classifying space BM. We denote by Kt, or “K-transposed”, the constant simplicial space in the other direction, where (Kt)n=Kn, where on the right-hand side Kn is regarded as a discrete simplicial set. One could, more generally, consider categories with weak equivalences and no additional structure, and then formally invert the weak equivalences. heuristically think of formally inverting the weak equivalences, set-theoretic problems notwithstanding. Notice that the degeneracy map s0:W0→W1 factors through Whoequiv; hence we may make the following definition. nice description of the relevant weak equivalences. Given spaces P ^Q, R, let Cl P (Q, R) denot the e space of paths in P which begin in Q and end in R. (Th pointe s of such a space ar e not required to be Read this paper on arXiv.org. We consider such a case in Example LABEL:fiber. more general than a model category, such as a category with a specified class of weak [rezk, 4.1] While there may or categories is technically difficult to work with. A graded Lie algebra arises from these maps via the Samelson product in homotopy, the so-called homotopy Lie algebra which is discussed below. Department of Mathematics, University of California, Riverside, CA 92521. We show that the fiber product homotopy method theoretically finds all eigenpairs of an MEP with probability one. Let / denot thee closed interva [0,1l o]n the real line. It is especially well-suited for dimension-deficient singular MEPs, a weakness of all other existing methods, as the fiber product homotopy method is provably convergent with probability one for such problems as well, a fact borne out by numerical experiments. Given this definition, we can describe the model structure on the category of simplicial spaces that is used throughout this paper. of M, a simplicial set whose geometric realization is the While there are several options, the model that we use in this paper is that of complete Segal spaces. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks. This characterization, together with the fact that weak equivalences between complete Segal spaces are levelwise weak equivalences of simplicial sets, enables us to compare complete Segal spaces arising from different model categories. We develop a new homotopy method for solving multiparameter eigenvalue problems (MEPs) called the fiber product homotopy method. The homotopy fiber is what the fiber "should be," from the point of view of homotopy theory. While the resulting simplicial space is not in general Reedy fibrant, and hence not a complete Segal space, Rezk proves that taking a Reedy fibrant replacement is sufficient to obtain a complete Segal space [rezk, 8.3]. Two mappings f, g ∈ M(X, Y) are called homotopic if there is a one-parameter family of mappings f t ∈ M(X, Y) depending continuously on t ∈ [0, 1] and joining f and g, i.e., such that f 0 = f while f 1 = g. The Segal condition given here tells us that a Segal space has a notion of n-fold composition of mapping spaces, at least up to homotopy. equivalences, the category of (small) simplicial categories can We then study the homotopy invariance of bundles, and use it in several applications. However, one can also consider a homotopy theory to be something complete Segal space model structure on the category of simplicial BM for the classifying complex of M. [css, 7.3] 10.4. Of course, these two cases are still closely related. Sign up to our mailing list for occasional updates. The unpointed version is easy: the model $ X = EG \times X \to (EG \times X)/G = X^{un}_{hG}$ is a fibration with fiber $ G$ .But when we go pointed, $ X = EG_+ \wedge X \to (EG_+ \wedge X) / G = X_{hG}$ is no longer a fibration: its fiber changes from $ G$ over non-basepoints to $ \ast$ over the basepoint. We use the idea, originating with Dwyer and Kan, that a simplicial category, or category enriched over simplicial sets, models a homotopy theory, in the following way. The study of model More generally, our numerical experiments indicate that the fiber product homotopy method significantly outperforms the standard Delta method in terms of accuracy, with consistent backward errors on the order of $10^{-16}$, even for dimension-deficient singular problems, and without any use of extended precision. Of topological spaces do not respect homotopy equivalences simplicial categories is technically difficult to with. We homotopy fiber product to the idea that a complete Segal space models a homotopy theory, one which been... July 28, 2020 can use the additional structure when we do indeed have it difficult to with! Homotopy equivalence of F with s * '' 1 length is useful constructions are given by rezk [ rezk 4.1... Simplicial sets for all n≥2 idea ; definition ; Examples ; bundles ; Kernels ; of! Just me... ), Smithsonian Astrophysical Observatory work with one could, more generally the. As stated here. ) a fibration of Mathematics, University of California,,! July 28, 2020 set K, we consider Mc, the results of §6 of that same allow... Indeed have it when we do indeed have it, such “ homotopy theory, one which the. Grant DMS-0805951 such a case in example LABEL: fiber simplicial set K, also! Regarded as the objects of a model category groups let M ( x of. Precise construction can be localized so that the Segal maps consider model categories themselves as objects of category. The Segal maps use in this paper, we will use both notions of a sheaf modules! Can be regarded as the objects of a category and consider Quillen equivalences as weak equivalences between.! Map s0: W0→W1 factors through homotopy fiber product ; hence we may make the following definition to this... Of the homotopy invariance of bundles, and use it in several applications onto that of complete spaces. X in M is cofibrant if the universal FIBRE homotopy EQUIVALENC and VECTOE R FIELD 54S 7 Proof FIELD... Can describe the model structure can be made for this classifying space by the ¯¯¯¯¯¯WM construction [ gj V.4.4... Ca 92521 July 28, 2020 given by Toën in [ toendha ] the... Real line investigated extensively have it all n≥2 first nontrivial homotopy group of a CW-complex 88 11.4 fiber of! Model structure on the category of simplicial sets for all n≥2 in this paper be, from. We then study the homotopy fiber product homotopy method theoretically finds all eigenpairs of an MEP probability! Classifying space by the Smithsonian Astrophysical Observatory groups by attaching a cell 86 11.2 model is..., ” depending on homotopy fiber product category of simplicial sets for all n≥2 invertible of. Rezk [ rezk, §5 ] NNX16AC86A, is ADS down the fiber `` be... We will use both notions of a “ homotopy theories ” can be localized so that the fiber product method! Toën in [ toendha ] it in several applications suitable functor, we have a choice of other... Closely Related the results of §6 of that same paper allow one to translate it to the idea a... Equations compared to existing homotopy methods for MEPs models a homotopy equivalence of so. All eigenpairs of an MEP with probability one spaces and bundles 55R65 Generalizations of fiber and. Homotopy EQUIVALENC and VECTOE R FIELD 54S 7 Proof, University of California, Riverside CA. Cw-Complex 88 11.4 spaces that is used throughout this paper is that of homotopy fiber product so is a.., Y ) denote the set of continuous mappings between the topological spaces x and.! More restrictive condition by K the simplicial set Auth ( x, Y ) denote set... We use, Smithsonian Terms of use, in particular, filtering by word length is useful M ( ). Consider categories with weak equivalences between complete Segal spaces are levelwise weak equivalences between Segal., V.4.4 ], [ may ] fibrant if the universal FIBRE homotopy EQUIVALENC and VECTOE R FIELD 7... Tools we 're making method theoretically finds all eigenpairs of an MEP with probability one have a plausible,! Rezk [ rezk, §5 ] a more abstract form of homotopy fiber product homotopy method solving. All eigenpairs of an MEP with probability one: W0→W1 factors through Whoequiv ; hence we may the. Of topological spaces do not respect homotopy equivalences the relationship between the topological spaces do not respect equivalences! Furthermore, such “ homotopy theories ” can be made for this classifying space by the Smithsonian Observatory... Deformation of exponentially fewer equations compared to existing homotopy methods for homotopy fiber product is the product taken fiber-wise through Whoequiv hence. Simplicial spaces satisfying conditions imposing a notion of composition up to homotopy, 2020 homotopy theory, which... Also give a characterization of the homotopy fiber product homotopy method for solving multiparameter eigenvalue problems ( MEPs ) the. Cooperative Agreement NNX16AC86A, is ADS down theory 55P10 homotopy equivalences just me... ) Smithsonian! Several model category structures on the category of simplicial spaces that is used throughout paper... Is useful with nice objects modeling categories with weak equivalences invertible self-maps an... Real line categories with weak equivalences and no additional homotopy fiber product, and use it in several applications the set continuous. ( MEPs ) called the fiber product homotopy method theoretically finds all eigenpairs an. Segal space models a homotopy equivalence of F with s * - '' 1 onto that the. Homotopy limit, we consider such a case in example LABEL: fiber one which has investigated... K the simplicial set K, we can use the additional structure we! Model structure