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2  Higher categories and universes

Definition 2.1 (Universe and decoded objects) Work in an external cartesian closed \((\infty,\infty)\)-category \(\mathcal K\) with pullbacks and terminal object \(*\). Fix a monoidal closed \((\infty,\infty)\)-category \(\mathcal U\) internal to \(\mathcal K\), and use its canonical self-enrichment. Fix a universe fibration \[ p_{\mathcal U}\colon\widetilde{\mathcal U}\longrightarrow\mathcal U. \] A morphism \(q\colon E\to B\) in \(\mathcal K\) is \(\mathcal U\)-small if there is an arrow \(\chi_q\colon B\to\mathcal U\) and a cartesian square

The universe fibration classifies these morphisms: base change induces an equivalence of higher categories \[ [B,\mathcal U]_{\mathcal K} \simeq \mathcal K^{\mathcal U\text{-small}}_{/B}, \] where the right-hand side is the replete full higher subcategory of \(\mathcal K_{/B}\) on the \(\mathcal U\)-small morphisms.

For a point \(h_A\colon *\to\mathcal U\), define its decoded object by the cartesian square

Assume that these pullbacks extend to a decoding functor \[ \operatorname{Dec}_{\mathcal U}\colon\mathcal U\longrightarrow\mathcal K, \qquad \operatorname{Dec}_{\mathcal U}(h_A)=A, \] which preserves the base-change squares used below. An object of the internal category \(\mathcal U\) is specified by a point \(b\colon *\to\mathcal U\); below the same symbol \(b\) denotes that internal object, while \(\operatorname{Dec}_{\mathcal U}(b)\) denotes its external decoding in \(\mathcal K\). A point of \(A\) is a \(1\)-cell \(a\colon *\to A\) in \(\mathcal K\).

For points \(x,y\colon *\to\mathcal U\), the self-enriched hom functor supplies a classifying point \[ h_{[x,y]_{\mathcal U}}:=[-,-]_{\mathcal U}(x,y)\colon *\longrightarrow\mathcal U. \] Its external decoding, denoted \([x,y]_{\mathcal U}\), is defined by the cartesian square

The size convention, the universe fibration, and the decoding functor are interpreted in the fixed external category \(\mathcal K\).

Definition 2.2 (Families and classifiers) For points \(b,c\colon *\to\mathcal U\), a family over \(b\) is a \(1\)-cell \[ G\colon c\longrightarrow b \] in \(\mathcal U\). Equivalently, it is a point \(g\colon *\to[c,b]_{\mathcal U}\). Writing \(B:=\operatorname{Dec}_{\mathcal U}(b)\) and \(C:=\operatorname{Dec}_{\mathcal U}(c)\), its decoded arrow is \[ G^{\sharp}:=\operatorname{Dec}_{\mathcal U}(G)\colon C\longrightarrow B. \] Let \(\mathcal M\) be a replete class of \(1\)-cells in \(\mathcal U\) that is stable under base change. An \(\mathcal M\)-family over \(b\) is a family \(G\colon c\to b\) that belongs to \(\mathcal M\). Write \(\mathcal M(b)\) for the replete full higher subcategory of \(\mathcal U_{/b}\) on the \(\mathcal M\)-families.

A classifier for \(\mathcal M\) is a \(1\)-cell \[ p_{\mathcal M}\colon e_{\mathcal M}\longrightarrow b_{\mathcal M} \] for which base change induces an equivalence \[ [b,b_{\mathcal M}]_{\mathcal U} \simeq \mathcal M(b) \] naturally in \(b\). Explicitly, an \(\mathcal M\)-family \(G\colon c\to b\) corresponds to a classifying \(1\)-cell \(F\colon b\to b_{\mathcal M}\) and a cartesian square in \(\mathcal U\)

The representing equivalence includes arrows and all higher cells between families, not only the existence of \(F\).

