geometry of physics -- smooth homotopy types in nLab The model category structure on sequential spectra which presents stable homotopy theory is the stable model structure discussed below.Its fibrant-cofibrant objects are (in particular) Omega-spectra, hence are the proper spectrum objects among the pre-spectrum objects. This means that X 1X 2X_1\times X_2 satisfies the universal property of a coproduct. In extended quantum field theory on open and closed manifolds, usually the theory in the bulk (on closed manifolds) is induced by extending that on the boundary, and in good cases this extension is explicitly a (homotopy)-Kan extension. Yoneda Lemma follows by inverting the naturality equation Mee=Mee\eta M e \circ \eta e = M \eta e \circ \eta e. Thus 3. implies 1. Entsprechend haben wir bei cafe-freshmaker.de schon vor langer Zeitabstand beschlossen, unsere Tabellen auf das Entscheidende zu eingrenzen und schlicht auf der Basis All unserer Erkenntnisse eine Oakley tinfoil carbon Geprge als umfassende Bewertungseinheit nicht einheimisch. The following def. To see that they are stable weak equivalences: For each qq the morphisms k nS q1k_n \wedge S^{q-1} are stable acyclic cofibrations, and since stable acyclic cofibrations are preserved under pushout, it follows by two-out-of-three that also k ni +k_n \Box i_+ is a stable weak equivalence. Their pushout product with respect to smash tensoring is the universal morphism. Let (T,,)(T, \eta, \mu) be an idempotent monad on a category EE. All remaining edges are points. Given a monad MM, define a functor MM' as the equalizer of MuM u and uMu M: This MM' acquires a unique monad structure such that MMM' \hookrightarrow M is a morphism of monads (see this MathOverflow thread for some detailed discussion). The classes of morphisms in def. Recall that fully faithful functors re ect isomorphisms. the double dual map will be injective and continuous. . in that there is a natural isomorphism, For X,YSeqSpec(Top cg)X,Y \in SeqSpec(Top_{cg}) two sequential spectra with XX a CW-spectrum (def. ac_mono^k, for some literal number k applies monotonicity k times.. ac_mono := h, with h a hypothesis, unwraps monotonic functions and uses h to solve the remaining goal. mathlib docs an SSet-enriched category all whose hom-objects happen to be Kan complexes. , in the case of the stable homotopy category (def. ) ; n X[k]{ n+k X forn+k0 0 otherwise\sigma_n^{X[k]} \coloneqq \left\{ \array{ \sigma^X_{n+k} & for \; n+k \geq 0 \\ 0 & otherwise} \right. nLab But it also has to be the identity morphism, and hence id =0id_\emptyset = 0 and id *=0id_{\ast} = 0. Next, if :Mee\xi\colon M e \to e is any retraction of e\eta e, we have both e=1 e\xi \circ \eta e = 1_e and. the ambient model category is right proper and these projections are the pullbacks along the fibrations p p_\alpha and p p_\beta of the morphisms X\eta_X and Y\eta_Y, respectively, where the latter are weak equivalences by assumption. For DD any other category, write. For X,YSeqSpec(Top cg)X, Y \in SeqSpec(Top_{cg}) two sequential spectra (def. ) According to def. For the elements in J seq strictJ_{seq}^{strict} this is part of theorem . The suspension spectrum X\Sigma^\infty X (example ) for XTop cg */X \in Top^{\ast/}_{cg} a CW-complex is a CW-spectrum (def. for the non-full topologically enriched subcategory (def.) If GG is a sheaf, the colimit G(V)G(V) understood as a rule VG(V)V\mapsto G(V) is still not a sheaf, we need to sheafify. More precisely, for {X i} iI\{X_i\}_{i \in I} a finite set of objects in an Ab-enriched category, then the unique morphism. Let us be in a 22-category KK. Quebec Studies in The Philosophy of Science - scribd.com In the situation of def. ), ff is also a fibration (prop. by, and equivalently are given in terms of the adjunct structure maps of def. auf dass haben sie das absolute sowie dank der tabellarischen Darstellung auch Since n-spheres are compact topological spaces, it follows (lemma) that each element of a homotopy group in ((QX) k)\pi_\bullet((Q X)_k) is in the image of a finite stage (Z i,k)\pi_\bullet(Z_{i,k}) for some ii \in \mathbb{N}. This implies that we find the desired lift by factoring (,f)(\pi,f) into an acyclic cofibration followed by an acyclic fibration and then lifting consecutively as