Homotopy coherent centers versus centers of homotopy categories. (English) Zbl 1411.55012
Kitchloo, Nitya (ed.) et al., New directions in homotopy theory. Second Mid-Atlantic Topology Conference, Johns Hopkins University, Baltimore, MD, USA, March 12–13, 2016. Proceedings. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 707, 121-142 (2018).
The center of a small category \(\mathbb{C}\) is the (necessarily commutative) endomorphism monoid of its identity functor – equivalently, the monoid of natural operations on the objects of \(\mathbb{C}\). When \(\mathbb{C}\) is simplicially enriched, it has a simplicial (strict) center \(Z(\mathbb{C})\) (defined as a certain equalizer), which is a generalized Eilenberg-Mac Lane space. However, the more natural (homotopy-invariant) notion is the homotopy coherent center \(\mathbb{Z}(\mathbb{C})\), defined as the corresponding homotopy limit: this is essentially the space of natural transformations \(\Phi\) of the identity, up to coherently chosen homotopies \(\Phi_{y}f\sim f\Phi_{x}\) for each \(f:x\to y\) in \(\mathbb{C}\), higher homotopies for composites \(gf\) in \(\mathbb{C}\), and so on. The homotopy coherent center is no longer a strict monoid, but the author shows that it does have a homotopy commutative \(A_{\infty}\)-structure.
When \(\mathbb{C}=G\) is a simplicial group, \(\mathbb{Z}(G)\) is just the homotopy fixed points of the action of \(G\) on itself by conjugation (equivalent to \(\Omega\operatorname{map}(BG,BG)_{\mathrm{Id}}\)). Thus even though in general we would not expect the natural map \(Z(\mathbb{C})\to\mathbb{Z}(\mathbb{C})\) (from the limit to the homotopy limit) to be an equivalence, in the special case when \(G\) is a compact Lie group this map induces an isomorphism in \(\mathbb{F}_p\)-homology, by a theorem of W. G. Dwyer and C. W. Wilkerson [Contemp. Math. 181, 119–157 (1995; Zbl 0828.55009)].
Finally, the author describes an obstruction theory for deciding whether the natural map \(\mathbb{Z}(\mathbb{C})\to\pi_{0}\mathbb{Z}(\mathbb{C})\to\pi_{0}\mathbb{C}=:Z(\mathrm{Ho}\mathbb{C})\) is surjective. In the special case when \(\mathbb{C}\) is a simplicial groupoid (correpsonding to a topological space \(X\) with \(\mathbb{C}=GX\) for the Dwyer-Kan \(G\)-functor), \(\mathrm{Ho}\mathbb{C}\) is the fundamental groupoid \(\Pi_{1}X\), and \(\mathbb{Z}(\mathbb{C})\) is just \(\Omega\operatorname{map}(X,X)_{\mathrm{Id}}\). Thus classical results on self-maps of spheres yield examples where this second natural map is not injective.
For the entire collection see [Zbl 1394.55002].
When \(\mathbb{C}=G\) is a simplicial group, \(\mathbb{Z}(G)\) is just the homotopy fixed points of the action of \(G\) on itself by conjugation (equivalent to \(\Omega\operatorname{map}(BG,BG)_{\mathrm{Id}}\)). Thus even though in general we would not expect the natural map \(Z(\mathbb{C})\to\mathbb{Z}(\mathbb{C})\) (from the limit to the homotopy limit) to be an equivalence, in the special case when \(G\) is a compact Lie group this map induces an isomorphism in \(\mathbb{F}_p\)-homology, by a theorem of W. G. Dwyer and C. W. Wilkerson [Contemp. Math. 181, 119–157 (1995; Zbl 0828.55009)].
Finally, the author describes an obstruction theory for deciding whether the natural map \(\mathbb{Z}(\mathbb{C})\to\pi_{0}\mathbb{Z}(\mathbb{C})\to\pi_{0}\mathbb{C}=:Z(\mathrm{Ho}\mathbb{C})\) is surjective. In the special case when \(\mathbb{C}\) is a simplicial groupoid (correpsonding to a topological space \(X\) with \(\mathbb{C}=GX\) for the Dwyer-Kan \(G\)-functor), \(\mathrm{Ho}\mathbb{C}\) is the fundamental groupoid \(\Pi_{1}X\), and \(\mathbb{Z}(\mathbb{C})\) is just \(\Omega\operatorname{map}(X,X)_{\mathrm{Id}}\). Thus classical results on self-maps of spheres yield examples where this second natural map is not injective.
For the entire collection see [Zbl 1394.55002].
Reviewer: David Blanc (Haifa)
MSC:
| 55U40 | Topological categories, foundations of homotopy theory |
| 18G40 | Spectral sequences, hypercohomology |
| 18G50 | Nonabelian homological algebra (category-theoretic aspects) |
| 55S35 | Obstruction theory in algebraic topology |
Keywords:
centers; homotopy coherence; homotopy limit problems; spectral sequences; obstruction theoryCitations:
Zbl 0828.55009References:
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