Philippe Gaucher
In this talk, I will explain the importance of the homotopy branching space functor (and of the homotopy merging space functor) in dihomotopy theory. The paper is a detailed abstract of math.AT/0304112 and math.AT/0305169.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Withdrawn paper because the results are recycled in several other papers and a new definition of T-homotopy is proposed in math.AT/0505152.
We check that there exists a model structure on the category of flows whose weak equivalences are the S-homotopy equivalences. As an application, we prove that the generalized T-homotopy equivalences preserve the branching and merging homology theories of a flow. The method of proof is completely different from the one of the third part of this series of papers.
This series explores a new notion of T-homotopy equivalence of flows. The new definition involves embeddings of finite bounded posets preserving the bottom and the top elements and the associated cofibrations of flows. In this third part, it is proved that the generalized T-homotopy equivalences preserve the branching and merging homology theories of a flow. These homology theories are of interest in computer science since they detect the non-deterministic branching and merging areas of execution paths in the time flow of a higher dimensional automaton. The proof is based on Reedy model catego…
Branching space of precubical set
本文提出前立方集的分支空间概念，并证明其在同伦等价下的不变性，同时推导出分支同调的不变性。
A functor is constructed from the category of globular CW-complexes to that of flows. It allows the comparison of the S-homotopy equivalences (resp. the T-homotopy equivalences) of globular complexes with the S-homotopy equivalences (resp. the T-homotopy equivalences) of flows. Moreover, it is proved that this functor induces an equivalence of categories from the localization of the category of globular CW-complexes with respect to S-homotopy equivalences to the localization of the category of flows with respect to weak S-homotopy equivalences. As an application, we construct the underlying ho…
This series explores a new notion of T-homotopy equivalence of flows. The new definition involves embeddings of finite bounded posets preserving the bottom and the top elements and the associated cofibrations of flows. In this fourth part, it is proved that the generalized T-homotopy equivalences preserve the underlying homotopy type of a flow. The proof is based on Reedy model category techniques.
We construct a small realization as flow of every precubical set (modeling for example a process algebra). The realization is small in the sense that the construction does not make use of any cofibrant replacement functor and of any transfinite construction. In particular, if the precubical set is finite, then the corresponding flow has a finite globular decomposition. Two applications are given. The first one presents a realization functor from precubical sets to globular complexes which is characterized up to a natural S-homotopy. The second one proves that, for such flows, the underlying ho…
A flow is homotopy continuous if it is indefinitely divisible up to S-homotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy. Thus, the category of cofibrant homotopy continuous flows provides an implementation of Whitehead's theorem for the full dihomotopy relation, and not only for S-homotopy as in previous works of the author. This fact is not the consequence of the existence of a model…
The notion of reparametrization category is incorrectly axiomatized and it must be adjusted. It is proved that for a general reparametrization category $\mathcal{P}$, the tensor product of $\mathcal{P}$-spaces yields a biclosed semimonoidal structure. It is also described some kind of objectwise braiding for $\mathcal{G}$-spaces.
This paper proves that the q-model structures of Moore flows and of multipointed $d$-spaces are Quillen equivalent. The main step is the proof that the counit and unit maps of the Quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). As an application, we provide a new proof of the fact that the categorization functor from multipointed $d$-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. The new proof sheds light on the internal structure of the categorization functor which is neither a …
We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories.
This paper is the second part of a series of papers about a new notion of T-homotopy of flows. It is proved that the old definition of T-homotopy equivalence does not allow the identification of the directed segment with the 3-dimensional cube. This contradicts a paradigm of dihomotopy theory. A new definition of T-homotopy equivalence is proposed, following the intuition of refinement of observation. And it is proved that up to weak S-homotopy, a old T-homotopy equivalence is a new T-homotopy equivalence. The left-properness of the weak S-homotopy model category of flows is also established i…
Globular CW-complexes and flows are both geometric models of concurrent processes which allow to model in a precise way the notion of dihomotopy. Dihomotopy is an equivalence relation which preserves computer-scientific properties like the presence or not of deadlock. One constructs an embedding from globular CW-complexes to flows and one proves that two globular CW-complexes are dihomotopic if and only if the corresponding flows are dihomotopic. This note is the first one presenting some of the results of math.AT/0201252.
