Dependencies

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.

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 “\(\rightsquigarrow \).”

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 ⁵ just when it represents an isomorphism in the homotopy category. By Lemma 1.2.28 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. ⁷

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 1.3.2 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.

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 1.3.1 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 \({\mathord {\mathsf{h}}}{A}\) 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

  

\(\quad \)

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 \({\mathord {\mathsf{Fun}}}(A,B)\).

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 ⁶ functors

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: functors satisfying the displayed right lifting property:

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 \({\mathord {\mathcal{K}}}\).