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Stream: theory: algebraic topology

Topic: intuition for simplicial sets

Leopold Schlicht (Oct 14 2021 at 19:32):

I'm trying to extract the ideas of the following formal mathematical concepts:

• simplicial set,
• the standard $n$-simplex, $\Delta^n$,
• simplicial subsets,
• the simplicial subsets of the form $\{v\}$, where $v$ is a vertex of a simplicial set,
• the boundary $\partial \Delta^n$ of $\Delta^n$,
• the horns $\Lambda^n_i$.

All this is covered in Kerodon, but I'm missing the intuitive understanding.

Leopold Schlicht (Oct 14 2021 at 19:32):

Here is how I currently think about simplicial sets: a simplicial set $S_\bullet$ is given by a family $\{S_n\}_{n\geq 0}$ of sets, a family $\{d_i\colon S_n\to S_{n-1}\}_{0\leq i\leq n}$ of face maps, and a family $\{s_i\colon S_n\to S_{n+1}\}_{0\leq i\leq n}$ of degeneracy maps. I visualize the elements of $S_0$ as points, the elements of $S_1$ as line segments, the elements of $S_2$ as triangles, the elements of $S_3$ as tetrahedrons, and so on. To a first degree of approximation, I visualize $S_\bullet$ as the disjoint union of all these shapes. But this isn't true, because they somehow have to get glued together in a way encoded by the face and degeneracy maps. For instance, when I want to glue two triangles $A$ and $B$ together, I take $S_0=\{1, 2, 3, 3'\}$ ($1$, $2$, and $3$ are the vertices of $A$, while $1$, $2$, and $3'$ are the vertices of $B$), $S_1=\{a, b, c, b', c'\}$ (where $a$, $b$, and $c$ are the edges of $A$ and $a$, $b'$, and $c'$ are the edges of $B$), and $S_2=\{A, B\}$. Then I set, say, $d_0(A)=d_0(B)=a$, i.e., $A$ and $B$ share the edge $a$, and $d_0(a)=1$ and $d_1(a)=2$. I would set $S_n=\emptyset$ for all $n>2$.

Leopold Schlicht (Oct 14 2021 at 19:33):

Now let's try to define the standard $n$-simplex. For example, consider $\Delta^2$. With my current understanding it would make sense to set $\Delta^2_0=\{1, 2, 3\}$, $\Delta^2_1=\{a, b, c\}$, and $\Delta^2_2=\{A\}$, where $d_0(A)=a$, $d_1(A)=b$, and $d_2(A)=c$, and so on. For $j>2$ I would set $\Delta^2_j=\emptyset$, since I don't see any higher dimensional shapes when imagining a triangle.

The official definition is quite different: $\Delta^n$ is defined to be the representable presheaf $\hom_\Delta(-, [n])$. In particular, $\Delta^n$ always contains simplices in any dimension. My first question is: why does that make sense? Is that definition somehow equivalent to the definition I gave? I guess not, but what's wrong with my definition? (I guess this has something to do with the fact that I don't really understand the degeneracy maps and my general understanding of what the description of a simplicial set means isn't complete.)

Leopold Schlicht (Oct 14 2021 at 19:33):

Above I gave the data of a simplicial set. But actually these data have to satisfy the simplicial identities, such as $d_i\circ d_j=d_{j-1}\circ d_i$. What's the intuition behind this simplicial identity? Why should the $i$-th face of the $j$-th face be the $j-1$-th face of the $i$-th face? I mean, using the definition of a simplicial set as a presheaf on the simplex category, I am able to verify these identities, but I don't have any intuition for them.

Leopold Schlicht (Oct 14 2021 at 19:33):

The boundary of $\Delta^n$ is the simplicial subset $\partial \Delta^n$ with $m$-simplices given by $m$-simplices of $\Delta^n$ which are not surjective. Why does that formal definition implement the idea of removing the interior of $\Delta^n$?

Similarly, the $i$-th horn in $\Delta^n$, $\Lambda^n_i$, consists of the $m$-simplices of $\Delta^n$ for which there is a $j\not = i$ doesn't lying in the image. Why does this implement the idea of removing the face of $\Delta^n$ opposite its $i$-th vertex?

