nLab recognition of relative loop spaces

Idea

The recognition principle for finite relative loop spaces states that, for $1 \leq N \lt \infty$, a pair of pointed topological spaces is of the homotopy type of a relative $N$-loop space pair of some $(N-2,N-1)$-connected inclusion map if and only if it is a grouplike algebra over the Swiss cheese operad $\mathcal{SC}_N$, in the sense of an algebra over an operad.

The recognition principle for infinite relative loop spaces states that a pair of pointed topological spaces is of the homotopy type of a relative $\infty$-loop space pair of some 1-shifted morphism of connective spectra if and only if it is a grouplike algebra over $\mathcal{SC}_\infty\coloneqq \colim_{N\to\infty}\mathcal{SC}_N$.

These are relative versions of May recognition theorem (May 72, May 74) of loop spaces.

The recognition principle for relative $N$-loop spaces was proved in (Vieira 2020) for $3\leq N\leq \infty$. The proof there works in the cases $N=1,2$ if we assume the spaces connected. The general case for $N=1$ was proved in (Hoefel, Livernet, Stasheff 2016). The unconnected case for $N=2$ remains open.

Notoriously the loop space functors that appear in these recognition theorems are not right adjoint, thus they cannot be proved as an equivalence of homotopy categories induced by a Quillen equivalence. May’s original strategy is to define what essentially constitutes adjunction units up to resolutions induced by the two-sided bar construction. The strategy of (Vieira 2020) is to define a generalization of Quillen adjunction that allow for the units and counits to exist only up to functorially determined resolutions, providing a natural model theoretical axiomatization of the essential elements of May’s original proof. For the full proof, homotopical versions of the concepts of idempotent monad and idempotent adjunction require similar generalizations. This is because we are interested in the homotopy subcategories of grouplike algebras and $(N-2,N-1)$-connected spaces/connective spectra.

Bellow the proof of the relative recognition of $N$-loop spaces for $3\leq N\leq \infty$ using the machinery of idempotent quasiadjunctions in (Vieira 2020) is sketched.

In (Vieira 2023) idempotent quasiadjunctions were used to prove a recognition theorem of $\infty$-loop pairs of commutative algebra spectra over commutative ring spectra. There it is shown that the multiplicative structure induced by the linear isometries operad is compatible with the model theoretical machinery presented here.

Idempotent weak Quillen quasiadjunctions

Weak Quillen quasiadjunction

To construct the unit and counit natural transformations of an adjuction between homotopy categories of model categories it suffices to construct a unit natural span and counit natural cospan at the model categories level, plus some additional compatibility conditions.

Definition

(Vieira 2020, Definition 2.1.1) Let $\mathcal{T}$ and $\mathcal{A}$ be model categories. A weak Quillen quasiadjunction, or just quasiadjunction, between $\mathcal{T}$ and $\mathcal{A}$, denoted by

$$(S \dashv_{\ \mathscr{C},\mathscr{F}}\Lambda):\mathcal{T}\rightleftharpoons \mathcal{A},$$

is a quadruple of functors

$$\mathscr{C}:\mathcal{T} \to \mathcal{T} ,\quad \mathscr{F}:\mathcal{A} \to \mathcal{A} ,\quad S:\mathcal{T} \to \mathcal{A} ,\quad \Lambda:\mathcal{A} \to \mathcal{T}$$

with $S$ the left quasiadjoint and $\Lambda$ the right quasiadjoint, equipped with a natural span in $\mathcal{T}$ and a natural cospan in $\mathcal{A}$

$$\eta':\mathscr{C}\xRightarrow{\sim} Id_{\mathcal{T}} ,\quad \eta:\mathscr{C}\Rightarrow \Lambda S ,\qquad \epsilon:S \Lambda \Rightarrow \mathscr{F}, \quad \epsilon':Id_{\mathcal{A}}\xRightarrow{\sim}\mathscr{F}$$

such that

(i) $S$ is left derivable;

(ii) $\Lambda$ is right derivable;

(iii) $\mathscr{C}$ and $\mathscr{F}$ preserve cofibrant and fibrant objects;

(iv) $\eta'$ and $\epsilon'$ are natural weak equivalences;

(v) If $X\in\mathcal{T}$ is cofibrant then $\epsilon_{SX}S\eta_X\simeq \epsilon'_{SX}S\eta_X'$;

(vi) If $Y\in\mathcal{A}$ is fibrant then $\Lambda\epsilon_Y \eta_{\Lambda Y}\simeq \Lambda\epsilon'_Y\eta'_{\Lambda Y}$.