Comparing cubical and globular directed paths
本文构造了一个不依赖于余纤维替代选择的新实现函子，将预立方集映射为流，并证明其执行路径空间与几何实现中的非恒定 d-路径空间同伦等价。
Homotopy theory of Moore flows (I)
本文修正了重参数化范畴的公理化，构造了Moore流的q-模型范畴，并证明其与流的q-模型范畴Quillen等价，为建立多基点d-空间与流之间的Quillen等价之字形迈出第一步。
The branching (resp. merging) space functor of a flow is a left Quillen functor. The associated derived functor allows to define the branching (resp. merging) homology of a flow. It is then proved that this homology theory is a dihomotopy invariant and that higher dimensional branchings (resp. mergings) satisfy a long exact sequence.
This paper proves that labelled flows are expressive enough to contain all process algebras which are a standard model for concurrency. More precisely, we construct the space of execution paths and of higher dimensional homotopies between them for every process name of every process algebra with any synchronization algebra using a notion of labelled flow. This interpretation of process algebra satisfies the paradigm of higher dimensional automata (HDA): one non-degenerate full $n$-dimensional cube (no more no less) in the underlying space of the time flow corresponding to the concurrent execut…
We construct a q-model structure, a h-model structure and a m-model structure on multipointed $d$-spaces and on flows. The two q-model structures are combinatorial and coincide with the combinatorial model structures already known on these categories. The four other model structures (the two m-model structures and the two h-model structures) are accessible. We give an example of multipointed $d$-space and of flow which are not cofibrant in any of the model structures. We explain why the m-model structures, Quillen equivalent to the q-model structure of the same category, are better behaved tha…
The previous paper of this series shows that the q-model categories of $\mathcal{G}$-multipointed $d$-spaces and of $\mathcal{G}$-flows are Quillen equivalent. In this paper, the same result is established by replacing the reparametrization category $\mathcal{G}$ by the reparametrization category $\mathcal{M}$. Unlike the case of $\mathcal{G}$, the execution paths of a cellular $\mathcal{M}$-multipointed $d$-space can have stop intervals. The technical tool to overcome this obstacle is the notion of globular naturalization. It is the globular analogue of Raussen's naturalization of a directed …
Comparing the non-unital and unital settings for directed homotopy
本文比较流的 q-模型结构与拓扑富集小范畴的 Ilias 模型结构，证明二者间的提升性质及恒等函子非 Quillen 伴随，并给出极小模型结构的同伦范畴。
Foundation of Floer homotopy theory I: Flow categories
本文构造了以流范畴为对象、流双模为态射的稳定无穷范畴，将Floer同伦类型实现为映射谱，并证明在标架配边下该范畴是谱范畴的模型，进而在Hamiltonian Floer理论中实现了此构造。
The Lipshitz-Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz-Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the flow category simplification techniques previously defined by the authors and Dan Jones. We give a purely combinatorial approach to calculating the second Steenrod square and Bockstein homomorphisms in Khovanov cohomology, and flow categories in general. The new method has been implemented in a computer program by the third author and applied to lar…
The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author's paper ``A model category for the homotopy theory of concurrency''. It is presented generalizations of the classical Whitehead theorem inverting weak homotopy equivalences between CW-complexes using weak factorization systems. It is also presented methods of calculation of homotopy limits and homotopy colimits using Quillen adjunctions and Reedy categories.
It is possible to translate a modified version of K. Worytkiewicz's combinatorial semantics of CCS (Milner's Calculus of Communicating Systems) in terms of labelled precubical sets into a categorical semantics of CCS in terms of labelled flows using a geometric realization functor. It turns out that a satisfactory semantics in terms of flows requires to work directly in their homotopy category since such a semantics requires non-canonical choices for constructing cofibrant replacements, homotopy limits and homotopy colimits. No geometric information is lost since two precubical sets are isomor…
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent index and handle cancellation which removes two critical points connected by a single flowline. A framed flow category is a way of encoding flow data such as that which may arise from the flowlines of a Morse function or of a Floer functional. The Cohen-Jones-Segal construction as…
Homotopy probability theory is a version of probability theory in which the vector space of random variables is replaced with a chain complex. A natural example extends ordinary probability theory on a finite volume Riemannian manifold M. In this example, initial conditions for fluid flow on M are identified with collections of homotopy random variables and solutions to the Euler equation are identified with homotopies between collections of homotopy random variables. Several ideas about using homotopy probability theory to study fluid flow are introduced.