Definition 2.3 (The category of higher categories) Let \(\mathcal M_{\mathrm{cocart}}\) be the replete base-change-stable class of \(1\)-cells \(G\) in \(\mathcal U\) for which \(\operatorname{Dec}_{\mathcal U}(G)\) is a \(\mathcal U\)-small cocartesian fibration in \(\mathcal K\). Assume that it has a classifier \[ p_{\mathcal M_{\mathrm{cocart}}}\colon e_{\mathcal M_{\mathrm{cocart}}} \longrightarrow b_{\mathcal M_{\mathrm{cocart}}}. \] Use the classifier objects as the internal codes \[ h_{\mathbf{Cat}_{\infty,\infty}} :=b_{\mathcal M_{\mathrm{cocart}}}, \qquad h_{\widetilde{\mathbf{Cat}}_{\infty,\infty}} :=e_{\mathcal M_{\mathrm{cocart}}}. \] Define their external decodings by \[ \mathbf{Cat}_{\infty,\infty} :=\operatorname{Dec}_{\mathcal U}(b_{\mathcal M_{\mathrm{cocart}}}), \qquad \widetilde{\mathbf{Cat}}_{\infty,\infty} :=\operatorname{Dec}_{\mathcal U}(e_{\mathcal M_{\mathrm{cocart}}}). \] These decodings are exhibited by the cartesian squares

and

Decode the universal classifier to obtain \[ p_{\mathcal V} := \operatorname{Dec}_{\mathcal U}(p_{\mathcal M_{\mathrm{cocart}}}) \colon \widetilde{\mathbf{Cat}}_{\infty,\infty} \longrightarrow \mathbf{Cat}_{\infty,\infty}. \] The representing equivalence determines \(h_{\mathbf{Cat}_{\infty,\infty}}\), and hence \(\mathbf{Cat}_{\infty,\infty}\), up to equivalence as the classifier of small cocartesian families.

A higher category is a point \[ h_C\colon *\longrightarrow\mathbf{Cat}_{\infty,\infty}. \] Its decoded higher category is defined by the cartesian square

An object of \(C\) is a point \(x\colon *\to C\). When \(C=\mathbf{Cat}_{\infty,\infty}\), such a point is itself the classifying point of a higher category, which is decoded by the same pullback. More generally, a functor \(R\colon C\to\mathbf{Cat}_{\infty,\infty}\) sends a point \(x\colon *\to C\) to the classifying point \(R\circ x\) of a higher category.

Definition 2.4 (Categorical families and categories of elements) Let \(G\colon c\to b\) be a family in \(\mathcal U\), and write \[ C:=\operatorname{Dec}_{\mathcal U}(c), \qquad B:=\operatorname{Dec}_{\mathcal U}(b), \qquad G^{\sharp}:=\operatorname{Dec}_{\mathcal U}(G)\colon C\to B. \] It is an \(\mathcal M_{\mathrm{cocart}}\)-family precisely when it corresponds to a classifying \(1\)-cell \[ F\colon b\longrightarrow b_{\mathcal M_{\mathrm{cocart}}}. \] Write \[ F^{\sharp}:=\operatorname{Dec}_{\mathcal U}(F) \colon B\longrightarrow\mathbf{Cat}_{\infty,\infty}. \] Define its category of elements by the cartesian square

Decoding the classifier square identifies \(G^{\sharp}\) with \(q_{F^{\sharp}}\) over \(B\), and hence identifies \[ C\simeq_B\int_BF^{\sharp}. \] For a point \(s\colon *\to B\), the fiber over \(s\) is \(\int_*(F^{\sharp}\circ s)\). A point of the family over \(s\) is a point \[ x\colon *\longrightarrow\int_*(F^{\sharp}\circ s). \] For \(B=*\) and \(F^{\sharp}=h_C\), this recovers the points \(x\colon *\to C\) of Definition 2.3 and (nLab authors 2026).