If $\mathscr{C}$, $\mathscr{F}$, $\eta'$ and $\epsilon'$ are all identities and the homotopy equivalences in (v) and (vi) are equalities we recover the notion of a weak Quillen adjunction. If we further require $S$ to preserve cofibrations and $\Lambda$ to preserve fibrations we get a Quillen adjunction. Though the above information is weaker than that of a Quillen adjunction it suffices to construct an adjunction at the homotopy categories level.

Theorem

(Vieira 2020, Theorem 2.1.2) A quasiadjunction induces an adjunction

$$(\mathbb{L}S \dashv\mathbb{R}\Lambda):\mathcal{H}o \mathcal{T}\rightleftharpoons \mathcal{H}o \mathcal{A}$$

between the homotopy categories.

Idempotent quasi(co)monads

The following generalization of Quillen idempotent monads (see Bousfield-Friedlander theorem) was also introduced following the same principle of only requiring the existence of a unit natural span, and they also induce Bousfield localizations.

Definition

(Vieira 2020, Definition 2.3.1) Let $\mathcal{T}$ be a right proper model category. A Quillen idempotent quasimonad on $\mathcal{T}$, or simply an idempotent quasimonad, is a pair of endofunctors $Q,\overline{\mathscr{C}}:\mathcal{T}\rightarrow\mathcal{T}$ equipped with a natural span

$$\eta':\overline{\mathscr{C}}\xRightarrow{\sim}Id_{\mathcal{T}} ,\qquad \eta:\overline{\mathscr{C}}\Rightarrow Q$$

such that:

(i) $\eta'$ is a natural weak equivalence;

(ii) $Q$ preserves weak equivalences;

(iii) $Q\eta$ and $\eta_{Q}$ are natural weak equivalences;

(iv) If $f\in\mathcal{T}(X,B)$, $p\in F(E,B)$ and $\eta_E,\eta_B,Qf\in W$ then $Q(f^\ast p)\in W$;

(v) If $\iota\in C(\overline{\mathscr{C}} X,K)$ then $\iota_\ast \eta'\in W$.

Theorem

(Vieira 2020, Theorem 2.3.5 / Proposition 2.3.6) An idempotent quasimonad induces a left Bousfield localization $\mathcal{T}_Q=(\mathcal{T};W_Q,C_Q,F_Q)$ with $W_Q=Q^{-1}W$, $C_Q=C$ and $F_Q\subset F$ the subset composed of the fibrations $(p:E\twoheadrightarrow B)\in F$ such that

is a homotopy pullback.

The resulting homotopy category is the reflective subcategory

$$\mathcal{H}o \mathcal{T}_Q:=\{X\in \mathcal{H}o \mathcal{T}\mid (i_X:X\rightarrow X\sqcup_{\overline{\mathscr{C}} X}QX)\in W\}$$

of $Q$-fibrant objects.

The above definition can be dualized. The resulting idempotent quasicomonads induce right Bousfield localizations and associated coreflective homotopy subcategories.