Leopold Schlicht (Oct 14 2021 at 19:34):

The simplicial sets $\Delta^n$, $\partial \Delta^n$, and $\Lambda^n_i$ can be described by universal properties: the universal property of $\Delta^n$ is the Yoneda lemma, while the maps $\partial \Delta^n$ to $S_\bullet$ correspond to tuples $(\sigma_0, \dots, \sigma_n)$ of $n-1$-simplices in $S_\bullet$ such that for all $j<k$, $d_j(\sigma_k)=d_{k-1}(\sigma_j)$, similarly for the horns. Is there a geometric intuition for these universal properties?

James Deikun (Oct 14 2021 at 19:51):

Leopold Schlicht said:

Now let's try to define the standard $n$-simplex. For example, consider $\Delta^2$. With my current understanding it would make sense to set $\Delta^2_0=\{1, 2, 3\}$, $\Delta^2_1=\{a, b, c\}$, and $\Delta^2_2=\{A\}$, where $d_0(A)=a$, $d_1(A)=b$, and $d_2(A)=c$, and so on. For $j>2$ I would set $\Delta^2_j=\emptyset$, since I don't see any higher dimensional shapes when imagining a triangle.

The official definition is quite different: $\Delta^n$ is defined to be the representable presheaf $\hom_\Delta(-, [n])$. In particular, $\Delta^n$ always contains simplices in any dimension. My first question is: why does that make sense? Is that definition somehow equivalent to the definition I gave? I guess not, but what's wrong with my definition? (I guess this has something to do with the fact that I don't really understand the degeneracy maps and my general understanding of what the description of a simplicial set means isn't complete.)

Yeah at first this is kind of counterintuitive. It seems like what you were visualizing at first is more like a simplicial complex, and those were invented first for a reason.

Degenerate simplices are simplices that are 'flattened' -- they have at least one edge that is from a point to the same point. This only makes sense relative to a simplicial set they are contained in, i.e. not for the abstract simplicial simplices. A simplex can only be degenerate if it is a specific simplex in a simplicial set.

The degeneracy maps make, for each k-simplex S, k different degenerate k+1-simplices, putting the extra degenerate edge on each possible vertex. Every face of each of these degenerate simplices is either a degenerate simplex "over" a face of S, or S itself (twice). The mixed simplicial identities communicate this.

James Deikun (Oct 14 2021 at 19:54):

Leopold Schlicht said:

Above I gave the data of a simplicial set. But actually these data have to satisfy the simplicial identities, such as $d_i\circ d_j=d_{j-1}\circ d_i$. What's the intuition behind this simplicial identity? Why should the $i$-th face of the $j$-th face be the $j-1$-th face of the $i$-th face. I mean, using the definition of a simplicial set as a presheaf on the simplex category, I am able to verify these identities, but I don't have any intuition for them.

These identities communicate the shape of a simplex -- the way that its faces are all glued together to form its boundary.

John Baez (Oct 14 2021 at 19:56):

It probably helps to draw everything, first for a triangle and then for a tetrahedron.

John Baez (Oct 14 2021 at 19:58):

Topology is after all about space, so it's good to actually see what's going on in examples.

James Deikun (Oct 14 2021 at 19:58):

Leopold Schlicht said:

By the way, it seems the definition of a simplicial set allows one to glue simplices together "in the wrong dimension". For instance, I could glue two triangles together at one vertex. Also, what seems to be possible is to glue a triangle on a tetrahedron. This seems weird. Is this really allowed?

Yes, it's really allowed. It may seem weird when you do this in isolation, but it's necessary to allow some really basic constructions to exist. For example, if you glue 6 triangles to form a hexagon, many of the pairs are glued just at a corner.

In general, though, simplicial sets don't represent only spaces of uniform dimension like manifolds, but can represent a much greater variety of spaces, or other things entirely like categories.

John Baez (Oct 14 2021 at 20:01):

When someone says "simplicial set", an image sort of like this should pop into your mind:

John Baez (Oct 14 2021 at 20:02):

... except with arrows on all the edges, etc.

