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I could not come up with a better title for my question.

What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein):

the category of simplicial sets has always been to me something distinguished, on multiple counts:

to begin with, it is simple to describe, almost childish, yet it has a seemingly unfathomable richness.

Its homotopy theory, although usually presented via the geometric realization functor (which is historically correct), is in fact entirely self-contained, and purely combinatorial in character.

Also, simplicial sets plays a very special role in category theory (after all, categories are just some almost trivial example of simplicial sets) and, even more important, in higher dimensional cats.

Somehow, I have the lingering feeling that, in abstract homotopy theory, simplicial sets (or, more properly simplicial objects in some ambient category ) should be, at least for some suitable notion of "regularity" of homotopy theories, paradigmatic. I am thinking of something like:

If an homotopy theory is "regular...." (whatever that may mean, fill the dots, the "regular" would stand for combinatorial in essence), then it is representable in the homotopy theory of simplicial objects for some ambient category.

Am I dreaming or there is something along these lines?

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    $\begingroup$ The discussion at mathoverflow.net/questions/58497/… seems relevant. $\endgroup$ Jul 17, 2012 at 14:17
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    $\begingroup$ In particular, Peter Arndt's answer (I hadn't noticed this) describes a universal property. $\endgroup$ Jul 17, 2012 at 14:20
  • $\begingroup$ Thanks Qiaochu! Yes, Peter's answer seems to be (very) relevant, as it singles out the simplicial homotopy nicely. Perhaps that can lead to a full answer to my 'dream", need some time to think about it... $\endgroup$ Jul 17, 2012 at 14:35
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    $\begingroup$ A restatement of Peter Arndt's answer: the "universal property" of the $\infty$-category ("homotopy theory") $\mathcal{S}$ of spaces (Kan complexes, etc.) is that, for any $\infty$-category $\mathcal{C}$ admitting all colimits, there is an equivalence of $\infty$-categories $\mathrm{Fun}^L(\mathcal{S}, \mathcal{C})) \simeq \mathcal{C}$ given by evaluation on a point ($L$ means colimit-preserving). That is, spaces are the "free" cocomplete $\infty$-category on a single object, in the same way that sets are the free ordinary cocomplete category on a point. $\endgroup$ Jul 17, 2012 at 19:51
  • $\begingroup$ (Here $\infty$ should be $(\infty, 1)$.) $\endgroup$ Jul 17, 2012 at 19:52

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I will as usual act as Devil's advocate (or cubical advocate) by saying that there are some things we can do with cubical sets (with connections) which we cannot (maybe others can!) do with simplicial sets, namely:

  1. Algebraic inverses to subdivision (multiple compositions);

  2. Tensor products.

These properties are exploited and used (paradigmatically!) in the book Nonabelian algebraic topology: there is a Higher Homotopy Seifert-van Kampen Theorem, and some theorems on homotopy classification of maps (in the non simply connected case), and so rewrite a portion of algebraic topology without using singular homology or simplicial approximation. Also these results would not, I believe, even have been conjectured simplicially.

This is not to deny the advantages of the simplicial approach, which include its large literature, or some disadvantages of the cubical approach.

I started in the 1960s with drawing many times the diagram of a subdivided square pictured as

array (source)

(see also Cubical vs. simplicial singular homology ) and saying to myself: surely there should be some mathematics which expresses that? Then Ehresmann's book "Catégories et structure" came out and the answer was clear: double categories (or groupoids)! In these one can describe the above as an array $(a_{rs})$ or as a composition $[a_{rs}]$ and this makes sense for the singular cubical complex, and also for double categories or groupoids. So one expresses "algebraic inverses to subdivision", while the corresponding globular or simplicial expressions are difficult or not available.

This role of cubical sets for intuition and conjecture is also suggested by the fact that Dan Kan's first results were cubical. Then problems were found with realisations and so, instead of trying to fix the cubical, they simply gave it up.

We came across connections for cubical sets for other reasons, namely to describe commutative cubes in terms of double groupoids.

January 9, 2015: See also this presentation at Galway, Dec, 2014, on "A homotopical approach to algebraic topology via compositions of cubes". Note that a simplicial approach of a similar type seems a non starter. However many aspects of a cubical approach have not yet been developed, because of the apparent success of the simplicial methods.

May 24, 2020 I can now refer to a more developed presentation of the philosophy of, background to, and development of the cubical approach in a paper Modelling and Computing Homotopy Types: I preprint which appeared in Indagationes Math. in 2017 in a special issue to honor L.E.J. Brouwer. One main point is that the cubical theory has been enriched by the notion of connections which, it has been found, rescue it from some problems of the older theory, and which make the expanded theory applicable in places not available to the simplicial theory.