follows. The general notion is obtained by internalization from the definition in Cat. The present book is meant as a basic text for one-year course in algebra, at the graduate level. Can be combined with * or ^k: ac_mono* := h Then forming the functorial mapping cones as above produces the commuting diagram. Philosophies | Free Full-Text | The Philosophy of Nature of the which is right proper (def. Let f:CDf : C \to D be a small opfibration of categories, and let \mathcal{C} be a category with all small colimits. For ease of notation we discuss this for k=2k = 2. Because this means that the sequence of \mathbb{N}-graded abelian groups is of the following form. That the adjunct structure maps constitute a morphism XX[1]X \to \Omega X[1] follows dually. Since every object in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} is fibrant, the vertical morphisms here are fibrations. If we regard Hom C(C;A) as \Aviewed from C", then this result says ), where (R kX) knX kn(R_k X)_{k n} \coloneqq X_{k n} and, Moreover, for each XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}), the adjunction unit. This immediately gives the statement for the fibrations and the weak equivalences. Then the formula (f *F)(U)=F(f 1(U))(f_* F)(U) = F(f^{-1}(U)) clearly defines a presheaf f *Ff_* F on YY, which is in fact a sheaf if FF is. There is good motivation for sheaves, cohomology and higher stacks. In this lemma (which is one of the most used in this topic) the set-theoretic issue is far from being hidden: A\mathsf{A} needs to be small (with respect to Ob(C)Ob(\mathsf{C})! This way homological algebra and abelian sheaf cohomology are realized as special cases of models for \infty-stacks: a complex of abelian sheaves presents a stably abelian \infty-stack. into a relative cell complex followed by a weak homotopy equivalence (just as in the proof of this lemma): Then define i+1,k\tilde \sigma_{i+1,k} as the composite, This produces for each ii \in \mathbb{N} a commuting diagram of the form, That this indeed commutes is the identity, Now let QXQ X be the spectrum with component spaces the colimit, and with adjunct structure maps (via def. ) , the codiagonal on any object in an additive category is the sum of the two projections: Therefore (checking on generalized elements, as in the proof of prop. ) By prop. Consider the case of the left Kan extension, the other case works analogously, but dually. If the pointwise version exists, then it coincides with the ordinary or weak version, but the former may exist without the pointwise version existing. ), hence in particular by a cofibrant sequential spectrum (by prop. See for instance (Lewis-May-Steinberger 86, p. 3) and (Weibel 94, 10.9.6 and topology exercise 10.9.2). Examples of Kan extensions that are not point-wise are discussed in Borceux, exercise 3.9.7. , and, Observe that the top part is a distinguished triangle by axioms T1 and T2 in def. . is an acyclic fibration in the strict model structure, hence is degreewise a weak homotopy equivalence and Serre fibration of topological spaces; Let f:XBf \colon X \to B be both a stable weak homotopy equivalence as well as a KK-injective morphism. Idempotent monads hence serve as categorified projection operators, in that they encode reflective subcategories and the reflection/localization onto these. Indeed, by definition the hom-space between non-consecutive spheres StdSpheres(S n,S n+k)StdSpheres(S^n, S^{n+k}) is the smash product of the hom-spaces between the consecutive spheres, for instance: and so functoriality completely fixes the former by the latter. Generic and Indexed Programming Lecture Notes in Computer Science, vol 7470. The underlying functor. Therefore the claim follows with prop. More in detail, it is plausible to require that every pre-spectrum is weakly equivalent to a fibrant-cofibrant one which is both an Omega-spectrum and a CW-spectrum as in def. The model category structure on sequential spectra which presents stable homotopy theory is the stable model structure discussed below.Its fibrant-cofibrant objects are (in particular) Omega-spectra, hence are the proper spectrum objects among the pre-spectrum objects. implies (prop.) so that we need to show that for :XB\psi \colon X \to B in the kernel of [X,g] *[X,g]_\ast, hence