by the pullback along the diagonal

   

5. pullbacks and limits of towers of isofibrations are created by the forgetful functor \({\mathord {\mathcal{K}}}_{/B} \to {\mathord {\mathcal{K}}}\)

6. simplicial cotensor of \(p \colon E \twoheadrightarrow B\) with \(U \in {\mathord {\mathcal{sSet}}}\) is constructed by the pullback

   

7. and in which a map over \(B\)

   

   is an equivalence in the \(\infty \)-cosmos \({\mathord {\mathcal{K}}}_{/B}\) if and only if \(f\) is an equivalence in \({\mathord {\mathcal{K}}}\).

In any \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\), the following are equivalent and characterize what it means for a functor \(f \colon A \to B\) between \(\infty \)-categories to define an equivalence.

1. For all \(X \in {\mathord {\mathcal{K}}}\), the post-composition map \(f_* \colon {\mathord {\mathsf{Fun}}}(X,A) \rightsquigarrow {\mathord {\mathsf{Fun}}}(X,B)\) defines an equivalence of quasi-categories.

2. There exists a functor \(g \colon B \to A\) and natural isomorphisms \(\alpha \colon \textup{id}_A \cong gf\) and \(\beta \colon fg \cong \textup{id}_B\) in the homotopy 2-category.

3. There exists a functor \(g \colon B \to A\) and maps

   

   in the \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\).