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August 19, 2017

Simplicial Sets vs. Simplicial Complexes

Posted by John Baez

I’m looking for a reference. Homotopy theorists love simplicial sets; certain other topologists love simplicial complexes; they are related in various ways, and I’m interested in one such relation.

Let me explain…

There’s a category SSetSSet of simplicial sets and a category SCpxSCpx of simplicial complexes.
There is, I believe, a functor

F:SCpxSSetF: SCpx \to SSet

that takes the simplices in a simplicial complex, which have unordered vertices, and creates a simplicial set in which the vertices of each simplex are given all possible orderings.

There is a geometric realization functor

||:SSetTop| \cdot | : SSet \to Top

There’s also a geometric realization functor for simplicial complexes. I’d better give it another name… let’s say

[]:SCpxTop [ \cdot ] : SCpx \to Top

Here’s what I want a reference for, assuming it’s true: there’s a natural transformation α\alpha from the composite

SCpxFSSet||Top SCpx \stackrel{F}{\longrightarrow} SSet \stackrel{|\cdot|}{\longrightarrow} Top

to

SCpx[]Top SCpx \stackrel{[\cdot]}{\longrightarrow} Top

and for any simplicial complex XX

α X:|F(X)|[X] \alpha_X : |F(X)| \to [X]

is a homotopy equivalence. (I don’t think it’s a homeomorphism; a bunch of simplices need to be squashed down.)

Since this field is fraught with confusion, I’d better say exactly what I mean by some things. By a simplicial set I mean a functor X:Δ opSetX : \Delta^{op} \to Set , and a morphism between simplicial sets to be a natural transformation between them. That seems pretty uncontroversial. By a simplicial complex I mean what Wikipedia calls an abstract simplicial complex, to distinguish them from simplicial complexes that are made of concrete and used as lawn ornaments… or something like that. Namely, I mean a set XX equipped with a family U XU_X of non-empty finite subsets that is downward closed and contains all singletons. By a map of simplicial complexes from (X,U X)(X,U_X) to (Y,U Y)(Y,U_Y) I mean a map f:XYf: X \to Y such that the image of any set in U XU_X is a set in U YU_Y.

On MathOverflow the topologist Allen Hatcher wrote:

In some areas simplicial sets are far more natural and useful than simplicial complexes, in others the reverse is true. If one drew a Venn diagram of the people using one or the other structure, the intersection might be very small.

Perhaps this cultural divide is reflected in the fact that the usual definition of an abstract simplicial complex involves a set that contains the set of vertices, whose extra members are entirely irrelevant when it comes to defining morphisms! It’s like defining a vector space to consist of two sets, one of which is a vector space in the usual sense and the other of which doesn’t show up in the definition of linear map. Someone categorically included would instead take the approach described on the nLab.

Ultimately you get equivalent categories either way. And as Alex Hoffnung and I explain, the resulting category of simplicial complexes is a quasitopos of concrete sheaves:

A quasitopos has many, but not quite all, of the good properties of a topos. Simplicial sets are a topos of presheaves, so from a category-theoretic viewpoint they’re more tractable than simplicial complexes.

Posted at August 19, 2017 7:48 AM UTC

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Re: Simplicial Sets vs. Simplicial Complexes

The claim is not true. While there is always a map |F(X)|[X]|F(X)| \to [X] that forgets the ordering of vertices, it is almost never a homotopy equivalence. The first counterexample is when XX is the 1-simplex! In this case [X][X] is an interval and |F(X)||F(X)| is a circle. (The two orderings of the 1-simplex produce two different 1-simplices that combine to form the circle.)

The functor FF is much-studied. If XX is the nn-simplex with vertices 0,1,,n0,1,\ldots,n then |F(X)||F(X)| is called the “complex of injective words”. It is homotopy equivalent to a wedge of nn-spheres, and the number of spheres in the wedge is equal to the number of derangements of the set {0,1,...,n}\{0,1,...,n\}. (A derangement is a permutation with no fixed points.)

The complex of injective words is an important tool in proofs of homological stability for the symmetric group and for various types of configuration spaces. That’s why I’m interested in it. It is also of interest in combinatorial topology, though I don’t know why, and there’s apparently also a connection to knot theory. I’ll try to provide references later.

A web search for “complex of injective words” should prove helpful!

