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Local Homotopy Theory

John F. Jardine
Publication Date: 
Number of Pages: 
Springer Monographs in Mathematics
[Reviewed by
Michael Berg
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The opening sentence of the book’s preface tells you a lot, one way or the other:

The subject of this monograph is the homotopy theory of diagrams of spaces, chain complexes, spectra, and generalized spectra, where the homotopy types are determined locally by a Grothendieck topology.

So, it’s obviously about very specialized stuff, not your grandfather’s homotopy theory. It fits under the heading of algebraic topology (where one usually located homotopy) only if this subject’s scope is interpreted very liberally, or perhaps in the hypermodern sense. In a way this sort of thing is unavoidable in the wake of what Grothendieck has wrought, seeing that he and his school transformed so much of modern mathematics, including large swaths of algebraic topology. We go on to read, accordingly:

The main components of the theory are the local homotopy theories of simplicial presheaves and simplicial sheaves, local stable homotopy theories, derived categories, and non-abelian cohomology theory … The subject has broad applicability. It can be used to study presheaf and sheaf objects which are defined on the open subjects of a topological space, or on the open subschemes of a scheme, or on more exotic covers. Local homotopy theory is a foundational tool for motivic homotopy theory, and for the theory of topological modular forms in classical stable homotopy theory. As such, there are … applications … in topology, geometry, and number theory.

Note that the objects mentioned here include a number tied directly to Grothendieck, e.g., derived categories, schemes and subschemes. This having been said, it is misleading to suggest that what Jardine is up to is of a certain vintage:

Some of the ideas of local homotopy theory [do] go back to … the Grothendieck school in the 1960s [but the] present form of the theory started to emerge in the late 1980s, as part of a study of cohomological problems in algebraic K-theory … now almost completely resolved, with a fusion of ideas from homotopy theory and algebraic geometry that represents the modern face of both subjects … We also now have a good understanding of non-abelian cohomology theory and its applications, and this theory has evolved into the modern theories of algebraic stacks and higher categories.

So, this book is concerned with very specialized and hypermodern mathematics that informs a number of avant garde areas of contemporary scholarship. Certainly stacks are being pursued very actively far and wide (with apologies to Grothendieck: and, for the stouthearted, ) as is higher category theory (see e.g. ). Yes, it’s a Brave New World.

Although Jardine spends some fifty pages on preliminary material, it is by no means the case that Local Homotopy Theory is for the uninitiated. The subject matter, as described above, coming from Jardine’s Preface, is of sufficient austerity in itself to suggest as much. The author states explicitly:

The foundational ideas and results of local homotopy theory have been established in a series of well-known papers which have appeared over the last 30 years, but have not before now been given a coherent description in a single source. This book is designed to rectify this … at least for the basic theory [and] is intended for the members of the Mathematics research community, at the senior graduate student level and beyond, with interest in areas related to homotopy theory and algebraic geometry. The assumption is that the reader either has a basic knowledge of these areas, or has a willingness to acquire it.

Thus, don’t go at this book without (at least) Hartshorne (i.e. his Algebraic Geometry as well as his “Residues and Duality”) and Weibel (i.e. An Introduction to Homological Algebra) in your corner --- and that’s just the start…

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.