John Baez (Oct 14 2021 at 20:04):

Here we aren't drawing the degenerate simplexes, but you have to remember that they are there!

Naso (Oct 15 2021 at 03:07):

I'm also learning about simplicial sets... I found the following articles really helpful in gaining intuition:

'A leisurely introduction to simplicial sets' by Emily Riehl https://math.jhu.edu/~eriehl/ssets.pdf

'An elementary illustrated introduction to simplicial sets' by Greg Friedman https://arxiv.org/abs/0809.4221

'A Quick Tour of Basic Concepts in Simplicial Homotopy Theory' by John Baez https://math.ucr.edu/home/baez/calgary/homotopy.html

Patrick Nicodemus (Oct 16 2021 at 02:48):

Leopold;
Are you familiar with the geometric realization functor?
Simplicial sets can be interpreted as combinatorial schemas for constructing simplicial complexes as topological spaces.
If you study examples of this construction being computed it may be helpful.
A common introduction to simplicial sets is the one written by Greg Friedman https://arxiv.org/abs/0809.4221

Leopold Schlicht (Oct 16 2021 at 16:23):

James Deikun said:

Degenerate simplices are simplices that are 'flattened' -- they have at least one edge that is from a point to the same point. This only makes sense relative to a simplicial set they are contained in, i.e. not for the abstract simplicial simplices. A simplex can only be degenerate if it is a specific simplex in a simplicial set.

Thanks! However, there are still a lot of things unclear to me. What do you mean by "flattened"? Is my naive definition of $\Delta^n$ equivalent to the official "representable presheaf" definition or is my definition wrong? If it's wrong: why?

Why does the "representable presheaf" definition cover the intuition of an $n$-simplex?

The degeneracy maps make, for each k-simplex S, k different degenerate k+1-simplices, putting the extra degenerate edge on each possible vertex.

What do you mean by that? Why do we consider degeneracy maps and degenerate simplices at all? My definition of $\Delta^n$ doesn't use them, so why do we need them?

These identities communicate the shape of a simplex -- the way that its faces are all glued together to form its boundary.

What's the intuition behind the simplicial identity? In my posts I gave a concrete example: why should the $i$-th face of the $j$-th face be the $j-1$-th face of the $i$-th face?

Reid Barton (Oct 16 2021 at 16:29):

Doesn't seem very concrete yet, it still has these funny letters $i$ and $j$ in it!

Reid Barton (Oct 16 2021 at 16:30):

Leopold Schlicht said:

Now let's try to define the standard $n$-simplex. For example, consider $\Delta^2$. With my current understanding it would make sense to set $\Delta^2_0=\{1, 2, 3\}$, $\Delta^2_1=\{a, b, c\}$, and $\Delta^2_2=\{A\}$, where $d_0(A)=a$, $d_1(A)=b$, and $d_2(A)=c$, and so on.

It's exactly this, if you keep going.

Reid Barton (Oct 16 2021 at 16:30):

The existence or lack of degeneracies doesn't matter for the relations between the face maps.

Leopold Schlicht (Oct 16 2021 at 16:31):

John Baez said:

It probably helps to draw everything, first for a triangle and then for a tetrahedron.

Topology is after all about space, so it's good to actually see what's going on in examples.

When someone says "simplicial set", an image sort of like this should pop into your mind:

I appreciate your reply, but I'm a bit confused by your answer, because in the posts above I tried to explain how I currently think about simplicial sets and what I think are points I currently don't understand. In particular, I already described that I visualize the 2-simplices as triangles and the 3-simplices as tetrahedrons. It would be a bit more helpful if you could answer one of the specific questions I asked. For instance, why does the formal definition of $\partial \Delta^n$ implement the idea of removing the interior of $\Delta^n$?

Reid Barton (Oct 16 2021 at 16:34):

These $a$, $b$, $c$ are supposed to be the sides of a triangle--so their faces (endpoints) must be compatible in a certain way; and that's what the simplicial face identities encode.

Leopold Schlicht (Oct 16 2021 at 16:35):

Nasos Evangelou-Oost said:

I'm also learning about simplicial sets... I found the following articles really helpful in gaining intuition:
...