Brief moment of pickiness: It is most common to define F(X)F(X) as a semi-simplicial set. In this case it is easy to make your definition of F(X)F(X) precise: an nn-simplex of F(X)F(X) is an nn-simplex of XX, together with an ordering of its vertices, and the ii-th face of such a simplex is obtained by erasing the ii-th vertex. If you wanted to do it all simplicially then you would have to talk about degenerate sinplices and so on. Anyhow, the good news is that you could rephrase everything you wrote to be semi-simplicial, and that in the end |F(X)||F(X)| doesn’t change!

Posted by: Richard Hepworth on August 19, 2017 10:26 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

In the interest of sending John back to his project with something more hopeful, there should also be a “Simplicial Realization” functor //:SCpxSSet/\cdot/ : SCpx \to SSet, with /X/ n/X/_n all the sequences of vertices in XX that happen to be contained in some abstract-simplex of XX (ignoring which one, in case there are many). The SCpx 1-cell {{1},{2},{1,2}}\{ \{1\},\{2\},\{1,2\}\} has, in /X/ 1/X/_1 both (1,2)(1,2) and (2,1)(2,1) but also (1,1)(1,1) and (2,2)(2,2) and in /X/ 2/X/_2 any triangles you like, such as (1,1,2)(1,1,2) and (1,2,1)(1,2,1) and …

Posted by: Jesse C. McKeown on August 19, 2017 3:19 PM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Jesse wrote:

In the interest of sending John back to his project with something more hopeful, there should also be a “Simplicial Realization” functor //:SCpxSSet/\cdot/ : SCpx \to SSet, with /X/ n/X/_n all the sequences of vertices in XX that happen to be contained in some abstract-simplex of XX (ignoring which one, in case there are many).

Yes, that’s what I really should have used instead of my hastily described FF. And if you don’t mind, I’d prefer to call your functor G:SCpxSSetG: SCpx \to SSet, since it’s an alternative to my F:SCpxSSetF: SCpx \to SSet, and we already have two funky bracket-shaped functors.

And though I said

(I don’t think it’s a homeomorphism; a bunch of simplices need to be squashed down.)

I now believe that when we use this GG, all the simplices that need to be squashed down, like those triangles you mention, are degenerate already and thus automatically squashed down by ||:SSetTop|\cdot| : SSet \to Top.

So in fact I now believe that the composite

SCpxGSSet||Top SCpx \stackrel{G}{\longrightarrow} SSet \stackrel{|\cdot|}{\longrightarrow} Top

is naturally isomorphic to

SCpx[]Top SCpx \stackrel{[\cdot]}{\longrightarrow} Top

Someone must have figured this stuff out already. Does anyone know a reference?

Posted by: John Baez on August 19, 2017 3:47 PM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

It is not true that |G(X)||G(X)| is naturally isomorphic to [X][X].

For example, if we take X={{1},{2},{1,2}}X=\{\{1\},\{2\},\{1,2\}\} as in Jesse’s example, then G(X)G(X) contains simplices of the form (1,2,1,2,1,)(1,2,1,2,1,\ldots) that are non-degenerate (they do not have consecutive repeated vertices) and have dimension as large as you like. This means that |G(X)||G(X)| is infinite-dimensional, unlike [X][X] which is 1-dimensional.

However, it sounds plausible that |G(X)|[X]|G(X)|\to [X] is a homotopy equivalence. For example, in |G(X)||G(X)| the 2-simplex (1,2,1)(1,2,1) appears as a globular 2-cell with boundary the edges (1,2)(1,2) and (2,1)(2,1), so the problematic circle formed by (1,2)(1,2) and (2,1)(2,1) can now be contracted.

Posted by: Richard Hepworth on August 19, 2017 4:12 PM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

The usual way to get a simplicial set from an abstract simplicial complex is via the “nerve” functor N:SCpxSSetN\colon \mathrm{SCpx}\to\mathrm{SSet}, which sends (X,U X)(X,U_X) to the usual nerve of the poset U XU_X. In this case there is a very natural homeomorphism |N(X)|[X]|N(X)|\approx [X], which is easy to write down. Of course, there is no bijective correspondence between simplices of XX and non-degenerate simplices of N(X)N(X).

Posted by: Charles Rezk on August 19, 2017 3:57 PM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

This is very nice, but alas it’s probably not the relation between simplicial sets and simplicial complexes that I need to think about. I think I need the functor called GG above, which takes a simplicial complex XX and turns it into a simplicial set for which an nn-simplex is an ordered (n+1)(n+1)-tuple of vertices of a single simplex of XX. I’m hoping that someone has shown |G(X)||G(X)| is naturally homotopy equivalent to [X][X].

