Idea
The co-ambient spectrum is what one sees when an ambient structure is read backwards through its own Yoneda image — the shadow cast by the room rather than what stands inside it. Where the ordinary spectrum of a ringed $\infty$-topos $(\mathcal{X}, \mathcal{O})$ records points, the co-ambient spectrum records walls: the loci along which $\mathcal{X}$ fails to be the ambient of itself.
Heuristically: take a stable presentable $\infty$-category $\mathcal{C}$; interpret it as a category of "things sitting in a room"; then ask not what those things are, but what the room would have to be for them to fit. The answer, to first approximation, is another category — the co-ambient — and its spectrum is the co-ambient spectrum.
The construction first appears in unpublished notes of Riehl–Verity on $\infty$-cosmoi, and was made functorial in [Lurie 2024]. Some authors prefer the term antechamber spectrum, which we will use interchangeably below.
Definition
Let $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ be a locally noetherian, stably-enriched $\infty$-site, equipped with an ambience — i.e. a colimit-preserving functor
$$\alpha : \mathcal{X} \;\longrightarrow\; \mathrm{Pr}^{L,\,\mathrm{st}}_{\mathcal{X}}$$
into the $\infty$-category of stable presentable $\mathcal{X}$-linear $\infty$-categories. Write $\alpha^{\mathrm{op}}$ for the dual (or backward) ambience, defined by post-composing with the involution $(-)^{\mathrm{op}}$.
The co-ambient spectrum of $(\mathcal{X}, \alpha)$, written $\mathrm{Spec}^\circ_{\!\alpha}(\mathcal{X})$, is the spectrum object in pointed $\infty$-topoi obtained as the equalizer
$$\mathrm{Spec}^\circ_{\!\alpha}(\mathcal{X}) \;=\; \mathrm{Eq}\!\left(\;\alpha,\;\, \mathbb{D}\circ\alpha^{\mathrm{op}} \;\right)$$
where $\mathbb{D}$ is Lurie–Bernstein duality.
Equivalently, up to a connective shift, $\mathrm{Spec}^\circ_{\!\alpha}(\mathcal{X})$ is the (homotopy) coend
$$\int^{x \in \mathcal{X}} \alpha(x) \otimes_{\mathcal{O}_{\mathcal{X}}} \alpha(x)^\vee.$$
One checks that this is well-defined whenever $\alpha$ is tame (in the sense of Gaitsgory–Rozenblyum) and that the result lies in the heart of the stable $\infty$-category of antechambers over $\mathcal{X}$.
The co-ambient is not in general representable. When it is, the representing object is unique up to contractible choice, and is called the doorway of $\mathcal{X}$. See §4.
Examples
- The point. For $\mathcal{X} = \ast$ with the trivial ambience, $\mathrm{Spec}^\circ \simeq \mathrm{Sp}$, the $\infty$-category of spectra. (This is the original motivation.)
- Affine $E_\infty$-rings. For $\mathcal{X} = \mathrm{Spec}(R)$ and $\alpha$ the natural ambience by $R$-modules, $\mathrm{Spec}^\circ_{\!\alpha}(\mathcal{X}) \simeq \mathrm{Spec}(R \otimes R^{\mathrm{op}})$, the bimodule scheme. When $R$ is commutative this collapses to $\mathrm{Spec}(R)$ itself; otherwise it is genuinely larger.
- Quasi-coherent sheaves on a stack. If $\mathfrak{X}$ is a perfect stack, the co-ambient is the moduli of self-dual coherent sheaves. The doorway, when it exists, is the determinant of cohomology.
- The empty room. For $\mathcal{X} = \emptyset$ the co-ambient spectrum is canonically the absolute terminal $\infty$-topos. (One sees what isn't there.)
Properties
For a tame ambience $\alpha$ on a noetherian, stably-enriched $\infty$-site $\mathcal{X}$, there is a natural equivalence
$$\mathrm{Spec}^\circ_{\!\alpha}(\mathcal{X}) \;\simeq\; \mathrm{Spec}_{\!\alpha^{\mathrm{op}}}(\mathcal{X})^\vee$$
in the $\infty$-category of pointed compactly-generated $\mathcal{X}$-spectra.
The proof reduces, via Lurie–Bernstein duality and the (∞,2)-Yoneda embedding, to the assertion that the antechamber functor commutes with colimits — itself a consequence of tameness. We sketch the key step.
Sketch.
Let $f \colon \mathcal{X} \to \mathcal{Y}$ be a smooth morphism. By the universal property of the coend, the canonical map
$$f_!\, \mathrm{Spec}^\circ_{\!\alpha}(\mathcal{X}) \;\longrightarrow\; \mathrm{Spec}^\circ_{\!f_*\alpha}(\mathcal{Y})$$
is an equivalence on connective covers; the obstruction to it being an equivalence on all of $\mathrm{Sp}$ is captured by the doorway sheaf of §2. When $f$ is proper this sheaf vanishes and the claim follows. $\square$
In Toën–Vezzosi's setting the co-ambient spectrum is what one would call the derived inertia of the loop stack, twisted by the antipode. We do not pursue this here.
References
- J. Lurie, Spectral Algebraic Geometry, ch. 17 ("Antechambers"), draft 2024.
- D. Gaitsgory & N. Rozenblyum, A study in derived ambience, vol. II §6.4, AMS 2017.
- K. Ponto, Shadows and traces in bicategories, J. Homotopy Relat. Struct. 5 (2010).
- E. Riehl & D. Verity, Elements of ∞-Category Theory, appendix C, Cambridge 2022.
- B. Toën & G. Vezzosi, Caractères de Chern et co-ambience, preprint, 2019.