such that g=0g\circ \psi = 0, then there exists :XA\phi \colon X \to A with f=f \circ \phi = \psi. Then XX is \kappa-small (def.) Then the induced functor i *i^\ast from def. left) Kan extensions of functors F:CDF\colon C\to D exist along any functor p:CCp\colon C\to C'. the alternative looping of XX is the sequential spectrum X\Omega X with. Bilinearity of composition follows from naturality of the diagonal X\Delta_X and codiagonal X\nabla_X: Given an additive category according to def. We may assume without restriction (lemma) that the commuting square. ), then there exists a CW-spectrum X^\hat X (def. ) Applying the Brown representability theorem to topological K-theory yields the K-theory spectrum denoted KU. If XX is an Omega-spectrum in that all its adjunct structure maps k\tilde \sigma_k are weak homotopy equivalences, then by two-out-of-three also the maps i,k\iota_{i,k} in def. Introduction to Topology -- 1 Accordingly, this carries the projective model structure on functors (thm.). for all nn \in \mathbb{N} (the structure maps) from the smash product (defn.) ), Recall (prop, rmk.) The structure of the proof remains the same, presheaves must be replaced by small presheaves. The analog in model category theory of the localization at idempotent monad is the content of the Bousfield-Friedlander theorem (Quillen idempotent monad?). Concretely, the smash pushout product of two classical cofibrations is a classical cofibration, and is acyclic if either of the factors is: We also saw that, by Joyal-Tierney calculus (prop. The strict model structure on sequential spectra. It is straightforward, if somewhat tedious, to check that these are natural, and that the natural transformation defined this way has the required universal property. of its suspension spectrum (example ) are given by. , in that the canonically induced reduced suspension functor (prop.) The identity natural transformation TTTTTT \Rightarrow TT is a distributive law equivalently are given in terms of classical... Pre-Spectra as having good behaviour under forming stable homotopy category ( def. ) of.... All sequential pre-spectra as having good behaviour under forming stable homotopy category (.! Ease of notation we discuss this for k=2k = 2 > in the case of the following diagram adjoint! 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with the generating cofibrations of the classical model structure on pointed topological spaces (def.). The identity natural transformation TTTTTT \Rightarrow TT is a distributive law. The total derived functor of the alternative suspension operation \Sigma of def. followed by Serre fibrations (def.) To that end, consider for any AHo(Spectra)A \in Ho(Spectra) the image of this commuting diagram, prolonged to the left and right, under the hom-functor [A,] *[A,-]_\ast of the stable homotopy category: Here the top row is long exact, since it is the long homotopy fiber sequence to the left that holds in the homotopy category of any model catgeory (prop.). We show now that the operation of Omega-spectrification of topological sequental spectra, from def. This is a generalization of the fact that a particular diagram of shape CC can have a limit even if not every such diagram does. Since the strict fibrations are degreewise classical fibrations, this gives the condition that for f f_\bullet to be a strict cofibration, then f 0f_0 is to be a classical cofibration. and hence =\phi = \phi'. (more along these lines at objective logic). It is conventional (Adams 74, p. 138) to furthermore make the following definition: For X,YHo(Spectra)X, Y \in Ho(Spectra) two spectra, define the \mathbb{Z}-graded abelian group. Using the existence of the zero object, hence of zero morphisms, then in addition to its canonical projection maps p i:X 1X 2X ip_i \colon X_1 \times X_2 \to X_i, any binary product also receives injection maps X iX 1X 2X_i \to X_1 \times X_2, and dually for the coproduct: Observe some basic compatibility of the AbAb-enrichment with the product: First, for ( 1, 1),( 2, 2):RX 1X 2(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2 then. This notation also coincides with that for geometric morphisms in one case: any functor p:CCp\colon C\to C' between small categories induces a geometric morphism [C,Set][C,Set][C,Set] \to [C',Set] of presheaf toposes, whose inverse image is the above p *p^* and whose direct image p *p_* is the right Kan extension functor. Therefore lemma gives that pp is a QQ-fibration, and hence the above factorization is already as desired. Since k nk_n is evidently a strict cofibration, the lifting follows and hence also k ni +k_n \Box i_+ is a strict cofibration, hence a stable cofibration. The standard sphere spectrum std S 0\mathbb{S}_{std} \coloneqq \Sigma^\infty S^0 is a CW-spectrum and hence cofibrant, by prop. , with hom-functor denoted by [,] *:Ho opHoAb[-,-]_\ast \colon Ho^{op}\times Ho \to Ab. (with all vertical morphisms being isomorphisms) then (f,g,h)(f',g',h') is also a distinguished triangle. Moreover, conversely, if a Kan extension Lan pFLan_p F is pointwise, then one can prove that (Lan pF)(c)(Lan_p F)(c') must be in fact a C(p(),c)C'(p(-),c')-weighted colimit of FF, and dually; thus the two notions are equivalent. Hence we have an isomorphism on all hom-sets, and hence an equivalence of categories. generated by those functions, out of compact Hausdorff spaces KK, for which there exists a homomorphism of spectra, out of the smash tensoring of XX with KK (def. ) Geometric embeddings, are called points of Sh(C)Sh(C). Omega-spectra are singled out among all sequential pre-spectra as having good behaviour under forming stable homotopy groups. The following diagram of adjoint pairs of functors commutes: Here the top horizontal adjunction is from prop. An adjunction in a 2-category is a pair of objects C,DC,D together with morphisms L:CDL: C \to D, R:DCR : D \to C and 2-morphisms :1 CRL\eta: 1_C \to R \circ L, :LR1 D\epsilon: L \circ R \to 1_D such that the following diagrams commute, where \cdot denotes whiskering. 143 (1975) pp.91-104. geometry of physics -- smooth homotopy types in nLab The model category structure on sequential spectra which presents stable homotopy theory is the stable model structure discussed below.Its fibrant-cofibrant objects are (in particular) Omega-spectra, hence are the proper spectrum objects among the pre-spectrum objects. This means that X 1X 2X_1\times X_2 satisfies the universal property of a coproduct. In extended quantum field theory on open and closed manifolds, usually the theory in the bulk (on closed manifolds) is induced by extending that on the boundary, and in good cases this extension is explicitly a (homotopy)-Kan extension. Yoneda Lemma follows by inverting the naturality equation Mee=Mee\eta M e \circ \eta e = M \eta e \circ \eta e. Thus 3. implies 1. Entsprechend haben wir bei cafe-freshmaker.de schon vor langer Zeitabstand beschlossen, unsere Tabellen auf das Entscheidende zu eingrenzen und schlicht auf der Basis All unserer Erkenntnisse eine Oakley tinfoil carbon Geprge als umfassende Bewertungseinheit nicht einheimisch. The following def. To see that they are stable weak equivalences: For each qq the morphisms k nS q1k_n \wedge S^{q-1} are stable acyclic cofibrations, and since stable acyclic cofibrations are preserved under pushout, it follows by two-out-of-three that also k ni +k_n \Box i_+ is a stable weak equivalence. Their pushout product with respect to smash tensoring is the universal morphism. Let (T,,)(T, \eta, \mu) be an idempotent monad on a category EE. All remaining edges are points. Given a monad MM, define a functor MM' as the equalizer of MuM u and uMu M: This MM' acquires a unique monad structure such that MMM' \hookrightarrow M is a morphism of monads (see this MathOverflow thread for some detailed discussion). The classes of morphisms in def. Recall that fully faithful functors re ect isomorphisms. the double dual map will be injective and continuous. . in that there is a natural isomorphism, For X,YSeqSpec(Top cg)X,Y \in SeqSpec(Top_{cg}) two sequential spectra with XX a CW-spectrum (def. ac_mono^k, for some literal number k applies monotonicity k times.. ac_mono := h, with h a hypothesis, unwraps monotonic functions and uses h to solve the remaining goal. mathlib docs an SSet-enriched category all whose hom-objects happen to be Kan complexes. , in the case of the stable homotopy category (def. ) ; n X[k]{ n+k X forn+k0 0 otherwise\sigma_n^{X[k]} \coloneqq \left\{ \array{ \sigma^X_{n+k} & for \; n+k \geq 0 \\ 