SFB1085 - Higher Invariants - Recent changes [en]

Monadic resolutions for (generalized) spaces

Vom41941 — Fri, 01 May 2026 08:00:22 GMT

Created page with "Understanding the homotopy type of a space often benefits from studying its values under homology theories. Every homology theory determines a Bousfield localization on the category of spaces — for example, rational cohomology leads to rationalization, while $F_p$-homology induces $p$-completion. In favorable cases, these localizations admit explicit descriptions via monadic resolutions. A key example is the $p$-completion $L_p X$ of a nilpotent space $X$, which can..."

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Understanding the homotopy type of a space often benefits from studying its values under homology theories. Every homology theory determines a Bousfield localization on the category of spaces — for example, rational cohomology leads to rationalization, while $F_p$-homology induces $p$-completion.
In favorable cases, these localizations admit explicit descriptions via monadic resolutions.
A key example is the $p$-completion $L_p X$ of a nilpotent space $X$, which can be expressed as an inverse limit:
$L_p X = lim_n (\Omega^\infty F_p \otimes \Sigma^\infty)^{n+1}(X)$.
Similarly, any nilpotent space $X$ can be expressed as the inverse limit
$X = lim_n (\Omega^\infty MU \otimes \Sigma^\infty)^{n+1}(X)$ of its iterated $MU$-homology, giving rise to the unstable Adams—Novikov spectral sequence.
A fundamental ingredient in proving this equivalence is Bousfield and Kan's celebrated principal fibration lemma.

In this talk, I will discuss how these ideas extend to oo-topoi and explain their application in motivic homotopy theory. In particular, I will sketch a proof of the principal fibration lemma in this broader context.
This is joint work with Tom Bachmann and Anton Engelmann.

AG-Seminar WS2021/22:

Vom41941 — Fri, 01 May 2026 07:59:08 GMT

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	\| Klaus Mattis (Mainz)		\| Klaus Mattis (Mainz)
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	\|		\| Leor Neuhauser (Hebrew University of Jerusalem)
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Calabi—Yau structures on constructible sheaves categories

Vom41941 — Fri, 24 Apr 2026 16:48:12 GMT

Created page with "For X a compact oriented topological manifold and k a field, Verdier duality on locally constant sheaves of k-modules on X can be encoded in a non-commutative symplectic structure, called Calabi—Yau structure. Brav—Dyckerhoff also introduced a relative notion of such, which allows one to recover Verdier duality for manifolds with boundaries. If now X is equipped with a (nice enough) finite stratification P, we show that there exists a Calabi—Yau structure on the k-..."

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For X a compact oriented topological manifold and k a field, Verdier duality on locally constant sheaves of k-modules on X can be encoded in a non-commutative symplectic structure, called Calabi—Yau structure. Brav—Dyckerhoff also introduced a relative notion of such, which allows one to recover Verdier duality for manifolds with boundaries.
If now X is equipped with a (nice enough) finite stratification P, we show that there exists a Calabi—Yau structure on the k-linear stable infinity-category of constructible sheaves on (X,P). In this case, even if X itself has no boundary, we will get a relative Calabi—Yau structure, where the role of the boundary is replaced by the tubular neighborhoods of the strata. This is joint work with Merlin Christ.

AG-Seminar WS2021/22:

Vom41941 — Fri, 24 Apr 2026 16:47:38 GMT

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	\| ~~TBA~~		\| [[Calabi—Yau structures on constructible sheaves categories]]
	\| Enrico Lampetti (IMJ-PRG, Paris)		\| Enrico Lampetti (IMJ-PRG, Paris)
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AG-Seminar WS2021/22:

Buu10307 — Fri, 24 Apr 2026 08:32:01 GMT

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	\|		\|G. Raptis
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Presenting the stratified homotopy hypothesis

Vom41941 — Fri, 17 Apr 2026 19:48:52 GMT

Created page with "The stratified homotopy hypothesis proclaims an equivalence between a homotopy theory of stratified spaces, and the homotopy theory of such small (infinity,1)-categories in which every endomorphism is an isomorphism. In this talk, after an introduction into the homotopy theory of stratified spaces, I want to talk about an explicit presentation of this proclaimed equivalence in terms of a Quillen equivalence using Lurie’s construction of the infinity-category of exit-pa..."

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The stratified homotopy hypothesis proclaims an equivalence between a homotopy theory of stratified spaces, and the homotopy theory of such small (infinity,1)-categories in which every endomorphism is an isomorphism. In this talk, after an introduction into the homotopy theory of stratified spaces, I want to talk about an explicit presentation of this proclaimed equivalence in terms of a Quillen equivalence using Lurie’s construction of the infinity-category of exit-paths. Crucially, I will talk about a new semi-model structure on topological stratified spaces, which presents the aforementioned homotopy theory, and interacts particularly well with classical geometrical examples, such as Whitney stratified spaces – thus bridging the gap between the geometry and homotopy theory of stratified spaces.

AG-Seminar WS2021/22:

Vom41941 — Fri, 17 Apr 2026 19:48:34 GMT

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	\| ~~TBA~~		\| [[Presenting the stratified homotopy hypothesis]]
	\| Lukas Waas (Oxford University)		\| Lukas Waas (Oxford University)
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Complexes of stable ∞-categories and higher Segal conditions

Cid36224 — Wed, 15 Apr 2026 12:38:08 GMT

Created page with "Title: Complexes of stable ∞-categories and higher Segal conditions Abstract: There exists an equivalence of (∞,2)-categories between the (∞,2)-category of complexes of stable ∞-categories and that of 2-simplicial stable ∞-categories, established by Dyckerhoff, which categorifies the classical Dold–Kan correspondence. It is well known that every simplicial abelian group is in particular a Kan complex, i.e. it admits certain horn fillers. In this talk, I will..."

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Title: Complexes of stable ∞-categories and higher Segal conditions

Abstract: There exists an equivalence of (∞,2)-categories between the (∞,2)-category of complexes of stable ∞-categories and that of 2-simplicial stable ∞-categories, established by Dyckerhoff, which categorifies the classical Dold–Kan correspondence. It is well known that every simplicial abelian group is in particular a Kan complex, i.e. it admits certain horn fillers. In this talk, I will explain joint work with Dyckerhoff, Gödicke, and Walde, in which we show that every 2-simplicial stable ∞-category satisfies an analogous categorified Kan complex condition, and use this to characterize those 2-simplicial stable ∞-categories that satisfy the (higher) Segal conditions of Dyckerhoff–Kapranov.

AG-Seminar WS2021/22:

Cid36224 — Wed, 15 Apr 2026 12:37:46 GMT

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	\| ~~TBA~~		\| [[Complexes of stable ∞-categories and higher Segal conditions]]
	\| Fernando Abellan (Bonn)		\| Fernando Abellan (Bonn)
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AG-Seminar WS2021/22:

Vom41941 — Sun, 05 Apr 2026 21:48:33 GMT

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	\|		\| Klaus Mattis (Mainz)
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	\|5		\|5