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I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow. Besides, I know that there are a lot of people working in homotopy theory who have probably at least used the book as a reference. I know that, obviously, the main prerequisite is category theory(and algebra). What I'm interested in is whether any amount of algebraic topology is assumed? Or it is a self-contained introduction to the subject using simplicial sets?

Can it really be an alternative way into algebraic topology/homotopical algebra/homotopy theory?

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    $\begingroup$ I'm certainly not an authority on the topic, but I think for just algebraic topology (i.e. not homotopy theory) one should use a more elementary reference. $\endgroup$ Nov 25, 2015 at 14:08
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    $\begingroup$ I don't think it has any prerequisites per se, since all used notions are explained, however without familiarity with category theory and classical algebraic topology it can be too much to swallow. I would suggest starting with Hatcher's book on algebraic topology and first 4 chapters of Maclane's "Categories for...". It definitely can't serve as an introduction to topology. $\endgroup$ Nov 25, 2015 at 16:24

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As the commenters already argued, I would not regard this book as a self-contained introduction. For instance, from a brief browse through the introductory chapters:

  • The reader is assumed to be familiar with CW-complexes and several of the major theorems about them already which will be generalized (e.g. the Whitehead theorem).

  • The reader is assumed to be familiar with homotopy in the classical sense (e.g. they point out that "homotopy" isn't an equivalence relation on maps of simplicial sets as an implicit contrast to the case of spaces).

  • The reader is assumed to be familiar with other important tools: e.g. they say "Recall that the integral singular homology groups $H_*(X;\Bbb Z)$ of the space $X$ are defined to be ..." (this is on page 5) and they assume structural properties of it are known.

  • They describe the geometric realization of a simplicial set as $$ |X| = \varinjlim_{(\Delta^n \to X)\text{ in }\Delta \downarrow X} |\Delta^n| $$ which is certainly concise and categorically valid, but assumes that the reader already has some familiarity with the point of this construction and some of its basic properties. For example, more introductory references would discuss how each point of the realization is in the interior of exactly one n-cell, give a proof that the result is a CW-complex, etc.

Simplicial sets are a fundamental tool used basically everywhere in modern homotopy theory. However, the reason for this is that there are concrete technical problems which they solve. I realize that it might be tempting to try to skip ahead to get to the more advanced material, but it can be very difficult for a student to "get the point" without first understanding the more basic material.

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  • $\begingroup$ Thank you for the response! Can I ask you about a possible short prerequisiste book for this one? Would any basic algebraic topology course suffice? Or even a part of it? $\endgroup$
    – JDou9
    Nov 28, 2015 at 11:09
  • $\begingroup$ @JDou9 I think that most basic algebraic topology texts would suffice to give a start in cover the material above (e.g. Hatcher's). $\endgroup$ Dec 1, 2015 at 2:46

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