Definition 2.5 (Local hom-objects) For every decoded higher category \(C\) and points \(x,y\colon *\to C\), fix a classifying point \[ h_{[x,y]_C}\colon * \longrightarrow \mathbf{Cat}_{\infty,\infty} \] and write \[ [x,y]_C:=\int_*h_{[x,y]_C}. \] These hom-objects form a bifunctor \[ [-,-]_C\colon C^{\mathrm{op}}\times C \longrightarrow \mathbf{Cat}_{\infty,\infty}. \] Its value at \((x,y)\) is the classifying point \(h_{[x,y]_C}\). They have identity points \[ \operatorname{id}_x\colon *\longrightarrow[x,x]_C \] and composition functors \[ [y,z]_C\times[x,y]_C \longrightarrow [x,z]_C. \] The associativity and unit cells and their higher coherences are part of the \((\infty,\infty)\)-categorical structure.

For higher categories classified by \(h_C,h_D\colon *\to\mathbf{Cat}_{\infty,\infty}\), let \[ h_{[C,D]_{\mathbf{Cat}_{\infty,\infty}}} := [-,-]_{\mathbf{Cat}_{\infty,\infty}}\circ(h_C,h_D) \colon *\longrightarrow\mathbf{Cat}_{\infty,\infty} \] be the classifying point supplied by the local hom bifunctor, and write \[ [C,D]_{\mathbf{Cat}_{\infty,\infty}} := \int_*h_{[C,D]_{\mathbf{Cat}_{\infty,\infty}}} \] for the decoded functor higher category. Its points are functors \(C\to D\). For higher categories \(A,C,D\), closedness gives \[ [A\times C,D]_{\mathbf{Cat}_{\infty,\infty}} \simeq [A,[C,D]_{\mathbf{Cat}_{\infty,\infty}}]_{\mathbf{Cat}_{\infty,\infty}}. \]

Definition 2.6 (Higher cells) A \(0\)-cell of \(C\) is a point \(x\colon *\to C\). For \(0\)-cells \(x,y\), a \(1\)-cell is a point \[ f\colon *\longrightarrow[x,y]_C. \] For parallel \(1\)-cells \(f,g\), a \(2\)-cell is a point \[ \alpha\colon * \longrightarrow [f,g]_{[x,y]_C}. \] For parallel \(2\)-cells \(\alpha,\beta\), a \(3\)-cell is a point \[ \Gamma\colon * \longrightarrow [\alpha,\beta]_{[f,g]_{[x,y]_C}}. \] Iterating the same point and local hom-object constructions defines all higher cells.

Proposition 2.7 (Cells of an iterated hom-object) Write \([x,y]^{(1)}_C:=[x,y]_C\) and, for a choice of parallel \(k\)-cells \(u,v\) of \(C\), write \([u,v]^{(k+1)}_C\) for the local hom-object formed at the \(k\)-th stage of Definition 2.6. By construction the \(n\)-cells of \([u,v]^{(k)}_C\) are the \((n+k)\)-cells of \(C\) that are parallel over the chosen \(u,v\); in particular its \(0\)-cells are the \(k\)-cells of \(C\) between them. Passing from \(C\) to a local hom-object therefore lowers the dimension of every cell by one, and its adjoint direction is \(B_{01}\dashv\Omega_{01}\) (Definition 2.14).

Definition 2.8 (Sources and targets) A \(1\)-cell \(f\colon *\to[x,y]_C\) has source \(x\) and target \(y\); an \((n+1)\)-cell, being a \(1\)-cell of an iterated hom-object \([u,v]^{(n)}_C\), has source \(u\) and target \(v\). Source and target are defined for cells of positive dimension only, and \(0\)-cells have neither. For every cell of dimension at least \(2\), \[ s(s(\xi))=s(t(\xi)), \qquad t(s(\xi))=t(t(\xi)), \] since \(s(\xi)\) and \(t(\xi)\) are parallel cells of the same iterated hom-object.

Definition 2.9 (Equivalences) A structure of reversibility is a family \(\mcr\) of positive-dimensional cells in the iterated hom-objects such that, for every cell \[ f\colon *\longrightarrow[x,y]_C \] in \(\mcr\), there is a cell \[ g\colon *\longrightarrow[y,x]_C \] in \(\mcr\) and cells \[ \eta\colon * \longrightarrow [g\circ f,\operatorname{id}_x]_{[x,x]_C}, \qquad \epsilon\colon * \longrightarrow [f\circ g,\operatorname{id}_y]_{[y,y]_C} \] in \(\mcr\). A cell is invertible if it belongs to a structure of reversibility (Ozornova and Rovelli 2023, Definition 1.5.1).