Idempotent quasiadjunctions

A quasiadjunction $(S\dashv_{\ \mathscr{C},\mathscr{F}}\Lambda):\mathcal{T}\rightleftharpoons \mathcal{A}$ induces the following natural span on $\mathcal{T}$ and natural cospan on $\mathcal{A}$

$$cof \eta'_{Cof}:\mathscr{C} Cof\xRightarrow{\sim} Id_{\mathcal{T}}, \quad (\Lambda fib_S \eta)_{Cof}:\mathscr{C} Cof\Rightarrow \Lambda Fib S Cof$$

$$(\epsilon S cof_{\Lambda})_{Fib}: S Cof \Lambda Fib\Rightarrow \mathscr{F} Fib, \quad \epsilon'_{Fib}fib:Id_{\mathcal{A}}\xRightarrow{\sim} \mathscr{F} Fib$$

where $cof : Cof \xRightarrow{\sim} Id$ and $fib : Id \xRightarrow{\sim} Fib$ are the functorial (co)fibrant resolutions of the model structures.

Definition

(Vieira 2020, Definition 2.3.7) An idempotent quasiadjunction is a quasiadjunction such that the induced span and cospan are respectively an idempotent quasimonad and an idempotent quasicomonad.

Theorem

(Vieira 2020, Theorem 2.3.8) An idempotent quasiadjunction induces an equivalence between the associated (co)reflective homotopy subcategories.

$$(\mathbb{L}S\dashv\mathbb{R}\Lambda): \mathcal{H}o \mathcal{T}_{\Lambda Fib S Cof} \leftrightharpoons \mathcal{H}o \mathcal{A}_{S Cof \Lambda Fib}.$$

Finite relative recognition theorem

In this section fix some $1\leq N\lt \infty$. Let $\mathcal{SC}_N$ be the cubical Swiss cheese relative operad, a colored operad over the colors $\{c, o\}$, respectively referred to as the closed and open colors. It induces a monad $SC_N$ on the category $Top_*^{\{c, o\}}$ of pairs of pointed topological spaces (see algebra over an operad). We denote pairs of pointed spaces as $X=(X_{c},X_{o})$. Denote by $\mathcal{SC}_N[Top]$ the category of algebras over $SC_N$, which we equip with the mixed model structure transferred from the one on $Top_*^{\{c, o\}}$.

Denote by $Top_*^\rightarrow$ the category of pointed maps, equipped with the mixed projective model structure. We denote maps as $Y=(Y:Y_0\rightarrow Y_1)$.

Definition

The relative $N$-loop pair functor is

$$\Omega^N_2:Top_*^{\rightarrow}\rightarrow Top_*^{\{c,o\}},\qquad \Omega^N_2(Y)\coloneqq(Y_1^{\mathbb{S}^N},(Y_0\times_{Y_1} Y_1^I)^{\mathbb{S}^{N-1}}).$$

The relative $N$-suspension functor is

$$\Sigma^N_\to : Top_*^{\{c,o\}}\to Top_*^\to, \qquad \Sigma^N_\to(X)\coloneqq (X_{o} \wedge \mathbb{S}^{N-1} \to ((X_o \wedge I) \vee(X_c \wedge \mathbb{S}^1)) \wedge \mathbb{S}^{N-1} ).$$

Remark

See relative loop space for details on why we use this definition for relative loop spaces. Note that the above functor outputs both a relative loop space and a loop space of the total space, since the structure of interest here includes the action of $Y_1^{\mathbb{S}^N}$ on $(Y_0\times_{Y_1} Y_1^I)^{\mathbb{S}^{N-1}}$.

Theorem

(Vieira 2020, Proposition 2.2.3) We have a weak Quillen adjunction

$$(\Sigma^N_\to \dashv \Omega^N_2):Top_*^{\{c,o\}}\leftrightharpoons Top_*^\to.$$

This adjunction transfers a new model structure on $Top_*^\to$, with weak equivalences the commutative squares $f\in Top_*^\to(X,Y)$ such that $f_{0,*}:\pi_q X_0\to \pi_q Y_0$ are isomorphisms for all $q\geq N$ and $(f_0,f_1^I)_*:\pi_q(X_0\times_{X_1}X_1^I)\to \pi_q(Y_0\times_{Y_1}Y_1^I)$ are isomorphisms for all $q\geq N-1$. All objects of $Top_*^\to$ are fibrant in this model structure. We will say a pointed map $Y\in Top_*^\to$ is $(N-2,N-1)$-connected if $Y_0$ is $(N-2)$-connected and $Y_1$ is $(N-1)$-connected. The cofibrant objects are the $Y\in Top_*^\to$ that are homotopy equivalent to $(N-2,N-1)$-connected inclusions of relative CW-pairs. We denote the category of pointed maps equipped with this model structure as $Top^\to_{N-2,N-1}$.

