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Several years have passed since Jacob Lurie wrote the book. I would like to know how today the community of mathematicians looks at the theories contained in it and whether the book has served to achieve important results within the mathematical discipline. Thank you.

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What is your background? The language of quasi-categories is a widely spread tool to deal with homotopy theories and coherent models for these. I do not know if you tagged this question as related to general topology because you did not find a better pre-existing label or because this is the topic you know and you think it is the most similar related to quasi-categories? Since the title says "brought to the discipline" I would think you are asking what contributions were given in point-set topology by Lurie's book, which are none.

The topics dealt in Higher Topos Theory are not related to how sets endowed with a topology behave. Topological spaces are dealt sufficiently well with the point-set arguments developed between the twenties and the fifties. The work of Lurie is about giving a theory of well behaved categorical models for homotopy theories which is a question rised in the eighties-nineties, a lot of time later. This notion of homotopy I mentioned is inspired by the one in the context of topology but it is far more general.

EDIT: so I will assume a basic knowledge of topology and noting more for my answer. As I said briefly above the answer is: yes. Lurie's work had, or better has, deep and fruitful consequences in modern approach to many branches of abstract mathematics. But not in the field of point-set topology, nor in the terms this discipline was developed in 1920-50.

This kind of topology deals with sets with extra data in form of the choice of a collection of their subsets. This extra data lets you formulate a lot of useful and interesting properties (open, closure, compactness, connectiveness and so on) which can be spelled and examined in the language of sets axiomatized by Fermelo and Fraenke.

The next step was algebraic topology: basically this is the discipline in which we study qualitatively the "shape" of topological spaces. A basic example is the following: consider a point and the entire plane $\mathbb{R}^2$. They are clearly different spaces and they are not homeomorphic, but you can easily contract the plane to the origin via a continuous function $[0,1]\times \mathbb{R}^2 \ni (t,v)\mapsto tv$. Observe that the restriction at $t=1$ is the identity, while $t=0$ gives you the constant function. So $\mathbb{R}^2$ has basically the same qualitative shape of the point, even if it not exactly the same. We say that these two spaces are $\textit{homotopically equivalent}$. I do not know any good article explaining this discipline, but the main text reference for it is Hatcher's Algebraic Topology. You could try to read the introduction (chapter 0) which does not require a lot of background.

To formalise better some ideas of algebraic topology Eilenberg and Mac Lane introduced around 1945 the idea of category. A category is a collection of objects together with a collection of morphisms bewteen this objects. I wrote collection because these are not required to be sets and in general they are not. The objects are basically the type of mathematical constructs we want to study, while the morphisms are the maps between these constructs which are interesting and makes sense to consider. Example: the category of groups. The objects are just all the groups while the morphisms are the group homomorphism (that is, we are considering only the functions which are compatible with the multiplication in the source and target, we do not care about the functions which do not preserve the products). If in any context you heard of maps called something-morphism you can be sure that they are morphisms in some category.

This theory of categories is not the same mindset of set theory of Zeta-Fermelo, since you are not working with sets but proper classes sometimes. Also the objects of a category in general are not sets and there is not a notion of equality in a category. There are foundational technicalities which must be taken care but I won't discuss them here.

As we passed from point-set topology to algebraic topology by considering spaces qualitatively similar according this notion of $\textit{homotopy}$, the following idea arised: can we do the same on a general category? That is can we define a notion of homotopy in any category and say when two objects are "basically identical"? The short answer is yes. The appropriate answer here is: yes, but there are a lot of different methods to formalise this theory of homotopy in a category. And you could also prove that these different methods are compatible and provide you with similar results, so you do not have to worry that two of these modi operandi give you different answers.

This comparison is dealt fairly well in the book "The homotopy theory of $(\infty,1)$-categories" by Bergner. I do not think that with your background you can read it, but other more qualified readers could. The book is not very technical, it has not many proofs and is mostly a review of known results (and it includes Lurie' approach).

Luries's work is about providing one of these methods, which is very flexible and extensive at the same time. It is the language which is taking progressively the prominent position. The best notes I would recommend to start with would be https://arxiv.org/abs/1007.2925 but I think you will not be able to understand them.

I also found recently the following article: https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/?mc_cid=372397c7db&mc_eid=1a272e512a I hope it helps.

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  • $\begingroup$ I am an ITC guy with an undergraduate background in mathematics, focussing mainly on mathematics for CS. What I wanted to know is how the work within the mathematical world is considered and if it served to reach some new results. $\endgroup$
    – Yamar69
    Commented Oct 16, 2019 at 13:44

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