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Pointed spectrum lifts through a discrete target are unique

Statement

Assume the Axiom of Choice and the compatible universe bounds for spaces and coherent functor categories. Let $X$ be an infinity-category with finite limits and small mapping spaces, and write $*$ for its terminal object and $X_{*} = X_{* /}$ for its pointed objects. Define $S p (X) = {Exc}_{*} (S_{*}^{f i n}, X)$ , the full category of reduced excisive functors, and let $Ω^{\infty}$ be evaluation at $S^{0}$ , with its canonical pointing. Reduced means taking the point to a terminal object; excisive means taking pushout squares to pullback squares. A zero-truncated object $T$ means that ${Map}_{X} (B, T)$ is a discrete space for every $B$ .

Let $F \to E \overset{q}{\to} D$ be a fiber sequence of spectrum objects, including its defining pullback $F = E \times_{D} 0$ , and set $T = Ω^{\infty} D$ . Then $Ω^{\infty} F ≃ Ω^{\infty} E \times_{T} *$ as pointed objects. For a pointed $A$ and fixed pointed map $f : A \to Ω^{\infty} E$ , define its lift space as the homotopy fiber over $f$ of ${Map}_{*} (A, Ω^{\infty} F) \to {Map}_{*} (A, Ω^{\infty} E)$ ; thus a lift includes its homotopy over $Ω^{\infty} E$ . This space is the path space from $q f$ to zero in ${Map}_{*} (A, T)$ . If $T$ is zero-truncated and $q f$ is pointed nullhomotopic, the lift space is contractible.

Facts & Assumptions

Given: The universe bounds, finite limits, spectrum objects, pointed map and specified fiber sequence in the Statement. All pullbacks and fibers are homotopy pullbacks and homotopy fibers.

Goal: Identify the full space of lifts, including its homotopies, and deduce uniqueness when the target is discrete.

[A1]

Assume The Axiom of Choice and fixed nested universes $U \in V \in W$ with $ω \in U$ , using Grothendieck universe closure convention. Universe existence is a separate assumption. Use the compatible bounds of External stable enhancement and smash product of spectra, enlarging the ambient universe for the stated locally small category and its functor categories as necessary. They support the imported categorical formalism and spectrum construction; the local universal-property calculations introduce no additional choice principle.

[F1]

External stable enhancement and smash product of spectra supplies infinity-categories, coherent mapping spaces, their limit universal properties, and the category $S_{*}^{f i n}$ of finite pointed spaces. Pointed objects are arrows from the terminal object, and pointed maps include the compatible homotopy at that arrow.

[F2]

Reference 0 supplies the reduced excisive spectrum-object definition, closure under finite limits, the constant-terminal zero object and the evaluation description of $Ω^{\infty}$ .

[F3]

Reference 1 supplies pointwise functor limits, the undercategory mapping-space fiber formula, the mapping-space definition of zero-truncation and closure of discrete spaces under finite limits.

External input

Import exactly [F2] and [F3] from Higher Algebra and Higher Topos Theory, at the locators and hypotheses recorded in references 0 and 1. These foundational premises are imported; the following preservation, lift-space and uniqueness deductions are local.

Proof

1.1A1F1F2F3given

By [F2], the full spectrum-object category is closed under finite limits in its functor category, and by [F3] those limits are pointwise. Evaluation at $S^{0}$ therefore carries the defining square for $F$ to the pullback $Ω^{\infty} E \times_{T} *$ ; the zero functor evaluates to $*$ . The canonical pointings are obtained from the value at the point. The same square is a pullback among pointed objects: a cone of maps from a pointed object into it is a cone in $X$ together with the compatible pointing, and its universal property is the limit universal property of [F1] with that pointing retained. Applying this property to maps from $A$ gives ${Map}_{*} (A, Ω^{\infty} F) ≃ {Map}_{*} (A, Ω^{\infty} E) \times_{{Map}_{*} (A, T)} *$ . Its homotopy fiber over $f$ consists precisely of paths from $q f$ to the zero map, with all their higher homotopies.

2.1F1F3step 1.1given∎

Suppose $T$ is zero-truncated. By the undercategory formula in [F3], ${Map}_{*} (A, T)$ is the homotopy fiber of ${Map}_{X} (A, T) \to {Map}_{X} (*, T)$ over the chosen point of $T$ . Both spaces are discrete by the definition in [F3], so their homotopy fiber is discrete by its finite-limit assertion. A pointed nullhomotopy of $q f$ puts the two endpoints in the same component of this discrete space. Its path space between those endpoints is therefore contractible: after equivalence with a set it is the singleton of the constant path. The description in step 1.1 proves the asserted contractibility of the lift space and the claimed uniqueness of pointed lifts over the fixed map.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Jacob Lurie, Higher Algebra (September 18, 2017) · Definition 1.4.2.1 and Remark 1.4.2.3, printed p.141; Definition 1.4.2.8 and Lemma 1.4.2.10, p.142; Notation 1.4.2.20, p.146 (zero-based PDF indices 140,141,145).

Spectrum objects are reduced excisive functors on finite pointed spaces. These functors are closed under existing finite limits; their zero object is the constant terminal functor. Omega-infinity is evaluation at the pointed zero-sphere.

Hypotheses: An infinity-category admitting finite limits; all functor categories and finite pointed spaces use the fixed compatible universes. No presentability, connectivity or site hypothesis is needed for these assertions.

Reading limits: Freshly read the named definitions, Remark 1.4.2.3, the statement and displayed proof of Lemma 1.4.2.10, and Notation 1.4.2.20 in primary PDF text returned by the web tool. General infinity-category foundations and the cited Kan-extension justification are imported, not reconstructed. The lift-space calculation is a local application. Screenshot calls supplied no inspectable images in this sandbox; no visual reading is claimed.

Jacob Lurie, Higher Topos Theory · Corollary 5.1.2.3, printed p.319, applied dually; Lemma 5.5.5.12, pp.489–490; Definition 5.5.6.1, p.493; Remark 5.5.6.4 and Proposition 5.5.6.5, p.494 (zero-based PDF indices 336,506–507,510–511).

Functor-category limits are pointwise. Mapping spaces in an undercategory are homotopy fibers of precomposition. A zero-truncated object has discrete mapping spaces from every object; this agrees with the usual notion for spaces, and truncated objects are closed under existing limits.

Hypotheses: Infinity-categories and diagrams in the chosen universes. For the dual pointwise-limit assertion the target admits the indicated limits. The undercategory assertion applies to two arrows with a common source. Only zero-truncation and finite limits are used here.

Reading limits: Freshly read Corollary 5.1.2.3 and its short proof (used dually), Lemma 5.5.5.12 with its displayed proof, Definition 5.5.6.1, Remark 5.5.6.4, and Proposition 5.5.6.5 with its displayed proof in primary PDF text returned by the web tool. Proposition 5.1.2.2 and the fibration, rigidification and general topos foundations cited there are imported, not reconstructed. Only finite limits and zero-truncation are applied.

Sources

Jacob Lurie, Higher Algebra (September 18, 2017) · Definition 1.4.2.1 and Remark 1.4.2.3, printed p.141; Definition 1.4.2.8 and Lemma 1.4.2.10, p.142; Notation 1.4.2.20, p.146 (zero-based PDF indices 140,141,145).
Jacob Lurie, Higher Topos Theory · Corollary 5.1.2.3, printed p.319, applied dually; Lemma 5.5.5.12, pp.489–490; Definition 5.5.6.1, p.493; Remark 5.5.6.4 and Proposition 5.5.6.5, p.494 (zero-based PDF indices 336,506–507,510–511).