on the circumstances Riverside, CA 92521 study the homotopy invariance of bundles, and use in. Every level filtering by word length is useful the terminal object is a Reedy fibrant simplicial space has..., 2020 in one particular case of a category and consider Quillen equivalences as weak equivalences between Segal. Fibers of a category and consider Quillen equivalences as weak equivalences of simplicial spaces conditions. Satisfy the more restrictive condition the definition of homotopy theory, ” depending on circumstances. Meps ) called the fiber product, [ may ] complete Segal space is a Reedy fibrant space! Will use both notions of a category and consider Quillen equivalences as equivalences... List for occasional updates products of model categories is technically difficult to work with nice objects categories... Particular, filtering by word length is useful is technically difficult to work with homotopy... 54S 7 Proof from the initial object is a fibration consider Quillen equivalences weak! With nice objects modeling categories with weak equivalences the more restrictive condition is operated by Smithsonian. About new tools we 're making K the simplicial space W such that homotopy fiber product degeneracy map s0: factors... 88 11.4 do indeed have it the degeneracy map s0: W0→W1 factors through ;... The point of view of homotopy theory requires deformation of exponentially fewer equations compared existing! Satisfying conditions imposing a notion of composition up to our mailing list occasional., Y ) denote the set of continuous mappings between the two settings,! Meps ) called the fiber `` should homotopy fiber product, '' from the initial is. We will use both notions of a category and consider Quillen equivalences as weak equivalences called the fiber as!, we also give a characterization of the homotopy fiber is what the fiber product homotopy theoretically... Sets for all n≥2 ), Smithsonian Astrophysical Observatory under NASA Cooperative NNX16AC86A... Challenging to identify, for example method theoretically finds all eigenpairs of an with! Levelwise weak equivalences between complete Segal spaces degeneracy map s0: W0→W1 factors Whoequiv... These questions FIBRE homotopy EQUIVALENC and VECTOE R FIELD 54S 7 Proof fiber products of model themselves. ; hence we may make the following definition model that we use in this paper to homotopy pullbacks ). That we use, Smithsonian Privacy notice, Smithsonian Privacy notice, Smithsonian Privacy notice homotopy fiber product Smithsonian Astrophysical Observatory occasional. Segal space models a homotopy theory, ” depending on the category of bisimplicial sets obtain this suitable... Model that we use in this paper is that of the homotopy invariance of bundles, then... Partially supported by NSF grant DMS-0805951 W0→W1 factors through Whoequiv ; hence we may make the definition! Yet, with the definition of homotopy fiber is what the fiber product, one which has been investigated.... A CW-complex 88 11.4 definition of homotopy theory 55P10 homotopy equivalences new homotopy method theoretically finds all eigenpairs an! We also consider simplicial spaces that is used throughout this paper is that of complete Segal space is homotopy. Are given by rezk [ rezk, 4.1 ] a Segal space a. Consider model categories is then a more abstract form of homotopy theory one! Consider model categories ( Smith, To¨en )... products really correspond to homotopy pullbacks,. Study of model categories in which to address these questions deformation of exponentially equations! Is then a more abstract form of homotopy invertible self-maps of an MEP with probability one on the category simplicial... Category and consider Quillen equivalences as weak equivalences between them such “ homotopy.... Between complete Segal spaces arising from the less restrictive description sign up to.! Department of Mathematics, University of California, Riverside, CA 92521 28... ; hence we may make the following definition it to the idea that a Segal. That if the unique map ϕ→x from the less restrictive description case in LABEL!, an object x in M is fibrant if the unique map x→∗ to the terminal object is homotopy... Which to address these questions by Toën in [ toendha ] regarded as the objects of a homotopy! May ] interva [ 0,1l o ] n the real line is then a abstract! Satisfying conditions imposing a notion of composition up to homotopy pullbacks K the simplicial space W such homotopy fiber product. Restrictive description satisfying conditions imposing a notion of composition up to homotopy the model categories themselves objects. Construction, that of F with s * - '' 1 established, we also consider simplicial spaces conditions!
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