I’m trying to relate two pieces of existing work, and I think bringing in the nerve would distract from the way some people are using simplicial complexes where others are using simplicial sets: it seems the simplicial complex people are using XX where the simplicial set people would use G(X)G(X).

If nobody has ever thought about this functor GG, I’ll either have to figure it out or wriggle around it with some expository trick.

It’s kinda neat how this thread has unearthed 3 fairly interesting functors from SCpxSCpx to SSetSSet.

Posted by: John Baez on August 20, 2017 1:45 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

The classical quick reference for some of this is in the little early survey article by Curtis: Simplicial homotopy theory, Advances in Math. 6 (1971), 107 – 209. He attributes to Barratt a result that the geometric realisation of a simplicial set can be triangulated, and thus is the geometric realisation (in the simplicial complex sense) of a simplicial complex. The proof is sketched in Curtis and involves the double barycentric subdivision of the simplicial set. (The discussion is in the last section of the paper.)

Posted by: Tim Porter on August 20, 2017 6:52 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Thanks!

My own puzzle does not involve barycentric subdivisions, but Curtis’ paper (free online from the Evil Empire) is still relevant.

In Example 1.3, Curtis describes a way to get a simplicial set from a simplicial complex. This should be approximately functor I’m calling G:SCpxSSetG : SCpx \to SSet in my conversation with Richard and Jesse above. However, Curtis works with “geometrical” simplicial complexes rather than the “abstract” simplicial complexes I’d prefer to think about. So, I should be careful.

In Section 1.29, Curtis describes the usual geometric realization of a simplicial set XX, which he calls R(X)R(X) rather than the now-standard |X||X|. He then claims that if we take a geometrical simplicial complex XX, and let KK be the simplicial set we get from it by the procedure in Example 1.3, then R(K)R(K) is homeomorphic to XX.

This seems to contradict Richard’s claim (above) that that R(G(X))R(G(X)) is not homeomorphic to XX. Perhaps the distinction between “geometrical” and “abstract” simplicial complexes is a bigger deal than I imagined. Or conceivably Curtis is just wrong, since he doesn’t actually prove his claim. Or maybe Richard is wrong… or more likely, I’m just mired in confusion.

Nonetheless, I feel I’m making progress! So thanks!

Posted by: John Baez on August 20, 2017 7:28 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

My understanding is that to get from a simplicial complex XX to a simplicial set T(X)T(X) (I should denote it by something different) in a sensible way you choose a total order on the set of vertices then then the nn-simplices T(X)T(X) will be the n+1n+1-tuples x 0x 1x nx_0\leq x_1\leq \ldots \leq x_n. Richard’s (1,2,1,2,1,2)(1,2,1,2,1,2) is thus disallowed. (This construction is Curtis’ 1.3 I think.) This is NOT a functor from SCpxSCpx to SSetSSet as it requires a choice, but study what happens when one changes the relative position in the total order of two vertices. The study of that situation makes me think that the end results will have homeomorphic realisations and also that there is a specifiable `concordance’ in some sense between them, i.e., a simplicial complex structure or triangulation of |X|×I|X|\times I with one ordering on the top and the other on the bottom of the cylinder. (I used something like this in my papers on Dave Yetter’s TQFT.) In other words, the lack of well-definition of TT is controlled.

I will not explore the question the behaviour of TT with respect to simplicial maps, but that does need exploring.

Posted by: Tim Porter on August 20, 2017 11:09 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Tim wrote approximately:

My understanding is that to get from a simplicial complex XX to a simplicial set KK in a sensible way you choose a total order on the set of vertices; then the nn-simplices of T(X)T(X) will be the n+1n+1-tuples x 0x 1x nx_0\leq x_1\leq \ldots \leq x_n. Richard’s (1,2,1,2,1,2)(1,2,1,2,1,2) is thus disallowed. (This construction is Curtis’ 1.3 I think.) This is NOT a functor from SCpxSCpx to SSetSSet as it requires a choice.

Brilliant! Yes, this is the construction in Curtis’ Example 1.3, and he starts by randomly choosing a total order on the vertices of XX, destroying functoriality but eliminating the extra simplices Richard found.

So, the ‘paradox’ that was bothering me seems to be resolved. We seem to be getting this: any simplicial complex XX we are finding some simplicial set KK whose geometric realization |K||K| is homeomorphic to the geometrical realization [X][X].

This procedure is not functorial, but KK has the same set of vertices as XX, and I believe there’s a canonical (meaning: no further choices are required) bijection between the nn-simplices of XX and the nondegenerate nn-simplices of KK.

