Topological coHochschild Homology

• Topological coHochschild Homology is a spectrum-valued invariant defined via the cyclic cobar construction that dualizes classical coHochschild homology for coalgebra spectra.
• It exhibits robust homotopy and Morita–Takeuchi invariance, with categorical duality properties linking it to topological Hochschild homology via Spanier–Whitehead duality.
• coTHH provides explicit computational models, including spectral sequences that recover free loop space topology and facilitate analyses in stable homotopy theory.

Topological coHochschild homology (coTHH) is the spectrum-valued invariant that formalizes the dual notion to topological Hochschild homology (THH), operadically and categorically extending the classical coHochschild homology of (differential graded) coalgebras to the setting of stable homotopy theory. Constructed via the totalization of a canonical cosimplicial object associated to a coalgebra spectrum, coTHH not only recovers central phenomena in free loop space topology but also admits deep duality results, explicit computational models, and robust invariance properties analogous to those of THH. The theory has undergone significant recent developments—including Morita–Takeuchi invariance, spectral sequence computations, and connections to Spanier–Whitehead duality—across multiple categorical frameworks.

1. Cosimplicial Model and Definition

The definition of topological coHochschild homology is governed by the cyclic cobar construction. For a coaugmented coalgebra spectrum $(C, \Delta, \varepsilon)$ in a symmetric monoidal, stable model category (e.g., $k$-module spectra), one forms the cosimplicial object

$\mathcal{H}^n(C) = C^{\wedge_k (n+1)}$

with coface maps

$$d^0 = \rho \wedge \operatorname{id}, \quad d^i = \operatorname{id}^{\wedge (i-1)} \wedge \Delta \wedge \operatorname{id}^{\wedge(n-i)} \ (1 \leq i \leq n), \quad d^{n+1} = (\operatorname{id} \wedge T)\circ (\operatorname{id} \wedge \Delta \wedge \operatorname{id}^{\wedge(n-1)})$$

and codegeneracies arising from repeated application of $\varepsilon$. coTHH is the totalization of this cosimplicial spectrum: $$\operatorname{coTHH}(C) = \mathrm{Tot}\,\mathcal{H}^\bullet(C)$$ with model-categorical precautions (e.g., using Reedy-fibrant replacements) to guarantee homotopy invariance (Hess et al., 2018). This construction naturally dualizes the cyclic bar realization of THH.

When $C$ is a coalgebra over a commutative ring spectrum $R$ (i.e., $C \in \mathrm{CoAlg}(\mathrm{Mod}_R)$), the cyclic cobar realization extends to

$$\operatorname{coTHH}_R(C) = \big|\mathcal{B}^\bullet(C)\big| = \mathrm{Tot}\,\mathcal{B}^\bullet(C)$$

with

$\mathcal{B}^n(C) = C^{\otimes_R (n+1)}$

and the dual Hochschild face and degeneracy maps (Zha, 20 May 2025). For Thom spectrum coefficients, the construction generalizes to bicomodules, resulting in derived functoriality supported by the cosimplicial totalization (Zha, 16 Jan 2026).

2. Formal Properties and Invariance

coTHH enjoys several categorical and homotopical invariance properties:

• Homotopy invariance: Weak equivalences $C \simeq C'$ of cofibrant coalgebras induce weak equivalences $$\operatorname{coTHH}(C) \simeq \operatorname{coTHH}(C')$$ (Hess et al., 2018).
• Model-independence: The construction is preserved under strong symmetric-monoidal Quillen or $\infty$-categorical equivalences of ambient stable categories (Hess et al., 2018, Bayındır et al., 2020, Zha, 20 May 2025).
• Morita (Takeuchi) invariance: If coalgebras $C$ and $D$ are Morita–Takeuchi equivalent—i.e., their comodule categories are equivalent via suitable bicomodules—then their coTHH spectra are equivalent as coalgebras (Zha, 20 May 2025).
• Agreement principle: For a coalgebra $C$, $$\operatorname{coTHH}(C) \simeq \operatorname{THH}(\mathrm{Perf}(\Omega C))$$ where $\Omega C$ is the cobar construction, and $\mathrm{Perf}$ denotes compact right $\Omega C$-modules (Hess et al., 2018).
• Spanier–Whitehead duality: For quasi-proper coalgebras $C$, there is an equivalence $$\operatorname{coTHH}(C) \simeq D(\operatorname{THH}(D C))$$, where $D(-)$ denotes Spanier–Whitehead duality (Bayındır et al., 2020).

These formal properties ensure that coTHH is a robust invariant, well-behaved under quasi-equivalences and duality, extending the celebrated invariance and duality of THH.

3. Identification with Free Loop Spaces and Computations

A key topological manifestation of coTHH is its identification with the suspension spectrum of the free loop space. For a simply connected (or EMSS-good) pointed space $X$,

$$\operatorname{coTHH}(\Sigma_+^\infty X) \simeq \Sigma_+^\infty \mathcal{L} X$$

where $\mathcal{L} X = \operatorname{Map}(S^1, X)$ is the free loop space (Hess et al., 2018, Bohmann et al., 2021). The proof proceeds by identifying the cosimplicial levels $$\mathcal{H}^n(\Sigma_+^\infty X)\cong \Sigma_+^\infty X^{\times(n+1)}$$ with the $n$-simplices of the mapping space from $S^1$ to $X$, and then using the convergence of the Eilenberg–Moore spectral sequence to pass the suspension spectrum functor through totalization.