An equivalence between higher categories \(C\) and \(D\) is an invertible point \[ F\colon * \longrightarrow [C,D]_{\mathbf{Cat}_{\infty,\infty}}. \] Write \(C\simeq D\) when such an equivalence has been chosen.

Definition 2.10 (Underlying homotopy type and core) Let \(\mathcal S=\mathbf{Types}\) and let \[ i\colon\mathcal S\hookrightarrow\mathbf{Cat}_{\infty,\infty} \] be the full inclusion of groupoidal higher categories. Assume the adjunctions \[ \Pi_\infty\dashv i\dashv(-)^\simeq. \] The left adjoint \[ \Pi_\infty\colon\mathbf{Cat}_{\infty,\infty}\longrightarrow\mathcal S \] is the underlying homotopy type obtained by inverting every morphism. The right adjoint \(C\mapsto C^\simeq\) is the core obtained by retaining every object and only the equivalences (Lurie 2018, Tags 01DQ and 02F5).

Definition 2.11 (Limits, loops, and suspensions) The higher category \(\mathbf{Cat}_{\infty,\infty}\) has an initial object \(\varnothing\) and a terminal object \(*\), and it is bicomplete. For a pointed higher category \((X,x)\), where \(x\colon *\to X\), define \[ \Omega_xX:=*\times_X*. \] For a higher category \(A\), define its suspension by the pushout

The two coprojections \(*\to\Sigma A\) make \(\Sigma A\) bipointed.

Definition 2.12 (Spheres) Define \[ S^{-1}:=\varnothing, \qquad S^{n}:=\Sigma S^{n-1}\quad(n\geq0). \] Since the pushout of \(*\leftarrow\varnothing\to*\) is the coproduct, \(S^{0}=*\amalg*\), the bipointed object indexing the comma category of Definition 2.14.

Example 2.13 (The initial and terminal higher categories) Closedness makes \(-\times C\) a left adjoint (Definition 2.5), so it preserves colimits and in particular the initial object: \[ \varnothing\times C\simeq\varnothing, \qquad *\times C\simeq C, \qquad \varnothing\amalg C\simeq C. \] For hom-objects, \([\varnothing,C]\simeq*\) for every \(C\), since \(\varnothing\) is initial; \([C,\varnothing]\simeq\varnothing\) whenever \(C\) has a point, since evaluating a point of \([C,\varnothing]\) at a point of \(C\) would give a point of \(\varnothing\), and a higher category with no points is \(\varnothing\); and \([\varnothing,\varnothing]\simeq*\). The coproduct \(*\amalg*\) is neither initial nor terminal, so \(\mathbf{Cat}_{\infty,\infty}\) has no zero object.

Definition 2.14 (The $B_{01}$ construction) The higher category of bipointed higher categories is the comma category \[ ((*\amalg *)\downarrow\operatorname{id}_{\mathbf{Cat}_{\infty,\infty}}). \] Its objects are morphisms \[ x_0\amalg x_1\colon *\amalg *\longrightarrow X. \] The endpoint hom functor is \[ \Omega_{01}\colon ((*\amalg *)\downarrow\operatorname{id}_{\mathbf{Cat}_{\infty,\infty}}) \longrightarrow\mathbf{Cat}_{\infty,\infty}, \qquad (X;x_0,x_1)\longmapsto[x_0,x_1]_X. \] Assume that \(\Omega_{01}\) has a left adjoint \[ B_{01}\dashv\Omega_{01}. \] For a higher category \(A=\int_*h_A\), the bipointed higher category \(B_{01}A\) has distinguished objects \(0,1\) and hom-objects displayed by

Composition is given by the identity actions on \(A\) and the unique morphisms from \(\varnothing\). The suspension \(\Sigma A\) is the pushout in Definition 2.11. The object \(B_{01}A\) is characterized by the adjunction \(B_{01}\dashv\Omega_{01}\).

