Let \(\mathcal{C}\) be an abelian category. The nested subspace data (or SS data) for a spectral sequence at a single grading index consists of:

1. An ambient object \(V \in \mathcal{C}\).

2. A decreasing family of subobjects \(\{ Z_r\} _{r \in \mathbb {N} \cup \{ \infty \} }\) of \(V\), called \(r\)-cycles:

   \[ V = Z_0 \supseteq Z_1 \supseteq Z_2 \supseteq \cdots \supseteq Z_\infty . \]

3. An increasing family of subobjects \(\{ B_r\} _{r \in \mathbb {N} \cup \{ \infty \} }\) of \(V\), called \(r\)-boundaries:

   \[ B_0 \subseteq B_1 \subseteq B_2 \subseteq \cdots \subseteq B_\infty . \]

4. The containment \(B_r \subseteq Z_r\) for all \(r\).

5. \(Z_\infty \) is the greatest lower bound of the finite cycles: for any subobject \(X \subseteq V\), if \(X \subseteq Z_i\) for all \(i \in \mathbb {N}\), then \(X \subseteq Z_\infty \). Together with the monotonicity of \(Z\), this gives \(Z_\infty = \bigcap _{i \in \mathbb {N}} Z_i\).

6. \(B_\infty \) is the least upper bound of the finite boundaries: for any subobject \(X \subseteq V\), if \(B_i \subseteq X\) for all \(i \in \mathbb {N}\), then \(B_\infty \subseteq X\). Together with the monotonicity of \(B\), this gives \(B_\infty = \bigcup _{i \in \mathbb {N}} B_i\).

Together these form the nested chain

The \(r\)-th page at this index is defined as \(E_r = Z_r / B_r\). The \(E_\infty \)-page is \(E_\infty = Z_\infty / B_\infty \).

A spectral sequence in an abelian category \(\mathcal{C}\) with index type \(\iota \) (typically \(\mathbb {Z}^n\)) consists of:

1. A starting page index \(r_0 \in \mathbb {Z}\).

2. For each \(r \ge r_0\) and index \(\mathbf{k} \in \iota \), an object \(E_r^{\mathbf{k}} \in \mathcal{C}\) (the \(E_r\)-page at index \(\mathbf{k}\)).

3. A differential degree function \(\mathbf{d}: \mathbb {Z} \to \iota \), and differentials

   \[ d_r^{\mathbf{k}}: E_r^{\mathbf{k}} \to E_r^{\mathbf{k} + \mathbf{d}(r)} \]

   satisfying \(d_r \circ d_r = 0\) componentwise.

4. For each index \(\mathbf{k}\), nested subspace data \((V^{\mathbf{k}}, Z_r^{\mathbf{k}}, B_r^{\mathbf{k}})\) such that \(E_r^{\mathbf{k}} = Z_r^{\mathbf{k}} / B_r^{\mathbf{k}}\) for \(r \ge r_0\).

5. An isomorphism \(H(E_r, d_r) \cong E_{r+1}\) for each \(r\). By the nesting \(B_r \subseteq B_{r+1} \subseteq Z_{r+1} \subseteq Z_r\), the differential \(d_r\) has kernel \(Z_{r+1}/B_r\) and image \(B_{r+1}/B_r\), so

   \[ H(E_r, d_r) = \frac{Z_{r+1}/B_r}{B_{r+1}/B_r} \cong Z_{r+1}/B_{r+1} = E_{r+1} \]

   by the third isomorphism theorem.

The \(E_\infty \)-page of a spectral sequence \(E\) at index \(\mathbf{k}\) is given by the nested subspace data: \(E_\infty ^{\mathbf{k}} = Z_\infty ^{\mathbf{k}} / B_\infty ^{\mathbf{k}}\). For bounded or degenerate spectral sequences, \(E_\infty = E_r\) for all sufficiently large \(r\).

A morphism of spectral sequences \(f: E \to E'\) consists of a family of maps \(\varphi _{\mathbf{k}}: V^{\mathbf{k}} \to V'^{\mathbf{k}}\) on the underlying objects at each index, preserving the cycle and boundary subobjects:

for all \(r\), and commuting with differentials on the induced page maps. A morphism induces maps on all pages \(E_r \to E'_r\) and on the \(E_\infty \)-page.

