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Manfred Stelzer

Publications and source records attributed to Manfred Stelzer.

8 recordsLinked to original sources
arXiv2021

The cube axiom and resolutions in homotopy theory

We show that a version of the cube axiom holds in cosimplicial unstable coalgebras and cosimplicial spaces equipped with a resolution model structure. As an application, classical theorems in unstable homotopy theory are extended to this context.

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arXiv2021

The homotopy automorphisms of a marked n-stage

We show nilpotency and completeness results for the homotopy automorphisms of a marked n-stage for an unstable coalgebra. These objects figure in the moduli problem of unstable coalgebras. Our theorems extend classical work of Dror, Zabrodsky, Maruyama and M\o ller.

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arXiv2014

The realization space of an unstable coalgebra

Unstable coalgebras over the Steenrod algebra form a natural target category for singular homology with prime field coefficients. The realization problem asks whether an unstable coalgebra is isomorphic to the homology of a topological space. We study the moduli space of such realizations and give a description of this in terms of cohomological invariants of the unstable coalgebra. This is accomplished by a thorough comparative study of the homotopy theories of cosimplicial unstable coalgebras and of cosimplicial spaces.

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arXiv2013

Rectification of Weak Product Algebras over an Operad in Cat and Top and Applications

We develop an alternative to the May-Thomason construction used to compare operad based infinite loop machines to that of Segal, which relies on weak products. Our construction has the advantage that it can be carried out in $Cat$, whereas their construction gives rise to simplicial categories. As an application we show that a simplicial algebra over a $\Sigma$-free $Cat$ operad $\mathcal{O}$ is functorially weakly equivalent to a $Cat$ algebra over $\mathcal{O}$. When combined with the results of a previous paper, this allows us to conclude that up to weak equivalences the category of $\mathcal{O}$-categories is equivalent to the category of $B\mathcal{O}$-spaces, where $B:Cat\to Top$ is the classifying space functor. In particular, $n$-fold loop spaces (and more generally $E_n$ spaces) are functorially weakly equivalent to classifying spaces of $n$-fold monoidal categories. Another application is a change of operads construction within $Cat$.

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arXiv2013

Homotopy Colimits of Algebras Over Cat-Operads and Iterated Loop Spaces

We extend Thomason's homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the category of $n$-fold loop spaces and loop maps after localization with respect to some other class of weak equivalences. We recover Thomason's corresponding result about infinite loop spaces and obtain related results about braided monoidal categories and 2-fold loop spaces.

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