1.3 The homotopy category of a quasi-category, isofibrations, and equivalences

The homotopy category of a quasi-category admits a simplified description, which we build up to over a series of definitions. As the homotopy category functor is formalized in terms of 2-truncated simplicial sets, we introduce an auxiliary definition to isolate the structure of interest in a 2-truncated quasi-category.

Definition 1.3.1

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 $\sigma _1$.

Lemma 1.3.2

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

Proof ▶

Immediate from the definition by filling horns in dimensions 2 and 3.

We revisit Definition 1.2.19 in this setting.

Definition 1.3.3 homotopy relation on 1-simplices

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:

$\begin{tikzcd} [row sep=small, column sep=small] & y \arrow[dr, equals] & && & x \arrow[dr, "f"] \\ x \arrow[ur, "f"] \arrow[rr, "g"'] & & y & & x \arrow[ur, equals] \arrow[rr, "g"'] & & y \end{tikzcd}$

$\begin{tikzcd} [row sep=small, column sep=small] & y \arrow[dr, equals] \\ x \arrow[ur, "f"] \arrow[rr, "g"'] & & y \end{tikzcd}$

Lemma 1.3.4

If $A$ is a 2-truncated quasi-category then:

The left and right homotopy relations are reflexive.
The left and right homotopy relations are symmetric.
The left and right homotopy relations are transitive.
The left homotopy relation coincides with the right homotopy relation.

Proof ▶

Each statement follows from a single 3-dimensional horn filling, typically involving degenerate simplices.

As the left and right homotopy relations coincide in a 2-truncated quasi-category, in that setting we take right homotopy to be the default and refer to it simply as “homotopy” and denote it by “$\sim $” going forward.

Lemma 1.3.5

$\quad $

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

Proof ▶

For (i), fill the (3,2)-horn filling the path formed by a degenerate edge, followed by the given path edges, and using the given simplices as the 0th and 1st faces. The proofs of (ii) and (iii) are similar.

Corollary 1.3.6

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 $h'$.

Proof ▶

Apply the three conclusions of Lemma 1.3.5 one at a time to transform the given 2-simplex.

These results now combine to justify the following definition:

Definition 1.3.7 the homotopy category of a 2-truncated quasi-category

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

$\begin{tikzcd} [row sep=small, column sep=small] & a_1 \arrow[dr, "g"] \\ a_0 \arrow[ur, "f"] \arrow[rr, "h"'] & & a_2 \end{tikzcd}$

In other words, the hom-types are quotients of the hom-types of the underlying reflexive quiver of Definition 1.2.21 of a 2-truncated simplicial set, where the additional quotienting is by the homotopy relation, which is an equivalence relation by Lemma 1.3.4. Composition is defined by (2,1)-horn filling and is well-defined by Lemma 1.3.5.

Lemma 1.3.8 the homotopy category of a quasi-category

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.

Proof ▶

Given a 2-truncated quasi-category $A$, we can construct a natural isomorphism between its 2-truncated homotopy category ${\mathord {\mathsf{h}}}_2A$ in the sense of Definition 1.2.23 and its 2-truncated homotopy category ${\mathord {\mathsf{h}}}{A}$ in the sense of Definition 1.3.7 by showing the latter satisfies the same universal property of the former, as a quotient of the free category $FA$ on the underlying reflexive quiver.

By adjunction, to define a functor $q \colon FA \to {\mathord {\mathsf{h}}}{A}$, it suffices to define a refl prefunctor $q \colon A \to {\mathord {\mathsf{h}}}{A}$ from the one-truncation of $A$ to the underlying refl quiver of ${\mathord {\mathsf{h}}}{A}$. The objects of these quivers coincide while the homs in the latter and quotients of the homs in the former, defining a canonical quotient map. By construction, the corresponding functor $q \colon FA \to {\mathord {\mathsf{h}}}{A}$ respects the hom-relation that defines the homotopy category ${\mathord {\mathsf{h}}}_2{A}$, so the universal property of the latter quotient induces a comparison functor ${\mathord {\mathsf{h}}}_2{A} \to {\mathord {\mathsf{h}}}{A}$ which factors $q$ through the analogously defined functor $q \colon FA \to {\mathord {\mathsf{h}}}_2{A}$.

To see this is an isomorphism, we show that $q \colon FA \to {\mathord {\mathsf{h}}}{A}$ satisfies the same universal property. To that end, consider another functor $g \colon FA \to C$ respecting the hom-relation. In particular, $g$ respects the homotopy relation of Definition 1.3.3, since this is a special case of the hom-relation. Thus, on underlying refl prefunctors, $g$ factors uniquely through $q$ along a map $h \colon {\mathord {\mathsf{h}}}{A} \to C$. By Corollary 1.3.6, $h$ respects composition and thus lifts to define a functor. This gives the required factorization. Uniqueness follows because the the functor $U \colon {\mathord {\mathcal{Cat}}}\to {\mathord {\mathcal{rQuiv}}}$ is faithful.