Let \(E\) be a spectral sequence with \(n\)-grading and let \(A\) be an \((n-1)\)-graded object equipped with a decreasing filtration (Definition 0.14). We say \(E\) converges to \(A\), written \(E_{r_0} \Longrightarrow A\), if there exists an isomorphism of \(n\)-graded objects

where the \(n\)-grading on \(\mathrm{gr}\, A\) is obtained from the \((n-1)\)-grading of \(A\) together with the filtration index, possibly composed with a fixed reindexing map.

Given a reindexing rule, the category of converging spectral sequences has objects \((E, A, F, \varphi )\) where \(E\) is a spectral sequence, \(A\) is a filtered graded object, and \(\varphi : E_\infty \xrightarrow {\sim } \mathrm{gr}\, A\) is the convergence isomorphism. A morphism \((E_1, A_1) \to (E_2, A_2)\) consists of:

1. A morphism of spectral sequences \(E_1 \to E_2\) (Definition 0.5), inducing maps on \(E_\infty \)-pages.

2. A filtration-preserving map \(A_1 \to A_2\).

3. Compatibility: the induced \(E_\infty \)-map and the associated graded map commute with the convergence isomorphisms.

In the setting of Definition 0.6, the component of the \(n\)-grading on \(E\) that corresponds to the original \((n-1)\)-grading of \(A\) is called the stem (or topological degree).

The component of the \(n\)-grading on \(E\) that corresponds to the filtration on \(A\) is called the filtration degree.

Suppose \(E\) converges to \(A\). Let \(x \in A\) be an element with filtration at least \(r\), i.e., \(x \in F^r A\). Let \(\bar{x}\) denote the image of \(x\) in \(\mathrm{gr}^r A = F^r A / F^{r+1} A\), and let \(y \in E_\infty \) be the element corresponding to \(\bar{x}\) under the convergence isomorphism. We say \(y\) detects \(x\).

A filtered \(n\)-graded abelian group is an \(n\)-graded abelian group \(A\) equipped with a decreasing filtration: for each index \(\mathbf{k} \in \mathbb {Z}^n\), a family of subgroups

A morphism of filtered graded abelian groups is a graded homomorphism \(f: A \to B\) that preserves the filtration: \(f(F^s A^{\mathbf{k}}) \subseteq F^s B^{\mathbf{k}}\) for all \(s\) and \(\mathbf{k}\). Filtered graded abelian groups and their morphisms form a category.

A filtered chain complex is a chain complex \((C_*, d)\) in the category of filtered graded abelian groups: the differential \(d: C_n \to C_{n-1}\) preserves the filtration, i.e., \(d(F^s C_n) \subseteq F^s C_{n-1}\) for all \(s\) and \(n\).

Given a filtered graded abelian group \((A, F)\), the associated graded is

Given a filtered chain complex \((C_*, d, F^s)\), the associated graded complex is the bigraded object \(\mathrm{gr}^s C_n\) equipped with the induced differential \(\bar{d}: \mathrm{gr}^s C_n \to \mathrm{gr}^s C_{n-1}\) (well-defined since \(d\) preserves the filtration).

A filtered chain complex \((C_*, d, F^s)\) is exhaustive if \(\bigcup _s F^s C_n = C_n\) for all \(n\).

A filtered chain complex \((C_*, d, F^s)\) is Hausdorff (or separated) if \(\bigcap _s F^s C_n = 0\) for all \(n\).

A filtered chain complex \((C_*, d, F^s)\) is bounded if for each \(n\), there exist \(a \le b\) with \(F^a C_n = C_n\) and \(F^{b+1} C_n = 0\). A bounded filtration is both exhaustive and Hausdorff.

Given a filtered chain complex \((C_*, d, F^s)\), we construct an SSData structure. The spectral sequence starts at \(E_0\) with \((n+1)\)-grading.

1. Define \(V^{s,t} = \mathrm{gr}^s C_{s+t}\), the associated graded.

2. Define \(Z_r^{s,t}\) as the image in \(\mathrm{gr}^s C_{s+t}\) of those \(x \in F^s C_{s+t}\) with \(dx \in F^{s+r} C_{s+t-1}\).

3. Define \(B_r^{s,t}\) as the image in \(\mathrm{gr}^s C_{s+t}\) of those elements of the form \(dy\) with \(y \in F^{s-r} C_{s+t+1}\).