On the other hand, Charles has pointed out a way to construct a functorial way to construct a simplicial set N(X)N(X) from XX for which |N(X)||N(X)| is homeomorphic to [X][X], but which has more vertices, and in general more nn-simplices.

Ordinarily I’d prefer the functorial construction, but for what I’m doing — trying to connect up to other people’s work — the non-functorial construction may be best.

Posted by: John Baez on August 20, 2017 11:51 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

If I am not wrong, Charles’ construction gives the barycentric subdivision of the one I mentioned but does not involve a choice.

(Some of this was discussed in a paper by Abels and Holz: Higher generation by subgroups, J. Alg, 160, (1993), 311– 341. It also discusses some applications of these ideas in geometric and combinatorial group theory. These involve calculation of higher syzygies from presentations of subgroups. Great Fun!)

Posted by: Tim Porter on August 20, 2017 4:37 PM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Tim wrote:

If I am not wrong, Charles’ construction gives the barycentric subdivision of the one I mentioned but does not involve a choice.

That sounds right to me.

Posted by: John Baez on August 21, 2017 1:40 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Since I said it is easy to describe the homeomorphism |N(X)|[X]|N(X)|\xrightarrow{\sim} [X], I should probably describe it.

For a simplicial complex (X,U X)(X,U_X), the space |N(X)||N(X)| is a quotient of the space whose points are data:

(S 0S k,t 0,,t k) (S_0\subseteq \cdots \subseteq S_k, \; t_0,\dots, t_k)

where S iU XS_i\in U_X and t i[0,1]t_i\in [0,1] with t i=1\sum t_i=1. (The identifications are the “obvious” ones: (i) if t i=0t_i=0 then delete S iS_i and t it_i, or (ii) if S i=S i+1S_i=S_{i+1} then delete S i+1S_{i+1} and replace t i,t i+1t_i,t_{i+1} with t i+t i+1t_i+t_{i+1}. Thus, every point of |N(X)||N(X)| has a unique “canonical form”, in which all S iS i+1S_i\neq S_{i+1} and all t i0t_i\neq0.)

Per nLab, [X][X] is the space of probability measures on the set XX of vertices which are supported on some element of U XU_X.

The function |N(X)|[X]|N(X)|\to [X] sends a point ({S i},{t i})(\{S_i\}, \{t_i\}) to the measure μ\mu defined by

μ(A)= j=0 kt j|AS j||S j|. \mu(A) = \sum_{j=0}^k t_j \frac{|A\cap S_j|}{|S_j|}.

Conversely, given μ[X]\mu\in [X] we can recover ({S i},{t i})(\{S_i\}, \{t_i\}) as follows. Let

M μ={μ({x})|xX,μ({x})>0}, M_\mu = \{ \mu(\{x\})\; |\; x\in X,\; \mu(\{x\})\gt 0\},

and list the elements of M μM_\mu as 1m 0>>m k>01\geq m_0\gt \cdots \gt m_k\gt 0, with m k+1=0m_{k+1}=0. Set

S j={xX|μ({x})m j} S_j = \{ x\in X\; | \; \mu(\{x\})\geq m_j \}

and

t j=|S j|(m jm j+1). t_j = |S_j|(m_j - m_{j+1}).

Then it’s easy to see that ({S i},{t i})μ(\{S_i\}, \{t_i\}) \mapsto \mu, and not hard to see that this is the only point of |N(X)||N(X)| which is sent to μ\mu.

Posted by: Charles Rezk on August 20, 2017 5:58 PM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Nice! It’s neat how a spoonful of probability theory helps the medicine go down. Has anyone used this connection to do anything interesting?

The formula for μ(A)\mu(A) reminds me of the formula for conditional probability, and now one of the pictures in the Wikipedia article on conditional probability reminds me of something about simplicial complexes.

Is there a nice probabilistic interpretation of your map |N(X)|[X]|N(X)| \to [X], or am I just imagining things?

Posted by: John Baez on August 21, 2017 1:52 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

concrete simplicial complexes as lawn ornaments

Posted by: Allen Knutson on August 20, 2017 11:43 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

These concrete simplicial complexes are just as ugly as I’d imagined!

Posted by: John Baez on August 21, 2017 1:38 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Where is this by the way?

Posted by: Todd Trimble on August 21, 2017 4:56 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

The Juno Beach Centre in Normandy, apparently. Image appears here.

