2-Local Homotopy Types

• 2-local homotopy types are spaces considered after prime-2 localization, defined by algebraic invariants π₀, π₁, and π₂.
• Models including double groupoids, Θ₂-sets, and chain coalgebras provide computational frameworks and Quillen equivalences for these 2-local structures.
• Applications range from computing manifold homotopy groups to classifying gauge groups via algebraic formulations and Adams–cobar constructions.

A 2-local homotopy type is a homotopy type considered after localization at the prime 2, i.e., in a category where all spaces and maps are replaced by their 2-localizations. The paper of 2-local homotopy types is central to understanding the structure of spaces and classifying spaces up to equivalence with respect to invariants sensitive only to the prime 2. This notion interacts deeply with algebraic models for low-dimensional truncated homotopy types, especially homotopy 2-types—spaces whose homotopy groups vanish above dimension 2—and the corresponding algebraic and model-categorical frameworks that can describe them.

1. Algebraic Models for Homotopy 2-Types

The algebraic modeling of homotopy 2-types is achieved using double groupoids equipped with a filling condition. A double groupoid consists of a set of objects, horizontal and vertical groupoids on the objects, and a set of squares carrying both horizontal and vertical groupoid structures, all subject to compatibility axioms (identities, associativity, interchange law). The key structural property is the filling condition: every horn in the double nerve must admit a filler, corresponding to the ability to complete certain corner configurations with a square. This ensures that the geometric realization of the diagonal of the double nerve is a Kan complex, and that the higher homotopy groups vanish above dimension 2 (Cegarra et al., 2010).

The algebraic homotopy groups associated to such a double groupoid $G$ are defined purely algebraically:

• $\pi_0(G)$: connected components.
• $\pi_1(G,a)$: classes of certain (horizontal, vertical) pairs modulo sliding along squares.
• $\pi_2(G,a)$: squares with all four sides identity, with abelian group structure induced by horizontal/vertical pasting.
• $\pi_n(G)=0$ for $n\geq3$.

The classifying space $|G|$ of the double groupoid recovers these homotopy groups: $\pi_i(G,a)\cong\pi_i(|G|,|a|)$, providing a strict algebraic model for any homotopy 2-type.

2. Simplicial and Model-Categorical Approaches

Homotopy 2-types can also be modeled via presheaves on the category $\Theta_2$—so-called $\Theta_2$-sets—which generalize the nerve construction for higher categories. In this context, "groupoidal 2-quasi-categories" are defined as fibrant objects in a model structure with cofibrations as monomorphisms, and fibrant objects satisfying both "inner horn-filling" and invertibility conditions in both the underlying quasi-category and the mapping spaces (Brittes, 2022). The Quillen equivalence between this model structure and the Kan–Quillen model structure on simplicial sets establishes that 2-truncated groupoidal 2-quasi-categories model homotopy 2-types. These models recover invariants as follows:

• $\pi_0$ and $\pi_1$ are obtained from the underlying Kan complex and loop spaces,
• $\pi_2$ is identified with the first homotopy group of the loop space in the mapping simplicial sets.

3. 2-Local Homotopy Types Via Chain Coalgebras

The 2-local homotopy type of a connected space $X$ is completely determined by its simplicial cocommutative $\mathbb{F}_2$-coalgebra of singular chains under Adams–cobar equivalence (Rivera et al., 2020). The normalized chain complex $N_*(X; \mathbb{F}_2)$ with Alexander–Whitney coproduct provides a strictly cocommutative and coassociative coalgebra.

The Adams–cobar construction $\Omega N_*(X; \mathbb{F}_2)$ produces a differential graded algebra whose zeroth homology $H_0$ is a Hopf algebra isomorphic to the group algebra $\mathbb{F}_2[\pi_1(X)_{(2)}]$, with elements characterized by satisfying the quadratic coproduct identity $\nabla(\alpha)=\alpha\otimes\alpha$. The universal cover and homology with local coefficients are likewise computed purely algebraically via twisted tensor products and bar constructions over $\mathbb{F}_2[\pi_1(X)_{(2)}]$. The strict cocommutativity at $p=2$ eliminates sign ambiguities encountered in odd characteristic, making the determination of the 2-local homotopy type especially transparent.

4. Explicit Computations and Applications: Manifolds and Gauge Groups

For an $(n-1)$-connected $(2n+1)$-manifold $M$ with $H_n(M)\cong \mathbb{Z}^r \oplus G$ (finite torsion $G$), the 2-local homotopy groups and loop space are expressed as explicit direct sums and products of sphere and Moore space factors once $2$ is not a divisor of the torsion in $G$:

• $$\pi_k(M)_{(2)} \cong \bigoplus_{i}\pi_k(S^{m_i})_{(2)}$$ with multiplicities and degrees encoded by generating functions of the free rank $r$ (Basu, 2018).
• If $G$ has 2-torsion, $\Omega M_{(2)}$ acquires a Moore space factor for the 2-primary part.

For gauge groups, the 2-local homotopy type of the gauge group $\mathcal{G}_k$ of $G_2$-bundles over $S^4$ is classified by the residue class of $k$ modulo $4$, that is, $$\mathcal{G}_k \simeq_{(2)} \mathcal{G}_{k'} \iff (k,4) = (k',4)$$ (Kameko, 7 Dec 2025). The invariant here is ultimately determined by the 2-primary order of the Samelson product $\langle i_3,1\rangle \in [\Sigma^3G_2,G_2]_{(2)}$, which is shown to be exactly $4$. This yields precisely four 2-local types distinguished by $k\bmod4$.

5. Computational and Structural Corollaries

Double groupoids with filling condition, groupoidal 2-quasi-categories, and chain coalgebras provide not only classification but computational access to 2-local invariants. Specifically, 2-local algebraic van Kampen theorems hold for the double groupoid model, and all relevant invariants are encoded in finite horn-filling data in degree up to $2$. These algebraic models are strictly equivalent as categories (up to weak equivalence) to the category of 2-local homotopy 2-types.

The following table summarizes the main models:

Framework	Algebraic Object	2-Local/2-Type Invariants Encoded
Double groupoids	Double groupoid with filling condition	$\pi_0$, $\pi_1$, $\pi_2$ algebraically
$\Theta_2$-set models	Groupoidal 2-quasi-category	$\pi_0$, $\pi_1$, $\pi_2$ from mapping
Chain coalgebra models	$\mathbb{F}_2$-chain coalgebra, cobar functor	$\pi_1$ from quadratic equation, homology

Each of these frameworks yields explicit computations and structure theorems for 2-local homotopy types.

6. Connections, Generalizations, and Context

Double groupoids satisfying the filling condition generalize the notion of crossed modules and strict 2-groupoids: these sit as special cases or consequences in the broader modeling frameworks. There exist explicit reflector functors connecting bisimplicial nerves to double groupoids and adjunctions with crossed module constructions, providing algebraic alternatives for categorical van Kampen theorems and explicit Postnikov truncations.

Groupoidal 2-quasi-categories fit into the general context of quasi-category theory, with 2-type components extracted via localizations at appropriate boundaries in the model structure, and are Quillen equivalent to classical models of 2-types in both Kan-complex and bicategory language (Brittes, 2022).

Chain coalgebra approaches—in particular for the prime 2—resemble rational homotopy theory but exploit the simplification that occurs at $p=2$ (full cocommutativity/invariance under chain-level symmetry), with subtle differences for odd primes due to additional $E_\infty$-structure data (Rivera et al., 2020).

This framework for 2-local homotopy types thus mediates between explicit algebraic models, model-categorical localizations, and homological algebra, providing classification, computation, and further structural results for spaces at the prime 2.