Later we will require either of the following results:

Lemma 1.3.9

$\quad $

The functor ${\mathord {\mathsf{h}}}\colon {\mathord {\mathcal{sSet}}}\to {\mathord {\mathcal{Cat}}}$ preserves finite products.
The functor ${\mathord {\mathsf{h}}}\colon {\mathord {\mathcal{QCat}}}\to {\mathord {\mathcal{Cat}}}$ preserves small products.

Proof ▶

For the first statement, preservation of the terminal object is by direct calculation. By Proposition 1.2.25, preservation of binary products is equivalent to the statement that the canonical map $N({\mathord {\mathcal{D}}}^{\mathord {\mathcal{C}}}) \to N({\mathord {\mathcal{D}}})^{N{\mathord {\mathcal{C}}}}$ involving nerves of categories is an isomorphism. On $n$-simplices, this is defined by uncurrying, which is bijection since ${\mathord {\mathcal{Cat}}}$ is cartesian closed.

For the second statement, we have a canonical comparison functor from the homotopy category of the products to the product of the homotopy categories. It follows from Definition 1.3.7 and Lemma 1.3.8 that this is an isomorphism on underlying quivers, which suffices.

Definition 1.3.10 isomorphism in a quasi-category

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

$\begin{tikzcd} [row sep=small, column sep=small] & b \arrow[dr, "f^{-1}", dashed] & & & & a \arrow[dr, "f"] \\ a \arrow[ur, "f"] \arrow[rr, equals] & & a & & b \arrow[ur, "f^{-1}", dashed] \arrow[rr, equals] & & b \end{tikzcd}$

The properties of the isomorphisms in a quasi-category are somewhat technical to prove and will likely be a pain to formalize (see [ RV22 , § D ] ). Here we focus on a few essential results, which are more easily obtainable.

Definition 1.3.11

✓

The homotopy coherent isomorphism $I$, is the nerve of the free-living isomorphism.

Just as the arrows in a quasi-category $A$ are represented by simplicial maps $ [1]\to A$ whose domain is the nerve of the free-living arrow, the isomorphisms in a quasi-category can be represented by diagrams $I\to A$ whose domain is the homotopy coherent isomorphism:

Proposition 1.3.12

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

$\begin{tikzcd} \cattwo \arrow[r, "f"] \arrow[d, hook] & A \\ \iso \arrow[ur, dashed] \end{tikzcd}$

Remark 1.3.13

If this result proves too annoying to formalize without the general theory of “special-outer horn filling,” we might instead substitute a finite model of the homotopy coherent isomorphism for $I$.

Quasi-categories define the fibrant objects in a model structure due to Joyal. We use the term isofibration to refer to the fibrations between fibrant objects in this model structure, which admit the following concrete description.

Definition 1.3.14 isofibration

✓

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 $I$.

$\begin{tikzcd} \Lambda^k[n] \arrow[d, hook] \arrow[r] & A \arrow[d, two heads, "f"] & & \catone \arrow[d, hook] \arrow[r] & A \arrow[d, two heads, "f"] \\ \Delta[n] \arrow[ur, dashed] \arrow[r] & B & & \iso \arrow[ur, dashed] \arrow[r] & B \end{tikzcd}$

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

We now introduce the weak equivalences and trivial fibrations between fibrant objects in the Joyal model structure.

Definition 1.3.15 equivalences of quasi-categories

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 $g \colon B \to A$,

$\begin{tikzcd} & A & & & B \\ A \arrow[ur, equals] \arrow[dr, "gf"'] \arrow[r, "\alpha"] & A^\iso \arrow[u, "\ev_0"'] \arrow[d, "\ev_1"] & \text{and} & B \arrow[dr, equals] \arrow[r, "\beta"] \arrow[ur, "fg"] & B^\iso \arrow[u, "\ev_0"'] \arrow[d, "\ev_1"] \\ & A & & & B \end{tikzcd}$

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

Lemma 1.3.16

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

$\begin{tikzcd} \ho{A} \arrow[r, "\ho{\alpha}"] & \ho(A^\iso) \arrow[r] & (\ho{A})^\iso & \ho{B} \arrow[r, "\ho{\beta}"] & \ho(B^\iso) \arrow[r] & (\ho{B})^\iso \end{tikzcd}$

Definition 1.3.17

✓

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

$\begin{tikzcd} \partial\Delta[n] \arrow[r] \arrow[d, hook] & X \arrow[d, two heads, "\sim"', "f"] \\ \Delta[n] \arrow[r] \arrow[ur, dashed] & Y \end{tikzcd}$

for n ≥0

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

The notation “$\twoheadrightarrow $” is suggestive: the trivial fibrations between quasi-categories are exactly those maps that are both isofibrations and equivalences. This can be proven by a relatively standard although rather technical argument in simplicial homotopy theory [ RV22 , D.5.6 ] .