The images of $\Omega^N_2$ are naturally algebras over $\mathcal{SC}_N$, so we have an induced functor $\Omega^N_2:Top_*^\to\rightarrow \mathcal{SC}_N[Top]$. This functor is not a right adjoint, but it does have a left weak Quillen quasiadjoint induced by the two-sided bar construction.

Definition

The relative $N$-delooping functor is

$$B^N_\to:\mathcal{SC}_N[Top]\to Top_*^\to,\qquad B^N_\to(X)\coloneqq B(\Sigma^N_\to,SC_N,X).$$

Definition

The resolution of $\mathcal{SC}_N$-spaces functor is

$$\overline{B}_2:\mathcal{SC}_N[Top]\to\mathcal{SC}_N[Top],\qquad \overline{B}_2(X)=B(SC_N,SC_N,X).$$

Theorem

(Vieira 2020, Theorem 4.3.5, Theorem 4.3.8) We have an idempotent weak Quillen quasiadjunction

$$(B^N_\to\dashv_{\ \overline{B}_2,Id_{Top_*^\to}}\Omega^N_2):\mathcal{SC}_N[Top]\leftrightharpoons Top^\to.$$

If $N\geq 3$ then this quasiadjunction induces an equivalence

$$(\mathbb{L}B^N_\to\dashv\mathbb{R}\Omega^N_2):\mathcal{H}o\mathcal{SC}_N[Top]_{grp}\leftrightharpoons \mathcal{H}o Top^\to_{N-2,N-1}$$

between the homotopy subcategory of grouplike Swiss cheese algebras and the homotopy category of $(N-2,N-1)$-connected pointed maps.

Remark

In the reference Vieira 2020 a cofibrant resolution of $\mathcal{SC}_N$ is used. The fact that $\mathcal{SC}_N$ is a $\Sigma$-cofibrant operad (see model structure on operads) when we consider the mixed model structure of collections means this assumption is not necessary.

Remark

The reason the proof of the above theorem doesn’t extend to the cases $N=1,2$ is due to the fact that the constructed unit of the quasiadjunction is induced by a natural map $\alpha:SC_N\Rightarrow \Omega^N_2\Sigma^N_\to$ which is a natural pair of group completions if and only if $N\geq 3$.

Infinite relative recognition theorem

Let $Sp$ be the category of sequential spectra, and $Sp^{\nearrow}$ the category composed of pairs of spectra $Y_0,Y_1\in Sp$ equipped with a 1-shifted spectra map $Y:Y_0\to Y_1[1]$. Morphisms are pairs of spectra maps that commute with the shifted spectra maps in the appropriate sense. It admits a strict mixed model structure with weak equivalences the level-wise weak homotopy equivalences and with fibrations the level-wise Hurewicz fibrations.

Definition

The spectrification functor is

$$\widetilde{\Omega}:Sp\to Sp,\qquad \widetilde{\Omega}(Y)_\bullet\coloneqq \colim_{q\to\infty} \widetilde Y_{\bullet+q}^{\mathbb{S}^q}$$

where $\widetilde{Y}$ is a certain inclusion prespectrum constructed from $Y$, equipped with a quotient map $Y\to \widetilde{Y}$ (LMS 86, Appendix 1). If $Y$ is already an inclusion spectrum then $\widetilde Y=Y$.