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    $\begingroup$ @Ronnie without a Devil's Advocate the trial for the beatification of DELTA cannot proceed, so your answer is most welcome! Now, my immediate reaction is: BOX instead of DELTA, why not? But before going a step further, what is their reciprocal relation? I read in the nLab: -- The cube category may also be described as the subcategory of Set whose objects are powers 2n of 2={0,1}, n≥0, and whose morphisms are generated by degeneracy maps 2m→2n which delete a coordinate and face maps which insert a 0 or 1 without modifying the order of coordinates.-- So it looks like I can think of BOX as $\endgroup$ Jul 18, 2012 at 10:28
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    $\begingroup$ sitting in DELTA, in some fashion. That in turn would give me some conection between simplicial sets and cubical set... $\endgroup$ Jul 18, 2012 at 10:29
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    $\begingroup$ The further comment is: why do people have it in for pentagons? and what is wrong with rhombic dodecahedra? (to say nothing of Stasheff polyhedra). A possible general polyhedral framework was given in David Jones' thesis, see ncatlab.org/nlab/show/T-complex . I have recently learned that the key idea of "marked complex" has been independently discovered as "discrete vector field", by Robin Forman, for different reasons. There are also papers by Marco Grandis on cubical sets. Don't hand out an apple till you have seen and compared all the candidates! $\endgroup$ Jul 18, 2012 at 22:43
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    $\begingroup$ Mirco: have you looked at cheng.staff.shef.ac.uk/guidebook/index.html. This has some of the characteristices that you want perhaps. $\endgroup$
    – Tim Porter
    Jul 20, 2012 at 9:23
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    $\begingroup$ I should add some points about the polysets of David Jones (see the ncatlab on T-complexes for references and pdf). 1. The T-complex notion is oriented to groupoids and it is not clear how to do a categorical version. 2. The idea of "multiple compositions" has not been developed, or other applications. 3. The idea in the thesis of using a more general polycell to relate simplicial and cubical is not fully worked out. (that was a criticism of a referee for American Math Soc Mathematical Studies.) But I still think he did a great job, starting from very little! $\endgroup$ Jul 20, 2012 at 15:11
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Cisinski has some very detailed thoughts here explaining how the homotopy theory of simplicial sets arises naturally using derivators as the free completion of the trivial category by homotopy colimits. That derivators arise he also puts in a very natural way (and of course Heller knew all about). But perhaps all this is contained in the monograph that Tim linked to.

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    $\begingroup$ Excellent link and discussion! Yes, this is very relevant, it sounds like the homtop of simplicial sets is playing a role akin to free objects in universal algebra, so it does have a distinguished role. That in turn makes me even more optimist as far as my dream goes: perhaps other homotopy theories would then be obtainable by some kind of quotient operations... $\endgroup$ Jul 18, 2012 at 10:39
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This requires some topos theory I'm afraid, but gives, I think, a satisfactory explanation.

The category $\mathrm{Set}^{\Delta^{\mathrm{op}}}$ classifies intervals: the simplicial set $\Delta[1]$ is linearly ordered, with smallest and largest elements $0,1:\Delta[0]\to\Delta[1]$, and for any topos $\mathscr E$ and any linearly ordered object $I\in\mathscr E$ with distinguished smallest and largest elements $\bot,\top:1\to I$ there is a unique up to isomorphism $f_I:\mathscr E\to\mathrm{Set}^{\Delta^{\mathrm{op}}}$ with $f_I^*(\Delta[1])=I$. In particular, one can form various $\mathscr E$ containing one or other category of spaces, and for the corresponding interval $I$ there, $f_I^*(S)$ is the geometric realization of the simplicial set $S$, while for a space $X$, ${f_I}_*(X)$ is the singular simplicial set of $X$.

Thus simplicial sets are determined (up to equivalence of categories) by the fact that they classify all possible notions of a continuous path, hence of a homotopy.

Reference - e. g. in ncatlab; first occurrence of this I've seen is in Johnstone's first topos theory book, where it is attributed to Joyal.

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There is an interesting perspective on the role of simplicial sets in the Mem. AMS of Alex Heller (Homotopy Theories, number 383 in Memoirs Amer. Math. Soc. 1988). I suggest you look there as well as some of the more recent sources suggested in the previous answer mentioned above. His viewpoint has been superseded by the Cisinski-Maltsiniotis publications and Dugger's but it has the benefit of being fairly simple.

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    $\begingroup$ @Tim Alex was my PhD advisor...though not in homotopy theory (I was interested in categorical algebra and logic at the time). Alas, in spite of his almost legendary patience and clarity, he did not manage to infuse me with some understanding of abstract homotopy theory, due no doubt to my very modest talents and dedication. He did manage to convince me of two things, though: 1) that math would progress from category theory to homotopy theory, just like it did from sets to cats 2) that what is essential in homotopy is the existence of homotopy limits and colimits, not the particular $\endgroup$ Jul 17, 2012 at 16:05
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    $\begingroup$ way it is presented. As for simplicial sets, I will follow your advice and revisit his memoir (perhaps I will be finally able to digest it), as well as the later refs. I do remember though our last math conversation over the phone (2007) , in which I asked him more or less this exact question. He answered: -If you think that the simplicial homotopy cat is universal, PROVE IT!- That was Alex.... $\endgroup$ Jul 17, 2012 at 16:10
  • $\begingroup$ @MircoMannucci I tried to find his memoirs, but I couldn't access it anywhere. I tried to puchase it on "google books", but google says that my card number is invalid. Do you know how can I get a copy of this paper? Thank you $\endgroup$
    – Fernando
    Jul 31, 2015 at 20:40
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    $\begingroup$ @Fernando : ams.org/books/memo/0383 $\endgroup$ Jan 25, 2016 at 12:12

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