Left properness of Moore flows
该研究引入带割的重参数化范畴概念，证明此类范畴上P-空间的张量引理，进而得到P-流q-模型结构的左正则性，还证明了三类带割的重参数化范畴，其中终范畴的例子可还原普通流的对应左正则性。
Globular complexes were introduced by E. Goubault and the author in arXiv:math/0107060 to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CW-complex. We prove that there exists a combinatorial model category such that the cellular objects are exactly the globular complexes and such that the homotopy category is equivalent to the homotopy category of flows introduced in arXiv:math/0308054. The underlying category of this model category is a variant of M. Grandis'…
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $μ$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of $X$. Forman also proved the existence of a CW complex which is homotopy equivalent to $X$ and whose cells are in one-to-one correspondence with the critical cells of $μ$, but the construction is ad hoc and does not have a combin…
A contact homotopy type
本文将global Kuranishi charts调整至切触设定，基于Reeb orbits和伪全纯建筑模空间为非退化contact manifold构建流范畴，该构造提升了contact homology并具有自然性。
On the classifying space of a Morse flow category
本文展示了一个 tame Morse 函数的流形分类空间恢复了 M 的同伦类型，同时通过构造反例说明 tameness 假设的重要性，澄清了流形分类空间与同伦类型的关联。
Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the resulting complex, recovering the Floer cohomology as its singular cohomology. Such a framed flow category was produced, for example, by Lipshitz-Sarkar from the input of a knot diagram, resulting in a stable homotopy type generalizing Khovanov cohomology. In this paper we give mo…
Branching spaces of transverse sets
提出c-直范畴并证明其上的c-Reedy模型结构与投射模型结构一致；构造横向集的ε-支化空间，证明与旧定义一致且对余纤维对象同伦等价。
Branching spaces of multipointed d-spaces
引入短有向路径概念定义多点d-空间的分支空间，证明其与分类函子得到的q-余纤维流的分支空间同胚，并应用于分支空间和分支同调在球状细分下的不变性。
Globular subdivisions are dihomotopy equivalences
本文证明了球形细分在 $d$-空间中产生双同伦等价，并展示了相关流的同调理论和同伦类型的同构性。
We describe a calculus of moves for modifying a framed flow category without changing the associated stable homotopy type. We use this calculus to show that if two framed flow categories give rise to the same stable homotopy type of homological width at most three, then the flow categories are move equivalent. The process we describe is essentially algorithmic and can often be performed by hand, without the aid of a computer program.
Given a family of Dirac operators with vanishing spectral flow we construct a thin-invariant rank-one field theory in the sense of Turner and Willerton arXiv:math.AT/0201116. Our construction of the field theory generalizes the one of the index gerbe by Lott, arXiv:math.DG/0106177, and it also complements the relation between gerbes and thin-invariant rank-one field theories studied in arXive:math.AT/0201116.
Time-reversal homotopical properties of concurrent systems
本研究探讨了有向拓扑中代数不变量在时间反转下的性质，证明自然同伦与同调可配备代数结构产生对偶对象，并通过引入相对有向同伦进一步精细化自然同伦。
The notion of a homotopy flow on a directed space was introduced in \cite{Raussen:07} as a coherent tool for comparing spaces of directed paths between pairs of points in that space with each other. If all parameter directed maps preserve the homotopy type of path spaces, such a flow (and these parameter maps) are called inessential. For a directed space, one may consider various categories whose objects are pairs of reachable points and whose morphisms may be induced by these inessential d-maps. Localization with respect to subcategories with these inessential d-maps as morphisms can be combi…
Morse flow categories as exit path categories
本文证明光滑闭流形上 Morse-Smale 对 $(f,ξ)$ 诱导的拓扑流范畴 $\mathcal{M}$ 作为 $\infty$-范畴等价于 Lurie 的关于稳定流形分层的出口路径 $\infty$-范畴 $\mathrm{Sing}_A(X)$。
Flow type suspension and homotopy suspension agree for attractor-repellor homotopy data.The connection maps associated in Conley index theory to an attractor-repellor decomposition with respect to the direct flow and its inverse are Spanier-Whitehead duals in the stably parallelizable context and are duals modulo a certain Thom construction in general.
Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev's $Ω$-spectrum
本文提出通过Ω-谱IP*及其晶格对称性扩展，构建可逆间隙量子系统同调群的数学框架，并利用装配映射计算其同调群结构。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。