# Is there a sense in which the homotopy theory of simplicial sets is the “paradigmatic” one?

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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The discussion at mathoverflow.net/questions/58497/… seems relevant. – Qiaochu Yuan Jul 17 '12 at 14:17
In particular, Peter Arndt's answer (I hadn't noticed this) describes a universal property. – Qiaochu Yuan Jul 17 '12 at 14:20
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... – Mirco A. Mannucci Jul 17 '12 at 14:35
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. – Akhil Mathew Jul 17 '12 at 19:51
(Here $\infty$ should be $(\infty, 1)$.) – Akhil Mathew Jul 17 '12 at 19:52

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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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... – Mirco A. Mannucci Jul 18 '12 at 10:39

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 in the book "Nonabelian algebraic topology", to prove a Higher Homotopy Seifert-van Kampen Theorem, and to prove 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

(see also Cubical vs. simplicial singular homology ) and saying to myself: surely there should be some mathematics which expresses that? Then Ehresmann's book "Categori\'es 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.

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@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 – Mirco A. Mannucci Jul 18 '12 at 10:28
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! – Ronnie Brown Jul 18 '12 at 22:43
Mirco: have you looked at cheng.staff.shef.ac.uk/guidebook/index.html. This has some of the characteristices that you want perhaps. – Tim Porter Jul 20 '12 at 9:23
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! – Ronnie Brown Jul 20 '12 at 15:11
Another point is that in the index to the Chang and Lauda "Guidebook" (version 2004) the words "globular" and "simplicial" occur but not "cubical". Another point about David's thesis is that to get a good notion of "Poly-category" to define "Poly-sets", he gives a crucial notion of "marked complex", (every cell has a "marked" face) which turns out to be the same as the notion of complex with a discrete vector field given much later by Robin Forman, and used by Graham Ellis for computational purposes! – Ronnie Brown Jul 20 '12 at 21:30

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 superceded by the Cisisnski-Maltsiniotis publications and Dugger's but it has the benefit of being fairly simple.

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@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 – Mirco A. Mannucci Jul 17 '12 at 16:05
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.... – Mirco A. Mannucci Jul 17 '12 at 16:10
@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 – Fernando Jul 31 '15 at 20:40
@Fernando : ams.org/books/memo/0383 – Mirco A. Mannucci Jan 25 at 12:12

This requires some topos theory I'm afraid, but gives, I think, a satisfactory explanation.

The category $\mathrm{Set}^{\Delta^{\mathrm{op}}}$ $\textit{classifies intervals}$: 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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