Definition 2.15 (Walking arrows and ordinal categories) Define \[ [0]:=*, \qquad [1]:=B_{01}*. \] Write \(s,t\colon[0]\to[1]\) for the two distinguished objects. For each integer \(n\geq2\), define the ordinal category \([n]\), the walking chain of \(n\) composable arrows, by \[ [n]:= \underbrace{[1]\amalg_{[0]}[1]\amalg_{[0]}\cdots\amalg_{[0]}[1]}_{n\text{ copies}}, \] where each pushout identifies the target of one copy with the source of the next. For a higher category \(C\), the hom-object \[ [[n],C]_{\mathbf{Cat}_{\infty,\infty}} \] is the higher category of functors \([n]\to C\) and their higher cells.

Definition 2.16 (Walking-arrow presentation and mapping types) Precomposition with \(s\) and \(t\) gives \[ (s^*,t^*)\colon [[1],C]_{\mathbf{Cat}_{\infty,\infty}} \longrightarrow [[0],C]_{\mathbf{Cat}_{\infty,\infty}} \times [[0],C]_{\mathbf{Cat}_{\infty,\infty}} \simeq C\times C. \] For points \(x,y\colon*\to C\), the local hom-object \([x,y]_C\) is the cartesian fiber

Its underlying mapping type is \[ \operatorname{Map}_C(x,y):=\Pi_\infty[x,y]_C. \] Likewise, \[ \operatorname{Map}_{\mathbf{Cat}_{\infty,\infty}}(C,D) := \Pi_\infty[C,D]_{\mathbf{Cat}_{\infty,\infty}}. \]

Definition 2.17 (Initial, terminal, and contractible objects) An object \(t\colon*\to C\) is terminal if \(\operatorname{Map}_C(x,t)\) is contractible for every \(x\colon*\to C\). An object \(i\colon*\to C\) is initial if \(\operatorname{Map}_C(i,x)\) is contractible for every \(x\colon*\to C\). A higher category is contractible when it is equivalent to \(*\).

Definition 2.18 (Truncated objects and morphisms) Let \(C\) be a higher category. For an integer \(n\geq-2\), an object \(X\colon*\to C\) is \(n\)-truncated if \(\operatorname{Map}_C(Y,X)\) is an \(n\)-truncated type for every \(Y\colon*\to C\). The \((-2)\)-truncated objects are the terminal objects.

For an object \(Y\colon*\to C\), write \(C_{/Y}:=(\operatorname{id}_C\downarrow Y)\) for the slice. A morphism \(f\colon X\to Y\) is \(n\)-truncated if it is an \(n\)-truncated object of \(C_{/Y}\). Equivalently, for every \(Z\colon*\to C\) and every point of \(\operatorname{Map}_C(Z,Y)\), the corresponding fiber of \[ \operatorname{Map}_C(Z,X) \longrightarrow \operatorname{Map}_C(Z,Y) \] is an \(n\)-truncated type. A \((-2)\)-truncated morphism is an equivalence. When the required pullback exists and \(n\geq-1\), the recursive criterion is \[ f\text{ is }n\text{-truncated} \quad\Longleftrightarrow\quad \Delta_f\colon X\longrightarrow X\times_YX \text{ is }(n-1)\text{-truncated} \] (Lurie 2018, Tags 05F8 and 05FS).

For a type \(S\), \(n\)-truncation means \(\pi_k(S,s)=0\) for every point \(s\) and every \(k>n\). A \((-2)\)-truncated type is contractible, a \((-1)\)-truncated type is empty or contractible, and a \(0\)-truncated type is a set. When the inclusion of \(n\)-truncated objects of \(C\) is reflective, write its reflector as \(\tau_{\leq n}^C\).