Posted by: David Roberts on August 21, 2017 7:05 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Over on G+, Allen Knutson and I are having a parallel conversation that might be interesting. I could use some help!

I wrote:

I just explained the category of abstract simplicial complexes on Wikipedia, where they start the article with a truly wretched — but apparently widespread — definition of these objects.

He wrote:

Oh no. Are you to blame for all the “non-empty” on that page?

There should be a correspondence between abstract simplicial complexes on (say) {1,...,n}\{1,...,n\} and monomial ideals in k[x 1...x n]k[x_1...x_n]. Two such monomial ideals are 1\langle 1 \rangle, and x 1,...,x n\langle x_1,...,x_n \rangle.

I wrote:

I didn’t change the definition of abstract simplicial complex on Wikipedia, which is one of the standard ones. I just pointed out that it gives an equivalent category to another of the standard ones. I think you’re advocating a third nonequivalent one, which is good for Stanley–Reisner rings.

He wrote:

Here’s a nice construction that’s only reasonable if you allow the set VV of vertices to be larger than the set of 1-element faces. (I call the extra ones phantom vertices.)

Given a simplicial complex Δ\Delta (considered as a collection of subsets of the vertex set, closed under going down), define a new one Δ c\Delta^c where FF is a face of Δ c\Delta^c iff VFV \ F is a face of Delta. IIRC this is the basis of Alexander duality.

People are used to the idea that Delta might have “cone vertices”, that are in every face. Regardless of how you choose VV, if Δ\Delta has a cone vertex, then Δ c\Delta^c will have a phantom vertex. I see no reason to say “you’re not allowed to take Alexander dual unless you ditch all the cone vertices first”; it’s like Moore forbidding the empty set and having to always check, before taking an intersection, whether it’s nonempty.

I wrote:

I like your thing. I wonder if it deserves to be called an ‘augmented’ simplicial complex, since it reminds me slightly of an augmented simplicial set — a contravariant functor from the category of finite ordinals to Set, where we drop the usual eye-popping but ultimately reasonable condition that the ordinals be nonempty. An augmented simplicial set has (-1)-simplices as well as simplices of dimensions 0, 1, 2, ….

I’m not sure if your idea is connected to augmented simplicial sets or not! Over on the nn-Café we’ve been discussing functors (and a non-functorial process) relating simplicial complexes and simplicial sets. If these extend gracefully to functors relating your things and augmented simplicial sets, then you’re really talking about augmented simplicial complexes.

It’s then a separate argument whether the augmented things are the real things.

… and I’m less interested in that other argument, than in understanding Allen’s concept of simplicial complex and its relation, if any, to augmented simplicial sets.

Posted by: John Baez on August 21, 2017 8:35 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Maybe this is part of what’s going on. Let a ‘simplicial complex’ be the kind of simplicial complex I like. Then there’s another concept, possibly better in some ways, which I’ll call an ‘augmented’ simplicial complex.

Definition 1. A simplicial complex is set VV of vertices together with a collection Δ\Delta of nonempty finite subsets of VV such that:

  • Δ\Delta contains all singletons.

  • If FΔF \in \Delta and EFE \subseteq F is nonempty then EΔE \in \Delta.

For n1n \ge 1, the nn-element subsets in Δ\Delta are called (n1)(n-1)-simplices.

Definition 2. Let an augmented simplicial complex be a set SS together with a collection Δ\Delta of finite subsets of SS such that:

  • If FΔF \in \Delta and EFE \subseteq F then EΔE \in \Delta.

The elements vSv \in S such that {v}Δ\{v\} \in \Delta are called vertices, and the set of vertices is called VSV \subseteq S. For n0n \ge 0, the nn-element subsets in Δ\Delta are called (n1)(n-1)-simplices. The elements of SVS - V are called phantoms.

There’s a category InjInj consisting of nonempty finite sets and injections. A simplicial complex determines a presheaf

F:Inj opSetF : Inj^{op} \to Set

where F(n)F(n) is the set of (n1)(n-1)-simplices for n1n \ge 1. However, not every presheaf on InjInj arises this way.

Puzzle 1. Which presheaves on InjInj arise (up to isomorphism) from simplicial sets?

There’s also a category Inj +Inj_+ consisting of finite sets and injections. An augmented simplicial complex determines a presheaf

F:Inj + opSet F: Inj_+^{op} \to Set

where F(n)F(n) is the set of (n1)(n-1)-simplices for n0n \ge 0. However, not every presheaf on Inj +Inj_+ arises this way.