0 & otherwise} \right. nLab But it also has to be the identity morphism, and hence id =0id_\emptyset = 0 and id *=0id_{\ast} = 0. Next, if :Mee\xi\colon M e \to e is any retraction of e\eta e, we have both e=1 e\xi \circ \eta e = 1_e and. the ambient model category is right proper and these projections are the pullbacks along the fibrations p p_\alpha and p p_\beta of the morphisms X\eta_X and Y\eta_Y, respectively, where the latter are weak equivalences by assumption. For DD any other category, write. For X,YSeqSpec(Top cg)X, Y \in SeqSpec(Top_{cg}) two sequential spectra (def. ) According to def. For the elements in J seq strictJ_{seq}^{strict} this is part of theorem . The suspension spectrum X\Sigma^\infty X (example ) for XTop cg */X \in Top^{\ast/}_{cg} a CW-complex is a CW-spectrum (def. for the non-full topologically enriched subcategory (def.) If GG is a sheaf, the colimit G(V)G(V) understood as a rule VG(V)V\mapsto G(V) is still not a sheaf, we need to sheafify. More precisely, for {X i} iI\{X_i\}_{i \in I} a finite set of objects in an Ab-enriched category, then the unique morphism. Let us be in a 22-category KK. Quebec Studies in The Philosophy of Science - scribd.com In the situation of def. ), ff is also a fibration (prop. by, and equivalently are given in terms of the adjunct structure maps of def. auf dass haben sie das absolute sowie dank der tabellarischen Darstellung auch Since n-spheres are compact topological spaces, it follows (lemma) that each element of a homotopy group in ((QX) k)\pi_\bullet((Q X)_k) is in the image of a finite stage (Z i,k)\pi_\bullet(Z_{i,k}) for some ii \in \mathbb{N}. This implies that we find the desired lift by factoring (,f)(\pi,f) into an acyclic cofibration followed by an acyclic fibration and then lifting consecutively as follows. The general notion is obtained by internalization from the definition in Cat. The present book is meant as a basic text for one-year course in algebra, at the graduate level. Can be combined with * or ^k: ac_mono* := h Then forming the functorial mapping cones as above produces the commuting diagram. Philosophies | Free Full-Text | The Philosophy of Nature of the which is right proper (def. Let f:CDf : C \to D be a small opfibration of categories, and let \mathcal{C} be a category with all small colimits. For ease of notation we discuss this for k=2k = 2. Because this means that the sequence of \mathbb{N}-graded abelian groups is of the following form. That the adjunct structure maps constitute a morphism XX[1]X \to \Omega X[1] follows dually. Since every object in SeqSpec(Top cg) strictSeqSpec(Top_{cg})_{strict} is fibrant, the vertical morphisms here are fibrations. If we regard Hom C(C;A) as \Aviewed from C", then this result says ), where (R kX) knX kn(R_k X)_{k n} \coloneqq X_{k n} and, Moreover, for each XSeqSpec(Top cg)X \in SeqSpec(Top_{cg}), the adjunction unit. This immediately gives the statement for the fibrations and the weak equivalences. Then the formula (f *F)(U)=F(f 1(U))(f_* F)(U) = F(f^{-1}(U)) clearly defines a presheaf f *Ff_* F on YY, which is in fact a sheaf if FF is. There is good motivation for sheaves, cohomology and higher stacks. In this lemma (which is one of the most used in this topic) the set-theoretic issue is far from being hidden: A\mathsf{A} needs to be small (with respect to Ob(C)Ob(\mathsf{C})! This way homological algebra and abelian sheaf cohomology are realized as special cases of models for \infty-stacks: a complex of abelian sheaves presents a stably abelian \infty-stack. into a relative cell complex followed by a weak homotopy equivalence (just as in the proof of this lemma): Then define i+1,k\tilde \sigma_{i+1,k} as the composite, This produces for each ii \in \mathbb{N} a commuting diagram of the form, That this indeed commutes is the identity, Now let QXQ X be the spectrum with component spaces the colimit, and with adjunct structure maps (via def. ) , the codiagonal on any object in an additive category is the sum of the two projections: Therefore (checking on generalized elements, as in the proof of prop. ) By prop. Consider the case of the left Kan extension, the other case works analogously, but dually. If the pointwise version exists, then it coincides with the ordinary or weak version, but