Thom spectrum variants—where the coefficients are comodule Thom spectra over $R \otimes \Sigma_+^\infty X$—yield reductions to loop group cases or filtered spectral sequence models using the cell structure of $X$ (Zha, 16 Jan 2026). In these cases, the cellular filtration produces an associated spectral sequence computing $$\pi_*(A \otimes_R \operatorname{coTHH}^R(\operatorname{Th}f; R[\!X]))$$ with explicit $E^1$- and $E^2$-pages based on the (co)homology of the based loop space $\Omega X$.

For group coalgebras, $\operatorname{coTHH}_R(R[G])$ can be identified with $R \otimes_{R[G]^{\otimes 2}} R[G]$, recovering homotopy orbit spectra and linking to classical computations of free loop space (as in Bökstedt–Waldhausen's work) (Zha, 20 May 2025, Zha, 16 Jan 2026).

4. Algebraic and Spectral Sequence Structures

The coBökstedt spectral sequence, built from the tower of skeleta (Bousfield–Kan), is the primary computational tool: $$E_2^{s,t} = \operatorname{coHH}_s(H_t(C;k)) \implies H_{t-s}(\operatorname{coTHH}(C); k)$$ where $\operatorname{coHH}_*$ denotes classical coHochschild homology of graded $k$-coalgebras (Bohmann et al., 2017, Bohmann et al., 2021, Klanderman, 2021). This spectral sequence inherits strong algebraic structure:

• Coalgebra/Hopf structure: The $E_r$-pages admit a $\square_{H_*(C)}$-Hopf algebra structure and an antipode, mirroring the THH Bökstedt spectral sequence under coflatness assumptions (Bohmann et al., 2021, Klanderman, 2021).
• Differential constraints: The first nonzero differential must send an algebra indecomposable to a coalgebra primitive, heavily restricting potential differentials and guaranteeing collapses in many cases (Bohmann et al., 2017, Klanderman, 2021).

Explicit computations depend on the algebraic type of $H_*(C;k)$. For instance, for connected polynomial coalgebras on even degrees or exterior coalgebras on odd degrees, the $E_2$-page can be described explicitly, and collapse criteria yield final answers for the homology of coTHH (see Table).

Coalgebra Type	$E_2$-term for $H_*(C;k)$	Collapse?
$\Lambda_k[y_i]$ (odd)	$\Lambda_k[y_i]\otimes k[w_i]$	Always $E_2=E_\infty$
$k[x_i]$ (even, polynomial)	$k[x_i]\otimes \Lambda_k[z_i]$	Always $E_2=E_\infty$
$\Gamma_k[x_i]$ (divided-power)	$\Gamma_k[x_i]\otimes \Lambda_k[z_i]$	Always $E_2=E_\infty$

This gives powerful computational leverage for spaces such as products of projective spaces, classifying spaces of Lie groups, and spheres (Klanderman, 2021, Bohmann et al., 2021).

5. Higher Coalgebraic Structures and Duality

If $C$ is an $\mathbb{E}_k$–coalgebra, the coTHH construction reduces the coalgebraic structure index by $1$, i.e., $\operatorname{coTHH}_R(C)$ is naturally an $\mathbb{E}_{k-1}$–coalgebra, and for $\mathbb{E}_\infty$–coalgebras, coTHH yields a canonical $\mathbb{E}_\infty$–coalgebra structure (Zha, 20 May 2025). This shift parallels the behavior of THH with respect to $\mathbb{E}_n$–algebras.

Spanier–Whitehead duality establishes a formal bridge to THH: for quasi-proper coalgebras (those dualizable in the relevant sense), there is a natural equivalence

$$\operatorname{coTHH}(C) \simeq D(\operatorname{THH}(D C))$$

where $D(-)$ is the internal dual (Bayındır et al., 2020, Zha, 20 May 2025). Duality theory provides a computational tool for passing between coTHH of coalgebras and THH of their Koszul dual algebras and clarifies the conceptual symmetry underlying their invariants.

6. Applications and Examples

Topological coHochschild homology gives spectrum-level models for string topology, producing and classifying operations on free loop spaces via the coalgebraic structures of coTHH (Bohmann et al., 2021). Explicit calculations for suspension spectra $\Sigma^\infty_+ X$ recover the suspension spectrum of $\mathcal{L}X$; for group-coalgebras, the spectrum of based maps $S^1 \to BG$; and for Thom spectrum comodules, filtered models reducing to loop group coTHH (Hess et al., 2018, Zha, 16 Jan 2026). The duality and Morita invariance properties allow computations in new regimes, such as the Steenrod coalgebra, via THH of dual-algebraic objects (Bayındır et al., 2020).

Relative coTHH over a commutative ring spectrum $R$ extends the range of computations, and the relative coBökstedt spectral sequence enables calculations even for generalized Eilenberg–Mac Lane and polynomial or exterior coalgebras, with explicit coHochschild homology inputs and differentials constrained by primitive–indecomposable algebraic structure (Klanderman, 2021).

7. Extensions, Open Problems, and Further Developments

Open directions in the theory of coTHH include the extension of spectral sequence collapse results to broader classes of coalgebras (beyond exterior and polynomial), the analysis of multiplicative and Gerstenhaber structures on $E_\infty$-pages, connections to factorization cohomology of $E_n$–coalgebras, and the systematic interplay with string-topological operations (Bohmann et al., 2021, Bayındır et al., 2020). The structural duality with THH continues to guide both computational and conceptual advances, implicating new invariants for ring and coalgebra spectra, and strengthening the synthesis of algebraic and topological approaches to free loop space homology and stable homotopy theory.