Cospan of Differential Graded Manifolds

• Cospan of differential graded manifolds is a diagram X → Z ← Y that encodes derived intersections via homotopy pullbacks and captures gluing phenomena.
• The topic unites models such as graded locally ringed spaces, dg C∞-algebras, and curved L∞[1]-algebras to represent both geometric and algebraic structures in derived differential geometry.
• Homotopy-theoretic frameworks and path object constructions enable the replacement of strict pushouts with derived methods, ensuring the preservation of higher intersection data.

A cospan of differential graded manifolds is a diagram

$X \xrightarrow{f} Z \xleftarrow{g} Y$

in a category of dg-manifolds, $Q$-manifolds, or an equivalent derived-differential model. In the literature represented here, such diagrams are understood through several mutually compatible formalisms: graded locally ringed spaces equipped with a degree-$+1$ homological vector field, dg $C^\infty$-algebras taken contravariantly, and bundles of curved $L_\infty[1]$-algebras of finite positive amplitude. The central structural issue is that strict pushouts in a geometric category are often inadequate or unavailable, whereas homotopy pullbacks and homotopy pushouts encode the derived intersection and gluing phenomena that cospans are meant to capture (Carchedi et al., 2012, Carchedi, 2023).

1. Differential graded manifolds as the ambient objects

A $\mathbb Z$-graded manifold is a graded locally ringed space locally modeled on

$$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$

where $\overline{\mathbf S\mathcal W}$ is a completed graded symmetric algebra. The completion is essential because even nonzero degree coordinates need not be nilpotent, and without completion the stalks are not local graded rings in the required sense (Fairon, 2015, Vysoky, 2021). Morphisms are morphisms of graded locally ringed spaces, local on stalks.

A differential graded manifold, or $Q$-manifold, is a graded manifold equipped with a homological vector field $Q$ of degree $Q$0 satisfying

$Q$1

Equivalently, $Q$2 is a degree-$Q$3 derivation of the structure sheaf whose square vanishes, so the sheaf of functions becomes a sheaf of differential graded commutative algebras (Fairon, 2015). Morphisms of $Q$4-manifolds are graded-manifold morphisms $Q$5 such that

$Q$6

A second model uses bundles of curved $Q$7-algebras of finite positive amplitude. A derived manifold in this sense is a triple $Q$8, where $Q$9 is a finite-dimensional graded vector bundle over $+1$0 and $+1$1 makes each fiber a curved $+1$2-algebra. This model is equivalent to dg manifolds of positive amplitude (Behrend et al., 2020, Behrend et al., 2023).

A third model is algebraic: for $+1$3, the opposite of the category of differential graded $+1$4-algebras contains the category of differential graded manifolds as a full subcategory (Carchedi et al., 2012). In this picture, smooth manifolds sit inside the theory as discrete dg $+1$5-algebras concentrated in degree $+1$6.

2. Cospans and their algebraic duals

The most basic description of a cospan of dg-manifolds is the geometric diagram

$+1$7

Under the contravariant dg $+1$8-algebra model, this becomes the span

$+1$9

This duality is the standard translation used to study cospans homologically (Carchedi et al., 2012).

When a strict pushout exists in the geometric category, it is represented dually by the pullback dg algebra

$C^\infty$0

More importantly, the homotopy pushout of dg-manifolds is represented by the homotopy pullback of the corresponding span of dg $C^\infty$1-algebras: $C^\infty$2 This is the derived version of cospan composition and of derived intersection (Carchedi et al., 2012).

At the $C^\infty$3-categorical level, dg-manifolds and derived manifolds become equivalent after localization at weak equivalences: $C^\infty$4 Under this equivalence, a cospan in dg-manifolds corresponds to a cospan in derived manifolds, and its pullback in the $C^\infty$5-category is computed by a homotopy pullback in the dg-manifold model. Dually, the algebra of functions on that pullback is a homotopy pushout of dg $C^\infty$6-algebras (Carchedi, 2023).