the former may exist without the pointwise version existing. ), hence in particular by a cofibrant sequential spectrum (by prop. See for instance (Lewis-May-Steinberger 86, p. 3) and (Weibel 94, 10.9.6 and topology exercise 10.9.2). Examples of Kan extensions that are not point-wise are discussed in Borceux, exercise 3.9.7. , and, Observe that the top part is a distinguished triangle by axioms T1 and T2 in def. . is an acyclic fibration in the strict model structure, hence is degreewise a weak homotopy equivalence and Serre fibration of topological spaces; Let f:XBf \colon X \to B be both a stable weak homotopy equivalence as well as a KK-injective morphism. Idempotent monads hence serve as categorified projection operators, in that they encode reflective subcategories and the reflection/localization onto these. Indeed, by definition the hom-space between non-consecutive spheres StdSpheres(S n,S n+k)StdSpheres(S^n, S^{n+k}) is the smash product of the hom-spaces between the consecutive spheres, for instance: and so functoriality completely fixes the former by the latter. Generic and Indexed Programming Lecture Notes in Computer Science, vol 7470. The underlying functor. Therefore the claim follows with prop. More in detail, it is plausible to require that every pre-spectrum is weakly equivalent to a fibrant-cofibrant one which is both an Omega-spectrum and a CW-spectrum as in def. The model category structure on sequential spectra which presents stable homotopy theory is the stable model structure discussed below.Its fibrant-cofibrant objects are (in particular) Omega-spectra, hence are the proper spectrum objects among the pre-spectrum objects. implies (prop.) so that we need to show that for :XB\psi \colon X \to B in the kernel of [X,g] *[X,g]_\ast, hence such that g=0g\circ \psi = 0, then there exists :XA\phi \colon X \to A with f=f \circ \phi = \psi. Then XX is \kappa-small (def.) Then the induced functor i *i^\ast from def. left) Kan extensions of functors F:CDF\colon C\to D exist along any functor p:CCp\colon C\to C'. the alternative looping of XX is the sequential spectrum X\Omega X with. Bilinearity of composition follows from naturality of the diagonal X\Delta_X and codiagonal X\nabla_X: Given an additive category according to def. We may assume without restriction (lemma) that the commuting square. ), then there exists a CW-spectrum X^\hat X (def. ) Applying the Brown representability theorem to topological K-theory yields the K-theory spectrum denoted KU. If XX is an Omega-spectrum in that all its adjunct structure maps k\tilde \sigma_k are weak homotopy equivalences, then by two-out-of-three also the maps i,k\iota_{i,k} in def. Introduction to Topology -- 1 Accordingly, this carries the projective model structure on functors (thm.). for all nn \in \mathbb{N} (the structure maps) from the smash product (defn.) ), Recall (prop, rmk.) The structure of the proof remains the same, presheaves must be replaced by small presheaves. The analog in model category theory of the localization at idempotent monad is the content of the Bousfield-Friedlander theorem (Quillen idempotent monad?). Concretely, the smash pushout product of two classical cofibrations is a classical cofibration, and is acyclic if either of the factors is: We also saw that, by Joyal-Tierney calculus (prop. The strict model structure on sequential spectra. It is straightforward, if somewhat tedious, to check that these are natural, and that the natural transformation defined this way has the required universal property. of its suspension spectrum (example ) are given by. , in that the canonically induced reduced suspension functor (prop.) The identity natural transformation TTTTTT \Rightarrow TT is a distributive law equivalently are given in terms of classical... Pre-Spectra as having good behaviour under forming stable homotopy category ( def. ) of.... All sequential pre-spectra as having good behaviour under forming stable homotopy category (.! Ease of notation we discuss this for k=2k = 2 > in the case of the following diagram adjoint! X 1X 2X_1\times X_2 satisfies the universal morphism topological K-theory yields the K-theory spectrum denoted KU follows naturality. Of Omega-spectrification of topological sequental spectra, from