The differential \(d_r: E_r^{s,t} \to E_r^{s+r,t-r+1}\) is induced by \(d\): for \([x] \in Z_r^{s,t}\), define \(d_r[x] = [dx] \in E_r^{s+r,t-r+1}\).

A morphism of filtered chain complexes \(f: (C_*, F) \to (D_*, G)\) is a chain map \(f: C_* \to D_*\) preserving the filtration: \(f(F^s C_n) \subseteq G^s D_n\) for all \(s, n\).

Let \(E\) be a spectral sequence. A differential \(d_r(x) = y\) is called essential if \(y \ne 0\) on the \(E_r\)-page. Equivalently, \(d_r(x) = y\) is essential if \(x\) is not a boundary and \(y\) is not zero in \(E_r\). A differential is inessential if \(y = 0\) on the \(E_r\)-page.

A differential datum in a spectral sequence \(E\) is a triple \((r, x, y)\) where \(r \ge r_0\), \(x \in E_r^{\mathbf{k}}\), and \(y \in E_r^{\mathbf{k}+\mathbf{d}(r)}\) with \(d_r(x) = y\). We say the differential datum is essential if \(y \ne 0\), and denote it \(d_r(x) = y\).

Let \(f: V_1 \to V_2\) be a morphism of converging spectral sequences, and suppose \(d_n^f(x) = y\) is an \(f\)-extension differential with \(x \in E_\infty ^{s}(V_1)\) and \(y \in E_\infty ^{s+n}(V_2)\).

1. We say that \(d_n^f(x) = y\) has a crossing that hits filtration \(p\) for some \(p \le s+n\), if there exists \(x' \in E_\infty ^{s+a}(V_1)\) with \(a {\gt} 0\) and an essential differential \(d_{n-a}^f(x') = y'\) for \(0 \ne y' \in E_\infty ^{s+a+(n-a)}(V_2)\) such that

   \[ p \le \mathrm{Fil}(y') \le \mathrm{Fil}(y) = s + n. \]

2. We say that \(d_n^f(x) = y\) has no crossing that hits the range \(\mathrm{Fil} \ge p\) if there does not exist such \(x'\) and \(y'\) with \(p \le \mathrm{Fil}(y') \le \mathrm{Fil}(y)\).

3. We say that \(d_n^f(x) = y\) has no crossing if it has no crossing that hits the range \(\mathrm{Fil} \ge \mathrm{Fil}(x) + 1 = s + 1\).

Let \(f: V_1 \to V_2\) be a morphism of converging spectral sequences, with underlying map \(f_A: A_1 \to A_2\) on the target groups. The underlying filtered chain complex of \(f\) is the two-term chain complex

equipped with the filtrations on \(A_1\) and \(A_2\) inherited from the convergence data: \(F^s C_1 = F^s A_1\) and \(F^s C_0 = F^s A_2\). Since \(f_A\) preserves the filtration, this is a filtered chain complex in the sense of Definition 0.16.

The \(f\)-extension spectral sequence (\(f\)-ESS) is the spectral sequence

obtained by applying the filtered complex construction (Theorem 0.25) to the underlying filtered chain complex of Definition 0.34. By Theorem 0.26, if the filtrations on \(A_1\) and \(A_2\) are exhaustive and Hausdorff (which holds when they are bounded), then the \(f\)-ESS converges to the homology of the underlying complex, i.e., to \(\ker (f_A) \oplus \mathrm{coker}(f_A)\).

The \(f\)-extension differential \(d_n^f: E_\infty ^{s}(V_1) \to E_\infty ^{s+n}(V_2)\) is the unique nonzero component of the \(n\)-th differential of the \(f\)-ESS (Proposition 0.37). We say there is an \(f\)-extension from \(x \in E_\infty ^{s}(V_1)\) to \(y \in E_\infty ^{s+n}(V_2)\) if \(d_n^f(x) = y\). The extension is essential if \(y\) is nontrivial on the \(E_n\)-page; otherwise it is inessential.

Let \((E, A, F)\) be a converging spectral sequence with filtration \(F^\bullet A\) bounded below. For an integer \(s\), the \(s\)-truncation of \((E, A, F)\) is the converging spectral sequence \((E^{[\le s]}, A^{[\le s]}, F^{[\le s]})\) defined as follows:

1. The truncated spectral sequence: for each filtration degree \(p\), set

   \[ V^{[\le s],p} = \begin{cases} V^p & \text{if } p \le s, \\ 0 & \text{otherwise,} \end{cases} \]

   so that \(E_r^{[\le s],p} = E_r^p\) for \(p \le s\) and \(E_r^{[\le s],p} = 0\) for \(p {\gt} s\).