Puzzle 2. Which presheaves on Inj +Inj_+ arise (up to isomorphism) from augmented simplicial sets?

I feel I’m muddling some things here: for example, the phantoms versus the (1)(-1)-simplices. But I wanted to get this idea out in public, since I may not get around to thinking about it myself.

Posted by: John Baez on August 22, 2017 5:55 AM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

My understanding is that the 1-1-simplices are the connected components of the augmented simplicial set (which is not the same thing as being the connected components of the un-augmented simplicial set). This is because the connected components functor is left adjoint to the discrete inclusion of sets into augmented simplicial sets, and is therefore calculated by evaluating at the terminal object of the site. Since [1][-1] is the terminal object, the 1-1-simplices have to be the connected components. I’m not sure this is a good way to think about things – it’s more like “formal” connected components (since the 1-1-simplices are not necessarily the limit of all the other ones).

Do your phantom vertices relate to connected components in this way? I don’t immediately see a relation, but I haven’t sat down with it.

Posted by: David Jaz Myers on August 23, 2017 5:45 PM | Permalink | Reply to this

Re: Simplicial Sets vs. Simplicial Complexes

Yes, I think of the (1)(-1)-simplices as something like ‘formal connected components’, with the property that any number of actual connected components can belong to each formal connected component. I think they’re pretty different from the phantoms.

For those a bit intimidated by the adjunction you mentioned, a nice lowbrow way to understand all this is to note that there’s a face map

0:X 1X 0\partial_0 : X_1 \to X_0

assigning a (1)(-1)-simplex to each vertex of our augemented simplicial set, and the relations on face maps imply that if two vertices are connected by an edge, they must be assigned the same (1)(-1)-simplex. So, two vertices lying in the same connected component must belong to the same ‘formal connected component’, but not conversely.

Todd Trimble puts all this to good use in his note on bar constructions. I’ll quote a bit.


B.1. Acyclic structures 

We follow the algebraists' convention, taking the simplicial category 
to mean the category of finite ordinals (including the empty ordinal) 
and order-preserving maps. It is well-known that \Delta is initial 
amongst strict monoidal categories equipped with a monoid, which in 
\Delta is 1. This induces a monad 1 + - on \Delta, called the 
translation monad. By composition, this in turn induces a pullback 
comonad P on simplicial sets; by the Kan construction, it has a left
adjoint which is a monad C on simplicial sets, called the cone monad. 

To explain this terminology, we recall the topologists' convention, 
where \Delta is restricted to the full subcategory \Delta_+ of 
non-empty ordinals.  If S-set denotes the category of simplicial 
sets under the algebraists' convention, and S_{+}-set that under the
topologists' convention, then by restriction we get a functor 

S-set --> S_{+}-set, 

which has a left adjoint. The left adjoint augments a S_{+} set X by 
its set of path components \pi_{0}(X). Starting with X, we can apply 
the left augmentation, followed by the cone monad, followed by 
restriction S-set --> S_{+} set: this gives a monad which, passing 
to geometric realization, is the mapping cone of X --> \pi_{0}(X). 
The right adjoint to this monad carries a comonad structure; the
category of algebras over the monad is equivalent to the category 
of coalgebras over the comonad. This category of algebras/coalgebras
could be called the "acyclic topos": the algebras X are acyclic as
simplicial sets.  More to the point, the algebra structure CX --> X 
witnesses this acyclicity by providing a representative basepoint 
for each path component of X, together with a well-behaved simplicial
homotopy which contracts each component down to its basepoint. 

Definition: An acyclic structure is an algebra over C (or coalgebra 
over P). An S-acyclic structure is one augmented over a set S. 

A morphism between acyclic structures is just a C-algebra map; a 
morphism between S-acyclic structures is one whose component at S is 
the identity. 

It doesn't matter whether the monad C is taken under the algebraists' 
or topologists' convention: the category of C-algebras in S-set is
equivalent to the category of C-algebras in S_{+}-set, since given a
C-algebra in S-set, 

                      --> 
            ... X_{1} --> X_{0} --> X_{-1}, 

it follows from acyclicity that this portion of the simplicial 
structure is a split coequalizer, so that the augmentation map is 
the usual augmentation to its set of path components. 

In the sequel, it will be useful to regard an acyclic structure, 
as a functor X: \Delta^{op} --> Set, as a right coalgebra XT --> X 
over the translation comonad T: \Delta^{op} --> \Delta^{op}. 
Posted by: John Baez on August 24, 2017 2:40 AM | Permalink | Reply to this

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