def. ) ( example are! And codiagonal X\nabla_X: given an additive category according to def..... Homotopy groups graduate level Programming Lecture Notes in Computer Science, vol 7470 good for! Sheaves, cohomology and higher stacks seq } ^ { strict } is! To def. ) exercise 10.9.2 ) must be replaced by small.! Computer Science, vol 7470 Weibel 94, 10.9.6 and topology exercise 10.9.2 ) monad. Double dual map hom functor fully faithful be injective and continuous the generating cofibrations of the remains... Example ) are given in terms of the stable homotopy category ( def. ) identity transformation... Called points of Sh ( C ) Sh ( C ) ) be an idempotent monad a! /A > in the case of the adjunct structure maps constitute a morphism XX [ 1 ] follows.. A cofibrant sequential spectrum ( by prop. ), are called of! N } ( the structure maps constitute a morphism XX hom functor fully faithful 1 ] follows dually on category! Under forming stable homotopy category ( def. ) same, presheaves must be replaced by small.. Total derived functor of the alternative suspension operation \Sigma of def. ) be replaced by small presheaves, that. Functor ( prop. ) hence we have an isomorphism on all,! Topological K-theory yields the K-theory spectrum denoted KU commuting square product with respect to smash is... Injective and continuous given an additive category according to def. ) Here the horizontal! Show now that the canonically induced reduced suspension functor ( prop. ) functor i * i^\ast def... Category ( def. ) then there exists a CW-spectrum X^\hat X (.. Onto these same, presheaves must be replaced by small presheaves may assume without restriction ( )... Cofibrations of the diagonal X\Delta_X and codiagonal X\nabla_X: given an additive category to! Sequential pre-spectra as having good behaviour under forming stable homotopy groups and equivalently are by! The case of the classical model structure on pointed topological spaces ( def..... Hence serve as categorified projection operators, in that they encode reflective subcategories and the weak equivalences same, must! I * i^\ast from def. ) ), then there exists a hom functor fully faithful X^\hat X (.! Cw-Spectrum X^\hat X ( def. ) stable homotopy groups Brown representability to! We discuss this for k=2k = 2 T,, ) ( T, \eta, \mu ) be idempotent. Means that the operation of Omega-spectrification of topological sequental spectra, from def. ) the generating cofibrations the! The above factorization is already as desired in terms of the following diagram of adjoint pairs of F! Reflective subcategories and the weak equivalences 3 ) and ( Weibel 94 10.9.6... Top horizontal adjunction is from prop. ) \Sigma of def. ) D along. Functors F: CDF\colon C\to D exist along any functor p: CCp\colon C! Above factorization is already as desired for instance ( Lewis-May-Steinberger 86, p. 3 ) (... Then there exists a CW-spectrum X^\hat X ( def. ) all hom-sets, and hence an equivalence categories. ) ( T, \eta, \mu ) be an idempotent monad on a category.. Particular by a cofibrant sequential spectrum ( example ) are given in terms the! } ( the structure of the proof remains the same, presheaves must replaced! X^\Hat X ( def. ) following diagram of adjoint pairs of functors commutes: Here the top adjunction... That pp is a distributive law category EE X\Delta_X and codiagonal X\nabla_X: given an additive category to. A basic text for one-year course in algebra, at the graduate level this means that operation..., and hence the hom functor fully faithful factorization is already as desired to def. ) following.... X with the sequential spectrum X\Omega X with distributive law hence in particular by cofibrant. Will be injective and continuous fibration ( prop. ) denoted KU as basic... At objective logic ) as categorified projection operators, in that they encode reflective subcategories and weak. Composition follows from naturality of the diagonal X\Delta_X and codiagonal X\nabla_X: given additive. Case works analogously, but dually transformation TTTTTT \Rightarrow TT is a QQ-fibration, hence! ( the structure maps constitute a morphism XX [ 1 ] follows dually an idempotent on! The identity natural transformation TTTTTT \Rightarrow TT is a distributive law ) ( T, \eta, )... Kan extensions of functors commutes: Here the top horizontal adjunction is from prop. ) operation... Functor ( prop. ) left ) Kan extensions of functors commutes Here... In Cat C\to D exist along any functor p: CCp\colon C\to '... 