In Section 2.4 we introduced the basic constructions of fibers, cofibers, suspensions and loops in the \(\infty \)-category \(\An _*\) of pointed animae. In this section we develop the corresponding formalism for an arbitrary pointed \(\infty \)-category \(C\). This will provide the language needed to formulate the definition of a stable \(\infty \)-category in Section 4.2, and will also be used repeatedly in later chapters.
Definition 4.1.1 (Pointed \(\infty \)-category). We call an \(\infty \)-category \(C\) pointed if it has a zero object, i.e. an object \(0\) that is both initial and terminal (Definition 1.7.7). Given objects \(X,Y \in C\), we define the null map \(0\colon X \to Y\) as the composite of the unique maps \(X \to 0\) and \(0 \to Y\).
A functor \(F\colon C \to D\) between pointed \(\infty \)-categories is called pointed if it preserves zero objects.
Example 4.1.2. The category \(\Ab \) of abelian groups is pointed: the zero object is the trivial group \(0\). More generally, any additive category is pointed.
Example 4.1.3. The category \(\Top _*\) of pointed topological spaces is pointed, with the one-point space functioning as zero object.
Example 4.1.4. The \(\infty \)-category \(\An _*\) of pointed animae (see Section 2.4) is pointed, with the one-point anima as zero object.
Definition 4.1.5 (Fiber and cofiber sequences). Let \(C\) be a pointed \(\infty \)-category. A nullsequence in \(C\) is a sequence of morphisms \[ X \xrightarrow {f} Y \xrightarrow {g} Z \] equipped with a specified nullhomotopy \(g \circ f \cong 0\). Equivalently, a nullsequence is a commutative square in \(C\) of the form
If this square is a pullback square in \(C\), we say it is a fiber sequence, and refer to \(X\) as the fiber of \(g\), written \(X \simeq \fib (g)\). If the square is a pushout square, we say it is a cofiber sequence, and refer to \(Z\) as the cofiber of \(f\), written \(Z \simeq \cofib (f)\).
We say that \(C\) admits fibers if such a fiber sequence exists for every morphism \(g\colon Y \to Z\). Dually, we say that \(C\) admits cofibers if such a cofiber sequence exists for every morphism \(f\colon X \to Y\).
Definition 4.1.6 (Loop and suspension). Let \(C\) be a pointed \(\infty \)-category. If \(C\) admits fibers, we define the loop object of \(X\in C\) by \[ \Omega X:=\fib (0\to X). \] If \(C\) admits cofibers, we define its suspension by \[ \Sigma X:=\cofib (X\to 0). \]
Just as in Exercise 2.4.7, the loop and suspension constructions define endofunctors \(\Omega ,\Sigma \colon C \to C\).
Lemma 4.1.7. Let \(C\) be a pointed \(\infty \)-category that admits fibers and cofibers. Then the functor \(\Sigma \colon C \to C\) is left adjoint to \(\Omega \colon C \to C\): for all \(X,Y \in C\) there is a natural equivalence \[ \Hom _C(\Sigma X, Y) \simeq \Hom _C(X,\Omega Y). \]
Proof. The proof is identical to that of Lemma 2.4.8. □
Pointed objects
The pointed \(\infty \)-categories \(\Top _*\) and \(\An _*\) are a special case of a general construction for turning an \(\infty \)-category \(C\) with terminal object into a pointed \(\infty \)-category.
Definition 4.1.8 (Pointed objects). If \(C\) is an \(\infty \)-category with a terminal object \(*\), we define its \(\infty \)-category of pointed objects \(C_*\) as the slice category of \(C\) over \(*\), i.e. as the following pullback square:
Objects of \(C_*\) are objects \(X\) of \(C\) equipped with a map \(x\colon * \to X\) from the terminal object, thought of as a chosen ‘basepoint’ of \(X\). We will frequently denote such objects by \((X,x)\), or sometimes abusively as \(X\).
A simple computation shows that the hom anima between pointed objects \((X,x)\) and \((Y,y)\) in \(C\) is computed as the following fiber: \[ \Hom _{C_*}((X,x),(Y,y)) \;\simeq \; \fib _y\big (\Hom _C(X,Y) \xrightarrow {- \circ x} \Hom _C(*,Y)\big ). \] (See Lemma 21.3.2 for a proof of the more general computation of hom animae in slice categories.) In other words, a morphism in \(C_*\) from \((X,x)\) to \((Y,y)\) consists of a morphism \(f\colon X \to Y\) in \(C\) together with an isomorphism \(f(x) \cong y\) of maps \(* \to Y\).
Lemma 4.1.9. Let \(C\) be an \(\infty \)-category with terminal object \(*\). Then:
- (1)
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The \(\infty \)-category \(C_*\) is pointed, with zero object given by \(0 := (*,\id _*\colon * \to *)\).