2. The truncated abutment is the quotient

   \[ A^{[\le s]} = A \big/ F^{s+1} A, \]

   equipped with the induced filtration

   \[ F^{[\le s],p} A^{[\le s]} = \begin{cases} \big(F^p A + F^{s+1} A\big) \big/ F^{s+1} A & \text{if } p \le s, \\ 0 & \text{if } p {\gt} s. \end{cases} \]

   When \(F^p A = A\) (which holds for all sufficiently negative \(p\) by the bounded below assumption), this reduces to \(F^{[\le s],p} A^{[\le s]} = A^{[\le s]}\).

The truncation \((E^{[\le s]}, A^{[\le s]}, F^{[\le s]})\) is a converging spectral sequence with bounded (hence exhaustive and Hausdorff) filtration.

Let \(f: (E_1, A_1) \to (E_2, A_2)\) be a morphism of converging spectral sequences with filtrations bounded below (not necessarily bounded above). For each integer \(s\), let \(f^{[\le s]}: (E_1^{[\le s]}, A_1^{[\le s]}) \to (E_2^{[\le s]}, A_2^{[\le s]})\) denote the induced morphism of truncated converging spectral sequences. The unbounded \(f\)-extension spectral sequence is the spectral sequence (without converging data) defined as the inverse limit in the category of spectral sequences:

where each \(\mathrm{ESS}(f^{[\le s]})\) is the (bounded) extension spectral sequence of Definition 0.35. The inverse limit is well-defined as a spectral sequence by the stabilization property (Proposition 0.49).

Let \((E, A, F)\) be a converging spectral sequence with filtration bounded below. The completion of \(A\) with respect to the filtration \(F\) is the inverse limit

There is a natural map \(\iota : A \to \widehat{A}\) induced by the projections \(A \to A/F^{s+1}A\). The filtration is called complete if \(\iota \) is an isomorphism. The Hausdorff condition (Definition 0.21) ensures \(\iota \) is injective; completeness further requires \(\iota \) to be surjective.

A triangulated category is an additive category \(\mathcal{T}\) equipped with:

1. An additive autoequivalence \(\Sigma : \mathcal{T} \to \mathcal{T}\), called the shift (or suspension) functor.

2. A class of distinguished triangles \(X \to Y \to Z \to \Sigma X\), satisfying the axioms (TR1)–(TR4):

   • Every morphism \(f: X \to Y\) extends to a distinguished triangle \(X \xrightarrow {f} Y \to Z \to \Sigma X\). The triangle \(X \xrightarrow {\mathrm{id}} X \to 0 \to \Sigma X\) is distinguished.

   • A triangle \(X \to Y \to Z \to \Sigma X\) is distinguished if and only if \(Y \to Z \to \Sigma X \to \Sigma Y\) is distinguished.

   • Given a morphism of the first two terms of two distinguished triangles, there exists a (not necessarily unique) morphism of the third terms making the diagram commute.

   • The octahedral axiom.

In a triangulated category, a distinguished triangle \(X \xrightarrow {f} Y \to C_f \to \Sigma X\) is called a cofiber sequence, and \(C_f\) is the cofiber (or cone) of \(f\). By (TR1), every morphism admits a cofiber sequence.

A functorial cofiber on a triangulated category \(\mathcal{T}\) is an assignment that to each morphism \(f: X \to Y\) in \(\mathcal{T}\) associates a distinguished triangle \(X \xrightarrow {f} Y \to Cf \to \Sigma X\) that is functorial: given a commutative square

there is an induced map \(C\alpha \beta : Cf \to Cf'\) making the diagram of distinguished triangles commute.

A closed symmetric monoidal category is a category \(\mathcal{C}\) equipped with:

1. A unit object \(S\).

2. A bifunctor \((X, Y) \mapsto X \wedge Y\) (the smash product or tensor product) from \(\mathcal{C} \times \mathcal{C}\) to \(\mathcal{C}\), which is associative and commutative up to coherent natural isomorphism, with \(S \wedge X \cong X\) up to coherent natural isomorphism.