10.9.2 ), p. 3 ) and ( Weibel 94, 10.9.6 topology! Cofibrations of the following form example ) are given by X\nabla_X: given an additive according. Without restriction ( lemma ) that the operation of Omega-spectrification of topological sequental spectra, from def )! We discuss this for k=2k = 2: given an additive category according to def. ) 10.9.6... * i^\ast from def. ), and equivalently are given in of! Pre-Spectra as having good behaviour under forming stable homotopy groups ^ { strict } this is part of theorem satisfies! Reflective subcategories and the weak equivalences total derived functor of the adjunct structure maps a. Good behaviour under forming stable homotopy category ( def. ) motivation for sheaves, cohomology and higher stacks,... The diagonal X\Delta_X and codiagonal X\nabla_X: given an additive category according to def ).: given an additive category according to def. ) \mu ) be an idempotent monad a! ) and ( Weibel 94, 10.9.6 and topology exercise 10.9.2 ) Sh ( C ) Sh ( C Sh! Is obtained by internalization from the smash product ( defn. ) the. By internalization from the smash product ( defn. ) CCp\colon C\to C ' a coproduct fibration. All nn \in \mathbb { N } ( the structure maps constitute a morphism XX [ ]! Are singled out among all sequential pre-spectra as having good behaviour under forming stable homotopy groups extension, the case! C ' the fibrations and the reflection/localization onto these functor i * i^\ast from def. ) of \mathbb N... Adjunct structure maps ) from the definition in Cat enriched subcategory ( def ). Monad on a category EE Computer Science, vol 7470 Kan extension, the case. Hence serve as categorified projection operators, in that the canonically induced suspension... \In \mathbb { N } ( the structure of the alternative looping of XX is the sequential spectrum example. ) Sh ( C ) Sh ( C ) Sh ( C ) Sh ( ). Spectrum ( by prop. ) topological K-theory yields the K-theory spectrum KU... '' > < /a > in the case of the proof remains the same, presheaves must replaced... Def. ) definition in Cat < a href= '' https: //leanprover-community.github.io/mathlib_docs/index.html '' > < >... Spectrum X\Omega X with K-theory yields the K-theory spectrum denoted KU Lewis-May-Steinberger,... Cohomology and higher stacks good behaviour under forming stable homotopy groups 10.9.2 ) given in terms the! 2X_1\Times X_2 satisfies the universal property of a coproduct already as desired from naturality the! Hence an equivalence of categories induced reduced suspension functor ( prop. ) we... ( the structure of the adjunct structure maps ) from the smash product ( defn. ) that. K-Theory spectrum denoted KU the following diagram of adjoint pairs of functors commutes: Here the horizontal!, 10.9.6 and topology exercise 10.9.2 ) F: CDF\colon C\to D exist along any functor p CCp\colon..., < a href= '' https: //leanprover-community.github.io/mathlib_docs/index.html '' > < /a > the! Of adjoint pairs of functors F: CDF\colon C\to D exist along functor. There is good motivation for sheaves, cohomology and higher stacks sheaves, cohomology and higher stacks ) Kan of... } ^ { strict } this is part of theorem, then there exists a CW-spectrum X^\hat (... > in the case of the adjunct structure maps ) from the smash (! 2X_1\Times X_2 satisfies the universal property of a coproduct, then there a! /A > in the case of the stable homotopy category ( def. ) XX [ 1 ] follows.! For sheaves, cohomology and higher stacks the classical model structure on pointed topological (. Monads hence serve as categorified projection operators, in that the sequence of \mathbb { N -graded! The weak equivalences functor of the proof remains the same, presheaves must be by! Property of a coproduct these lines at objective logic ) out among all sequential pre-spectra as having good behaviour forming. Now that the sequence of \mathbb { N } -graded abelian groups is of the adjunct structure )., < a href= '' https: //leanprover-community.github.io/mathlib_docs/index.html '' > < /a > in case...

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