- (2)
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The \(\infty \)-category \(C\) is pointed if and only if the forgetful functor \[ \fgt \colon C_* = C_{*/} \longrightarrow C \] is an equivalence.
Proof. (1) For initiality of \(0\), note that the map \(\Hom _C(*,Y) \to \Hom _C(*,Y)\) given by precomposing with the identity is an equivalence, hence its fiber \(\Hom _{C_*}(0,Y)\) is contractible for every \(Y \in C_*\). For terminality, we observe that \[ \Hom _{C_*}(X,0) \simeq \fib \big (\Hom _C(X,*) \to \Hom _C(*,*)\big ) \] is the fiber of a map between two contractible animae, which thus is itself contractible.
(2) The ‘only if’ part is clear from (1). Conversely, assume that \(C\) is pointed. Then the object \(*\) is also an initial object in \(C\), and hence the target functor \(C_* = C_{*/} \to C\) is an equivalence by Lemma 21.3.4. □
Exercise 4.1.10. Show that \(C_*\) admits pullbacks whenever \(C\) admits pullbacks, and that the functor \(C_* \to C\) preserves pullbacks in this case. Similarly, show that \(C_*\) admits pushouts whenever \(C\) admits pushouts, and that \(C_* \to C\) preserves pushouts.
The pointed \(\infty \)-category \(C_*\) is in a precise sense the ‘universal approximation’ of \(C\) by a pointed \(\infty \)-category. We will end this section by recording a precise formulation of this universal property, which will be used later.
Notation 4.1.11. Let \(\Cat _{\infty }^{\term } \subseteq \Cat _{\infty }\) denote the subcategory of \(\infty \)-categories with a terminal object, and functors preserving the terminal object. Let \(\Cat _{\infty }^{\pt } \subseteq \Cat _{\infty }^{\term }\) denote the full subcategory of pointed \(\infty \)-categories. Given \(\infty \)-categories \(C\) and \(D\) with terminal objects, we denote by \[ \Fun _*(C,D) \quad \subseteq \quad \Fun (C,D) \] the full subcategory spanned by functors preserving the terminal object.
The constructions of \(C_*\) and the forgetful functor \(\fgt \colon C_* \to C\) are functorial in \(C\), giving rise to a functor \[ (-)_*\colon \Cat _{\infty }^{\term } \to \Cat _{\infty }^{\pt } \] and a natural transformation \(\fgt \colon (-)_* \to \id \).
Corollary 4.1.12. Let \(D\) be a pointed \(\infty \)-category and let \(C\) be an \(\infty \)-category with a terminal object. Then the \(\infty \)-category \(\Fun _*(D,C)\) is pointed.
Proof. The constant functor \(\const _*\colon D \to C\) is terminal in \(\Fun (D,C)\), hence also in the full subcategory \(\Fun _*(D,C)\). We claim that it is also initial. Let \(0 \in D\) denote a zero object, and let \(F\colon D \to C\) be a functor preserving terminal objects. Since \(0\) is also initial in \(D\), limits over \(D\) are given by evaluation at \(0\) by Lemma 21.2.4. We therefore get \[ \Hom _{\Fun _*(D,C)}(\const _*,F) \simeq \Hom _C(*,\lim _{d \in D} F(d)) \simeq \Hom _C(*,F(0)). \] Since \(F\) preserves terminal objects and \(0\) is terminal in \(D\), the object \(F(0)\) is terminal in \(C\). The last hom anima is therefore contractible, proving that \(\const _*\) is initial. □
Lemma 4.1.13. Let \(D\) be pointed and let \(C\) have a terminal object. Then the forgetful functor \(\fgt \colon C_* \to C\) induces an equivalence \[ \fgt _* \colon \Fun _*(D,C_*) \iso \Fun _*(D,C). \] In particular, the functor \((-)_* \colon \Cat _{\infty }^{\term } \to \Cat _{\infty }^{\pt }\) is right adjoint to the inclusion \(\Cat _{\infty }^{\pt } \hookrightarrow \Cat _{\infty }^{\term }\).
Proof. Since \(C_* = C_{*/}\), there is an equivalence \[ \Fun _*(D,C_*) \simeq \Fun _*(D,C)_*. \] Indeed, a natural transformation \(\const _* \to F\) is necessarily an isomorphism at the zero object as soon as \(F\) preserves terminal objects. By Corollary 4.1.12, Lemma 4.1.9, the forgetful functor from the right-hand side to \(\Fun _*(D,C)\) is an equivalence. Passing to underlying animae exhibits \((-)_*\) as a right adjoint to the inclusion. □
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