3. Function objects \(F(X, Y)\), functorial contravariantly in \(X\) and covariantly in \(Y\), satisfying the tensor-hom adjunction

   \[ [X, F(Y, Z)] \cong [X \wedge Y, Z] \]

   naturally in all three variables.

A closed symmetric tensor triangulated category is a triangulated category \(\mathcal{C}\) equipped with a closed symmetric monoidal structure (Definition 0.62) that is compatible with the triangulation:

1. The smash product preserves suspensions: there is a natural equivalence \(e_{X,Y}: \Sigma X \wedge Y \xrightarrow {\sim } \Sigma (X \wedge Y)\), compatible with the unit and associativity isomorphisms.

2. The smash product is exact: if \(X \xrightarrow {f} Y \xrightarrow {g} Z \xrightarrow {h} \Sigma X\) is an exact triangle, then for any object \(W\),

   \[ X \wedge W \xrightarrow {f \wedge 1} Y \wedge W \xrightarrow {g \wedge 1} Z \wedge W \xrightarrow {h \wedge 1} \Sigma (X \wedge W) \]

   is exact.

3. The functor \(F(X, Y)\) is exact in \(Y\); it is exact in \(X\) up to sign.

4. The smash product interacts with suspension in a graded-commutative manner: the twist map \(T: S^r \wedge S^s \to S^s \wedge S^r\) and the equivalence \(S^r \wedge S^s \simeq S^{r+s}\) satisfy \(T = (-1)^{rs}\).

This definition follows Hovey–Palmieri–Strickland  [ , Definition A.2.1.

A closed symmetric tensor triangulated category with functorial cofiber is a closed symmetric tensor triangulated category \(\mathcal{T}\) (Definition 0.63) equipped with:

1. A functorial cofiber (Definition 0.60).

2. A tensor-cofiber exchange: for any morphism \(f: X \to Y\) and any object \(W\), a natural isomorphism

   \[ C(f \wedge W) \cong C(f) \wedge W, \]

   where \(f \wedge W : X \wedge W \to Y \wedge W\) denotes the right-tensoring of \(f\) with \(W\) (i.e., \(f \otimes \mathrm{id}_W\)), compatible with the triangulated structure.

The stable homotopy category \(\mathcal{S}\) is a closed symmetric tensor triangulated category with functorial cofiber (Definition 0.64). Its objects are called spectra.

Given morphisms \(f: X \to Y\) and \(g: X \to Z\) in \(\mathcal{S}\), the pushout of the diagram \(Y \xleftarrow {f} X \xrightarrow {g} Z\) is the cofiber \(C(f, g)\) of the canonical morphism \((f, g): X \to Y \vee Z\), where \(Y \vee Z\) denotes the coproduct (wedge sum) in \(\mathcal{S}\).

The sphere spectrum \(\mathbb {S} \in \mathcal{S}\) is the unit object for the smash product.

For an integer \(n \in \mathbb {Z}\), define

the \(n\)-fold suspension of the sphere spectrum.

For a spectrum \(X \in \mathcal{S}\), the \(n\)-th homotopy group of \(X\) is

The smash product \(\wedge : \mathcal{S} \times \mathcal{S} \to \mathcal{S}\) is the symmetric monoidal product from the closed symmetric monoidal structure of \(\mathcal{S}\) (Definition 0.63), with unit \(\mathbb {S}\). It satisfies \(S^m \wedge S^n \simeq S^{m+n}\) and induces a pairing on homotopy groups:

For spectra \(X, Y \in \mathcal{S}\), the mapping spectrum \(F(X, Y)\) is the internal hom (function object) from the closed symmetric monoidal structure (Definition 0.63), adjoint to the smash product:

In particular, \(\pi _n F(X, Y) \cong [\Sigma ^n X, Y]\).

The Eilenberg–MacLane spectrum \(\mathrm{H}{\mathbb F}_2\) is the spectrum representing mod 2 ordinary cohomology. It is a commutative ring spectrum in \(\mathcal{S}\).

The mod 2 cohomology of a spectrum \(X\) is

The mod 2 homology of a spectrum \(X\) is

The full subcategory \({\mathcal S}^{fin} \subset {\mathcal S}\) of finite spectra is the smallest full subcategory containing the sphere spectrum \({\mathbb S}\) and closed under suspension, desuspension, and taking cofibers.

