Higher Topos Theory

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Higher Topos Theory is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory.[1] Since 2018, Lurie has been transferring the contents of Higher Topos Theory (along with new material) to Kerodon, an "online resource for homotopy-coherent mathematics"[2] inspired by the Stacks Project.

## Contents

* 1 Topics
* 2 History
* 3 External links
* 4 References

## Topics

Higher Topos Theory covers two related topics: ∞-categories and ∞-topoi (which are a special case of the former). The first five of the book's seven chapters comprise a rigorous development of general ∞-category theory in the language of quasicategories, a special class of simplicial set which acts as a model for ∞-categories. The path of this development largely parallels classical category theory, with the notable exception of the ∞-categorical Grothendieck construction; this correspondence, which Lurie refers to as "straightening and unstraightening",[3] gains considerable importance in his treatment.

The last two chapters are devoted to ∞-topoi, Lurie's own invention and the ∞-categorical analogue of topoi in classical category theory. The material of these chapters is original, and is adapted from an earlier preprint of Lurie's.[4] There are also appendices discussing background material on categories, model categories, and simplicial categories.

## History

Higher Topos Theory followed an earlier work by Lurie, On Infinity Topoi, uploaded to the arXiv in 2003.[4] Algebraic topologist Peter May was critical of this preprint, emailing Lurie's then-advisor Mike Hopkins "to say that Lurie’s paper had some interesting ideas, but that it felt preliminary and needed more rigor."[1] Lurie released a draft of Higher Topos Theory on the arXiv in 2006,[5] and the book was finally published in 2009.

Lurie released a second book on higher category theory, Higher Algebra, as a preprint on his website in 2017.[6] This book assumes the content of Higher Topos Theory and uses it to study algebra in the ∞-categorical context.

## External links

* http://ncatlab.org/nlab/show/Higher+Topos+Theory
* If I want to study Jacob Lurie's books "Higher Topoi Theory", "Derived AG", what prerequisites should I have?
* https://www.math.ias.edu/~lurie/
* https://kerodon.net/about

## References

1. ↑ 1.0 1.1 Hartnett, Kevin (2019-10-10). "With Category Theory, Mathematics Escapes From Equality". Quanta Magazine. https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/.
2. ↑ Lurie, Jacob (2022). "Kerodon". https://kerodon.net/.
3. ↑ Lurie, Jacob (2009). Higher Topos Theory. Princeton University Press. ISBN 978-0-691-14048-3.
4. ↑ 4.0 4.1 Lurie, Jacob (June 8, 2003). "On Infinity Topoi". arXiv:math/0306109v2.
5. ↑ Lurie, Jacob (August 2, 2006). "Higher Topos Theory". arXiv:math/0608040v1.
6. ↑ Lurie, Jacob (2017). Higher Algebra. https://people.math.harvard.edu/~lurie/papers/HA.pdf.

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