This functor induces a spectrification functor

$$\widetilde{\Omega}_{\nearrow}:Sp^{\nearrow}\to Sp^{\nearrow} ,\qquad \widetilde{\Omega}_{\nearrow}(Y)\coloneqq(\widetilde{\Omega}(Y):\widetilde{\Omega}(Y_0)\to \widetilde{\Omega}(Y_1)[1]).$$

We have a natural stable weak homotopy equivalence $\epsilon':Id_{Sp^{\nearrow}}\Rightarrow\widetilde{\Omega}_{\nearrow}$ which gives a Quillen idempotent monad structure on $\widetilde{\Omega}_{\nearrow}$. The stable mixed model structure on $Sp^{\nearrow}$ is the one induced by this idempotent monad, which is the left Bousfield localization of the strict mixed model structure on the pairs of stable weak homotopy equivalences. The fibrant objects in this model structure are the Omega-spectra.

Definition

The relative base pair of spaces functor is

$$\Lambda^\infty_2:Sp^{\nearrow}\to Top_*^{\{c,o\}}, \qquad \Lambda^\infty_2(Y)\coloneqq (Y_{10},Y_{00}\times_{Y_{11}}Y_{11}^I).$$

The relative $\infty$-loop pair of spaces functor is

$$\Omega^\infty_2:=\Lambda^\infty_2\widetilde{\Omega}_{\nearrow}:Sp^{\nearrow}\to Top_*^{\{c,o\}}.$$

The images of $\Omega^\infty_2$ are naturally algebras over $\mathcal{SC}_\infty$, so we have an induced functor $\Omega^\infty_2:Sp^{\nearrow}\rightarrow \mathcal{SC}_\infty[Top]$.

Definition

The relative $\infty$-delooping functor is

$$B^\infty_\nearrow:\mathcal{SC}_\infty[Top]\to Sp_\nearrow, \qquad B^\infty_\nearrow(X)_\bullet\coloneqq B(\Sigma^{\bullet+1}_\to,SC_{\bullet+1},X).$$

Definition

The resolution of $\mathcal{SC}_\infty$-spaces functor is

$$\overline{B}_2:\mathcal{SC}_\infty[Top]\to\mathcal{SC}_\infty[Top],\qquad \overline{B}_2(X)=B(SC_\infty,SC_\infty,X).$$

Theorem

(Vieira 2020, Theorem 4.3.5, Theorem 4.3.8) We have an idempotent weak Quillen quasiadjunction

$$(B^\infty_\nearrow\dashv_{\ \overline{B}_2,\widetilde{\Omega}_\nearrow}\Omega^\infty_2):\mathcal{SC}_\infty[Top]\leftrightharpoons Sp^\nearrow$$

which induces an equivalence

$$(\mathbb{L}B^\infty_\nearrow\dashv\mathbb{R}\Omega^\infty_2):\mathcal{H}o\mathcal{SC}_\infty[Top]_{grp}\leftrightharpoons \mathcal{H}o Sp^\nearrow_{con}$$

between the homotopy category of grouplike Swiss cheese algebras and the homotopy category of shifted spectra maps between connective spectra.

Related concepts

• relative loop space

• May recognition theorem

• Quillen equivalence

References

• Eduardo Hoefel, Muriel Livernet, Jim Stasheff, $A_\infty$-actions and recognition of relative loop spaces, Topology and its Applications 206 (2016) 126-147 [arXiv:1312.7155, doi:10.1016/j.topol.2016.03.023]

• L. Gaunce Lewis, Peter May, Mark Steinberger (with contributions by J.E. McClure), Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics 1213 (1986) [pdf, doi:10.1007/BFb0075778]

• Peter May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Springer 1972 (doi:10.1007/BFb0067491, pdf)

• Peter May, $E_\infty$-Spaces, group completions, and permutative categories, New Developments in Topology, London Math. Soc. Lecture Note Series 11 (1974) (pdf)

• Renato Vasconcellos Vieira, Relative recognition principle, Algebr. Geom. Topol. 20(3): 1431-1486 (2020) [arXiv:1802.01530, doi:10.2140/agt.2020.20.1431]

• Renato Vasconcellos Vieira, Recognition of connective commutative algebra spectra

  through an idempotent quasiadjunction_, Algebr. Geom. Topol. 23(1):arXiv:2101.03052, 295-338 (2023) & doi:10.2140/agt.2023.23.295]