A spectrum \(X\) is \(n\)-connected if \(\pi _i(X) = 0\) for all \(i \le n\).

A spectrum \(X\) is bounded below if it is \(n\)-connected for some \(n \in {\mathbb Z}\).

A spectrum \(X\) is of finite type if for each \(n\), there exists a finite spectrum \(X_n \in {\mathcal S}^{fin}\) and a map \(X_n \to X\) whose cofiber is \(n\)-connected.

The Adams spectral sequence induces a natural decreasing filtration on \(\pi _* X\), called the Adams filtration. An element \(\alpha \in \pi _n X\) has Adams filtration \(\ge s\) if it lies in \(F^s \pi _n X\).

The Adams filtration extends to maps between spectra. For \(X \in {\mathcal S}^{fin}\) and \(Y\) of finite type, a map \(f \in [X, Y]\) has Adams filtration \(\mathrm{AF}(f) \ge k\) if \(f\) factors as a composition

where each \(f_i\) induces the zero map on mod 2 homology: \((f_i)_* = 0: H_*(X_{i-1}; {\mathbb F}_2) \to H_*(X_i; {\mathbb F}_2)\). The Adams filtration \(\mathrm{AF}(f)\) is the largest such \(k\).

A map \(f \in [X, Y]\) has Adams filtration \(\infty \) if \(\mathrm{AF}(f) \ge n\) for all \(n\).

The category \(h\mathrm{Syn}\) of \(\mathrm{H}{\mathbb F}_2\)-synthetic spectra is a closed symmetric monoidal triangulated category with functorial cofiber (it arises as the homotopy category of a stable model category \(\mathrm{Syn}_{\mathrm{H}{\mathbb F}_2}\)). We denote its tensor product by \(\otimes \) and its unit object by \(\mathbf{1}\).

The category \(h\mathrm{Syn}\) admits a bigraded suspension \(\Sigma ^{m,n}\) for each \((m,n) \in \mathbb {Z}^2\), which is an autoequivalence of \(h\mathrm{Syn}\) satisfying

The functor \(\Sigma ^{1,0}\) is identified with the triangulated suspension functor.

There exists a natural transformation

in \(h\mathrm{Syn}\). For a synthetic spectrum \(X\), \(\lambda _X\) denotes the component \(\Sigma ^{0,-1}X \to X\).

For \(n \ge 0\), define \(\lambda ^n_X: \Sigma ^{0,-n}X \to X\) by induction:

1. \(\lambda ^0_X = \mathrm{id}_X: X \to X\).

2. \(\lambda ^{n+1}_X = \lambda _X \circ \Sigma ^{0,-1}(\lambda ^n_X): \Sigma ^{0,-n-1}X \to \Sigma ^{0,-1}X \to X\), using the composition isomorphism \(\Sigma ^{0,-n-1} \cong \Sigma ^{0,-1}\Sigma ^{0,-n}\).

We define \(X/\lambda ^n\) as the cofiber of \(\lambda ^n_X\). In particular, \(X/\lambda ^0 = 0\) and \(X/\lambda ^1 = X/\lambda \). The cofiber construction gives a distinguished triangle

The synthetic sphere \(S^{0,0}\) is defined as the monoidal unit \(\mathbf{1}\) of \(h\mathrm{Syn}\). The bigraded spheres are

The synthetic homotopy groups of a synthetic spectrum \(X\) are

These form a bigraded abelian group with a natural \(\mathbb {Z}[\lambda ]\)-module structure (Proposition 0.106).

The \(\lambda \)-action on synthetic homotopy groups is the map

defined as follows: given \(f: S^{m,n} \to X\), the element \(\lambda \cdot f \in \pi _{m,n-1}(X)\) is the composite

where the first arrow is the biShift composition isomorphism \(S^{m,n-1} = \Sigma ^{m,n-1}\mathbf{1} \cong \Sigma ^{0,-1}\Sigma ^{m,n}\mathbf{1} = \Sigma ^{0,-1}S^{m,n}\), and \(\lambda _{S^{m,n}}: \Sigma ^{0,-1}S^{m,n} \to S^{m,n}\) is the component of \(\lambda \) at \(S^{m,n}\).

There exists an additive functor

from the homotopy category of spectra to the homotopy category of synthetic spectra.