arXiv subjects

Kathryn Hess

Publications and source records attributed to Kathryn Hess.

46 records · Page 3Linked to original sources

arXiv2010

A general framework for homotopic descent and codescent

In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can be viewed as $\infty$-category-theoretic, as our framework is constructed in the universe of simplicially enriched categories, which are a model for $(\infty, 1)$-categories. We provide general criteria, reminiscent of Mandell's theorem on $E_{\infty}$-algebra models of $p$-complete spaces, under which homotopic (co)descent is satisfied. Furthermore, we construct general descent and codescent spectral sequences, which we interpret in terms of derived (co)completion and homotopic (co)descent. We show that a number of very well-known spectral sequences, such as the unstable and stable Adams spectral sequences, the Adams-Novikov spectral sequence and the descent spectral sequence of a map, are examples of general (co)descent spectral sequences. There is also a close relationship between the Lichtenbaum-Quillen conjecture and homotopic descent along the Dwyer-Friedlander map from algebraic K-theory to étale K-theory. Moreover, there are intriguing analogies between derived cocompletion (respectively, completion) and homotopy left (respectively, right) Kan extensions and their associated assembly (respectively, coassembly) maps.

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arXiv2009

Homotopic Hopf-Galois extensions: foundations and examples

Hopf-Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group $G$ are the Hopf-Galois extensions with respect to the dual of the group algebra of $G$. Rognes recently defined an analogous notion of Hopf-Galois extensions in the category of structured ring spectra, motivated by the fundamental example of the unit map from the sphere spectrum to $MU$. This article introduces a theory of homotopic Hopf-Galois extensions in a monoidal category with compatible model category structure that generalizes the case of structured ring spectra. In particular, we provide explicit examples of homotopic Hopf-Galois extensions in various categories of interest to topologists, showing that, for example, a principal fibration of simplicial monoids is a homotopic Hopf-Galois extension in the category of simplicial sets. We also investigate the relation of homotopic Hopf-Galois extensions to descent.

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arXiv2009

The loop group and the cobar construction

We prove that for any 1-reduced simplicial set X, Adams' cobar construction, ΩCX, on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX, opening up the possibility of applying the tools of homological algebra to transfering perturbations of algebraic structure from the latter to the former. In order to prove our theorem, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation retract for all 0-reduced simplicial sets.

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arXiv2008

CoHochschild homology of chain coalgebras

Generalizing work of Doi and of Idrissi, we define a coHochschild homology theory for chain coalgebras over any commutative ring and prove its naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra $C$ is itself a morphism of chain coalgebras up to strong homotopy, then the coHochschild complex $\cohoch (C)$ admits a natural comultiplicative structure. In particular, if $K$ is a reduced simplicial set and $C_{*}K$ is its normalized chain complex, then $\cohoch (C_{*}K)$ is naturally a homotopy-coassociative chain coalgebra. We provide a simple, explicit formula for the comultiplication on $\cohoch (C_{*}K)$ when $K$ is a simplicial suspension. The coHochschild complex construction is topologically relevant. Given two simplicial maps $g,h:K\to L$, where $K$ and $L$ are reduced, the homology of the coHochschild complex of $C_{*}L$ with coefficients in $C_{*}K$ is isomorphic to the homology of the homotopy coincidence space of the geometric realizations of $g$ and $h$, and this isomorphism respects comultiplicative structure. In particular, there a isomorphism, respecting comultiplicative structure, from the homology of $\cohoch(C_{*}K)$ to $H_{*}\op L|K|$, the homology of the free loops on the geometric realization of $K$.

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arXiv2007

An algebraic model for the loop space homology of a homotopy fiber

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy coalgebra maps given in (math.AT/0505559) to strongly homotopy morphisms of comodules over Hopf algebras. This operadic description is expressed in terms of a general theory of monoidal structures in categories with morphism sets parametrized by co-rings, which we elaborate here.

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arXiv2006

Rational homotopy theory: a brief introduction

These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings.

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arXiv2006

A chain coalgebra model for the James map

Let EK be the simplicial suspension of a pointed simplicial set K. We construct a chain model of the James map, $α_{K} : CK \to ΩCEK$. We compute the cobar diagonal on $ΩCEK$, not assuming that $EK$ is 1-reduced, and show that $α_{K}$ is comultiplicative. As a result, the natural isomorphism of chain algebras $TCK \cong ΩCK$ preserves diagonals. In an appendix, we show that the Milgram map, $Ω(A \otimes B) \to ΩA \otimes ΩB$, where A and B are coaugmented coalgebras, forms part of a strong deformation retract of chain complexes. Therefore, it is a chain equivalence even when A and B are not 1-connected.

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arXiv2006

Co-rings over operads characterize morphisms

Let M be a bicomplete, closed symmetric monoidal category. Let P be an operad in M, i.e., a monoid in the category of symmetric sequences of objects in M, with its composition monoidal structure. Let R be a P-co-ring, i.e., a comonoid in the category of P-bimodules. The co-ring R induces a natural ``fattening'' of the category of P-(co)algebras, expanding the morphism sets while leaving the objects fixed. Co-rings over operads are thus ``relative operads,'' parametrizing morphisms as operads parametrize (co)algebras. Let A denote the associative operad in the category of chain complexes. We define a ``diffracting'' functor that produces A-co-rings from symmetric sequences of chain coalgebras, leading to a multitude of ``fattened'' categories of (co)associative chain (co)algebras. In particular, we obtain a purely operadic description of the categories DASH and DCSH first defined by Gugenheim and Munkholm, via an A-co-ring that has the two-sided Koszul resolution of A as its underlying A-bimodule. The diffracting functor plays a crucial role in enabling us to prove existence of higher, ``up to homotopy'' structure of morphisms via acyclic models methods. It has already been successfully applied in this sense in a number of recent articles and preprints.

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arXiv2005

A canonical enriched Adams-Hilton model for simplicial sets

For any 1-reduced simplicial set $K$ we define a canonical, coassociative coproduct on $\Om C(K)$, the cobar construction applied to the normalized, integral chains on $K$, such that any canonical quasi-isomorphism of chain algebras from $\Om C(K)$ to the normalized, integral chains on $GK$, the loop group of $K$, is a coalgebra map up to strong homotopy. Our proof relies on the operadic description of the category of chain coalgebras and of strongly homotopy coalgebra maps given in math.AT/0505559.

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arXiv2004

An algebraic model for mod 2 topological cyclic homology

For any space X with the homotopy type of simply-connected, finite-type CW-complex, we construct an associative cochain algebra fls(X) whose cohomology algebra is isomorphic to that of LX, the free loop space on X. For certain X, we define a cochain map from fls(X) to itself that is a good model of the pth-power operation on LX. Under additional conditions on X, e.g., when X is a wedge of spheres, we define a cochain complex hos(X) by twisting together fls(X) and H*(BS^1) and prove that the cohomology of hos(X) is isomorphic to the Borel cohomology of LX. Finally, we define tc(X) to be the mapping cone of the composite of the projection map from hos(X) to fls(X) with the model of the pth-power map (for p=2), so that the mod 2 spectrum cohomology of TC(X;2), the topological cyclic homology of X at 2, is isomorphic to the mod 2 cohomology of tc(X). We conclude by calculating the mod 2 spectrum homology of TC(S^{2n+1}; 2).

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