3. Local geometry, curvature, and strict existence issues

The local theory shows why cospans of dg-manifolds cannot be treated purely formally. In the graded-manifold setting, fiber products and submanifolds require genuine geometric control, and strict categorical limits are not automatic. One source states explicitly that the category of graded manifolds and $C^\infty$7-manifolds is established, but that categorical limits, fiber products, and pushouts are outside its scope (Fairon, 2015). Another source proves that products exist and that fiber products of graded manifolds exist under transversality hypotheses, via inverse images of the diagonal (Vysoky, 2021).

A further local refinement is furnished by normal-form theory for $C^\infty$8-graded $C^\infty$9-manifolds. Such a $L_\infty[1]$0-manifold has a canonical curvature $L_\infty[1]$1. On any open set where $L_\infty[1]$2 is nowhere vanishing, there is a splitting in which

$L_\infty[1]$3

and the cohomology of $L_\infty[1]$4 vanishes. This means that, away from the zero locus of the curvature, the dg structure is locally contractible. The paper further constructs the zero-locus DGA and, when the negative part is of Koszul–Tate type, shows that the relevant structure is concentrated along the zero locus of $L_\infty[1]$5 (Kotov et al., 2022). This suggests that, up to weak equivalence, many cospan constructions are governed by derived data supported near classical or zero-curvature loci rather than by the entire ambient graded space.

The resulting misconception to avoid is that an arbitrary cospan of graded or dg manifolds should admit a strict pushout in the ordinary geometric category. The literature here instead points toward derived or homotopical replacements whenever transversality or strict representability fails (Fairon, 2015, Vysoky, 2021).

4. Homotopy-theoretic frameworks for cospans

The algebraic model of dg $L_\infty[1]$6-algebras carries a Quillen model structure whenever the underlying super Fermat theory admits integration. In the smooth case, weak equivalences are quasi-isomorphisms and fibrations are surjective maps of dg $L_\infty[1]$7-algebras. The same framework supports simplicial enrichment by differential forms on simplices (Carchedi et al., 2012).

For dg manifolds of finite positive amplitude, the homotopy theory is formulated geometrically: these objects form a category of fibrant objects. In that structure, a morphism is a fibration when the base map is a submersion and the linear part on graded fibers is degreewise surjective; a morphism is a weak equivalence when it induces a bijection on classical loci and quasi-isomorphisms on tangent complexes at classical points (Behrend et al., 2020, Behrend et al., 2023).

The decisive ingredient is factorization of the diagonal by path objects. For each dg manifold $L_\infty[1]$8, there is a factorization

$L_\infty[1]$9

with the first map a weak equivalence and the second a fibration. In the positive-amplitude model, $\mathbb Z$0 is constructed from actual path spaces and then reduced to finite dimensions using the Fiorenza–Manetti homotopy transfer method for curved $\mathbb Z$1-algebras (Behrend et al., 2020, Behrend et al., 2023).

Once path objects exist, homotopy fiber products are defined in the usual Brown sense. For a cospan $\mathbb Z$2, the homotopy pullback is modeled by

$\mathbb Z$3

and its image in the localized $\mathbb Z$4-category is the pullback of the cospan. This is the homotopically correct operation underlying cospan composition in the derived setting (Behrend et al., 2023, Carchedi, 2023).

5. Derived intersections as canonical examples

Derived intersections are the model example of cospans of dg-manifolds. If $\mathbb Z$5 and $\mathbb Z$6 are submanifolds of a smooth manifold $\mathbb Z$7, regarded as derived manifolds with trivial higher structure, their derived intersection is the homotopy fiber product

$\mathbb Z$8

Using a path object of $\mathbb Z$9, one obtains a quasi-smooth derived manifold whose classical locus is $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$0 and whose virtual dimension is

$$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$1

When $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$2 and $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$3 are transverse, the derived intersection is equivalent to the classical intersection (Behrend et al., 2020, Behrend et al., 2023).