Definition 2.19 (The truncation tower) For each integer \(n\geq-2\), assume an idempotent truncation endomorphism \[ T_{\leq n}\colon \mathbf{Cat}_{\infty,\infty} \longrightarrow \mathbf{Cat}_{\infty,\infty}. \] Define \[ \mathbf{Cat}_n :=\operatorname{EssIm}(T_{\leq n}) =:T_{\leq n}\mathbf{Cat}_{\infty,\infty}. \] Let \[ \iota_n\colon \mathbf{Cat}_n \longrightarrow \mathbf{Cat}_{\infty,\infty} \] be the full inclusion, and write \[ \tau_{\leq n}\colon \mathbf{Cat}_{\infty,\infty} \longrightarrow \mathbf{Cat}_n \] for the corestriction of \(T_{\leq n}\) to its essential image. Assume \[ \tau_{\leq n}\dashv\iota_n. \] If \(\iota_n\) also has a right adjoint, write \[ \iota_n\dashv\tau_{\leq n}^{\mathrm R}. \] The tower is \[ \mathbf{Cat}_{-2}\longrightarrow \mathbf{Cat}_{-1}\longrightarrow \mathbf{Cat}_0\longrightarrow \mathbf{Cat}_1\longrightarrow\cdots\longrightarrow \mathbf{Cat}_n\longrightarrow\cdots\longrightarrow \mathbf{Cat}_{\infty,\infty}. \] An \(n\)-category structure on a higher category \(C=\int_*h_C\) is a lift

together with a specified equivalence \(\iota_n\circ\widetilde h_C\simeq h_C\).

For \(n\geq0\), a second construction begins with the full replete subcategory \[ \mathbf{Cat}_n^{\mathrm{loc}} \subseteq \mathbf{Cat}_{\infty,\infty} \] on the locally \((n-1)\)-truncated higher categories (Lurie 2018, Tag 05EA). Its inclusion supplies \(\iota_n\) after an equivalence \(\mathbf{Cat}_n^{\mathrm{loc}}\simeq\mathbf{Cat}_n\) has been established. Its left and right adjoints, when they exist, supply \(\tau_{\leq n}\) and \(\tau_{\leq n}^{\mathrm R}\).

For higher categories \(C,D\), define \[ [C,D]_{\mathbf{Cat}_{\infty,\infty}}^{\leq n} \hookrightarrow [C,D]_{\mathbf{Cat}_{\infty,\infty}} \] to be the full replete subcategory on the \(n\)-truncated objects of \([C,D]_{\mathbf{Cat}_{\infty,\infty}}\). These objects are the \(n\)-truncated morphisms \(C\to D\) internal to the mapping higher category.

Example 2.20 (The bottom of the tower) By Definition 2.18 the \((-2)\)-truncated objects of \(\mathbf{Cat}_{\infty,\infty}\) are its terminal objects, so \(\mathbf{Cat}_{-2}\) is spanned by \(*\) and is contractible. Both \(\varnothing\) and \(*\) are \((-1)\)-truncated: \(\operatorname{Map}(Y,*)\) is contractible for every \(Y\), and \(\operatorname{Map}(Y,\varnothing)\) is empty unless \(Y\simeq\varnothing\), in which case it is contractible. So \(\mathbf{Cat}_{-1}\) contains the two truth values \(\varnothing\) and \(*\), and the map \(\mathbf{Cat}_{-2}\to\mathbf{Cat}_{-1}\) of the tower is the inclusion of \(*\).

Definition 2.21 (Ordinary categories) An ordinary-category structure on a higher category \(C=\int_*h_C\) is a \(1\)-category structure on \(C\): a point \[ \widetilde h_C\colon*\longrightarrow\mathbf{Cat}_1 \] together with a specified equivalence \(\iota_1\circ\widetilde h_C\simeq h_C\). An ordinary category may be presented by the point \(\widetilde h_C\); its underlying higher category is decoded from \(\iota_1\circ\widetilde h_C\).

Definition 2.22 (Equality, isomorphism, and equivalence) Definitional equality is judgmental equality in the chosen formal language. An isomorphism in an ordinary category is a morphism with a two-sided inverse. An equivalence is a morphism satisfying Definition 2.9.

The notation \(a=b\), \(a\cong b\), and \(a\simeq b\) records these three claims. No univalence principle is assumed. Chosen comparison maps are treated in Chapter 7.