In dg $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$4-algebra language, the same construction is expressed by a homotopy pullback of algebras. A typical model is a Koszul dg algebra arising from a span

$$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$5

whose homotopy pullback encodes the derived intersection of $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$6 and $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$7 inside $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$8 (Carchedi et al., 2012).

Derived zero loci of sections give another canonical class. In the positive-amplitude theory, a quasi-smooth derived manifold of amplitude $$\mathcal C^\infty(U)\otimes \overline{\mathbf S\mathcal W},$$9 is simply a triple $\overline{\mathbf S\mathcal W}$0 where $\overline{\mathbf S\mathcal W}$1, and it is interpreted as a derived zero locus. The path-space machinery shows that these zero loci are weakly equivalent to suitable derived intersections of the section with the zero section (Behrend et al., 2020, Behrend et al., 2023).

These examples clarify a second common misconception: the role of a cospan apex is not merely to mediate ordinary gluing. In derived differential geometry, the apex may encode excess intersection data, higher obstructions, or nontransverse behavior, and those features are detected by tangent complexes and quasi-isomorphism classes rather than by ordinary manifold structure alone (Behrend et al., 2020).

6. Higher-categorical and linear enrichments

The homotopy theory of dg $\overline{\mathbf S\mathcal W}$2-algebras is simplicially enriched by differential forms on simplices: $\overline{\mathbf S\mathcal W}$3 For cofibrant-fibrant objects this mapping simplicial set is a Kan complex (Carchedi et al., 2012). This provides the mapping spaces needed to treat cospans up to homotopy rather than as rigid 1-categorical diagrams.

Several classes of algebraic structures are already encoded by $\overline{\mathbf S\mathcal W}$4-manifolds: Lie algebras, $\overline{\mathbf S\mathcal W}$5-algebras, Lie algebroids, and Lie $\overline{\mathbf S\mathcal W}$6-algebroids appear as homological vector fields on suitable graded manifolds (Fairon, 2015). A cospan of dg-manifolds can therefore function simultaneously as a cospan of geometric objects and as a correspondence between higher algebraic structures.

The linear theory over $\overline{\mathbf S\mathcal W}$7-manifolds sharpens this picture. Differential graded modules over a $\overline{\mathbf S\mathcal W}$8-manifold, adjoint and coadjoint modules, and VB-Lie $\overline{\mathbf S\mathcal W}$9-algebroids furnish linear data that can be transported along $Q$0-morphisms. In particular, there is an equivalence between the category of VB-Lie $Q$1-algebroids over a Lie $Q$2-algebroid $Q$3 and the category of $Q$4-term representations up to homotopy of $Q$5 (Papantonis, 2021). This suggests that cospans of dg-manifolds may be enhanced by compatible dg-modules, representations up to homotopy, or VB-structures on their legs and apexes.

At the level of abstract cospan calculus, double-category theory shows that a genuine double category of cospans requires pushouts in the underlying object category, while richer adjunctions between a double category $Q$6 and $Q$7 are characterized by companions, conjoints, and $Q$8-cotabulators (Niefield, 2012). A plausible implication is that any strict double-category theory of cospans of dg-manifolds must either restrict to a setting with suitable strict pushouts or pass to a homotopical or $Q$9-categorical enhancement where homotopy pushouts replace strict ones.

In this sense, the modern theory of cospans of differential graded manifolds is less a single category than a family of equivalent derived descriptions. The geometric diagram $Q$0, the opposite span of dg $Q$1-algebras, the path-object construction of homotopy pullbacks, and the $Q$2-bundle model of positive-amplitude dg-manifolds all describe the same phenomenon: gluing in differential geometry becomes mathematically stable only after passing from strict intersections and strict pushouts to homotopy-correct, derived constructions.