Thanks for the literature suggestions, but I came up with these questions after reading a lot of literature (these are things I feel are not covered well in the literature). So I'd be more happy to talk about the questions I asked. :grinning_face_with_smiling_eyes:

Leopold Schlicht (Oct 16 2021 at 16:41):

Reid Barton said:

Doesn't seem very concrete yet, it still has these funny letters $i$ and $j$ in it!

I mean "concrete" in the sense that I not just wrote "I don't understand the concept of a simplicial set, please explain it to me", but rather "here are a few things I don't understand: ..." :grinning_face_with_smiling_eyes:

Leopold Schlicht said:

Now let's try to define the standard $n$-simplex. For example, consider $\Delta^2$. With my current understanding it would make sense to set $\Delta^2_0=\{1, 2, 3\}$, $\Delta^2_1=\{a, b, c\}$, and $\Delta^2_2=\{A\}$, where $d_0(A)=a$, $d_1(A)=b$, and $d_2(A)=c$, and so on.

It's exactly this, if you keep going.

What do you mean by keep going? I would go on with $S_n=\emptyset$ for $n>2$, but this seems to be wrong for some reason I don't understand.
Reid Barton said:

These $a$, $b$, $c$ are supposed to be the sides of a triangle--so their faces (endpoints) must be compatible in a certain way; and that's what the simplicial face identities encode.

Alright, thanks. But I still don't understand why the $i$-th face of the $j$-th face should be the $j-1$-th face of the $i$-th face.

Leopold Schlicht (Oct 16 2021 at 16:42):

John Baez said:

... except with arrows on all the edges, etc.

Why do the arrows have a direction?

Leopold Schlicht (Oct 16 2021 at 16:42):

John Baez said:

Here we aren't drawing the degenerate simplexes, but you have to remember that they are there!

But why are they there?

Reid Barton (Oct 16 2021 at 16:46):

Historically "simplicial set" used to mean what we now call a semisimplicial set, which is something that only has the face maps (satisfying the face identities) but not the degeneracies, which is exactly the sort of thing you called $\Delta^2$ originally. It's probably a good idea to try to build intuition for that first, since a lot of it will transfer to full simplicial sets.

Reid Barton (Oct 16 2021 at 16:47):

Leopold Schlicht said:

Leopold Schlicht said:

Now let's try to define the standard $n$-simplex. For example, consider $\Delta^2$. With my current understanding it would make sense to set $\Delta^2_0=\{1, 2, 3\}$, $\Delta^2_1=\{a, b, c\}$, and $\Delta^2_2=\{A\}$, where $d_0(A)=a$, $d_1(A)=b$, and $d_2(A)=c$, and so on.

It's exactly this, if you keep going.

You said what the faces of $A$ are, but you didn't say what the faces of $a$, $b$, $c$ are.

Reid Barton (Oct 16 2021 at 16:48):

And those are exactly what the simplicial face identities talk about

Leopold Schlicht (Oct 16 2021 at 17:05):

Reid Barton said:

It's probably a good idea to try to build intuition for that first, since a lot of it will transfer to full simplicial sets.

Yeah, and I think I already have a good intuition for semisimplicial sets. And now I'm asking about the intuition for full simplicial sets! :smile:

James Deikun (Oct 16 2021 at 17:32):

An n-simplex has n+1 faces; each face differs from each other by the inclusion of one single vertex. Each face is glued to each of the others along its own face that does not include the differing vertex. There is a nice consistent numbering for faces where the facet that glues the i-th face to the j-the face is the i-th face of the j-th face and the j-1-th face of the i-th face. This is a good thing to work out on the triangle and the tetrahedron.

James Deikun (Oct 16 2021 at 17:38):

A degenerate simplex is "flat" or "flattened" in that it has a face that includes every vertex of the entire simplex. It is, for example, a triangle with only 2 distinct vertices, so that it is collapsed into a line, or a tetrahedron with only 3 distinct vertices, so that it is collapsed into a triangle. There are other equivalent characterizations, like 'it has 2 faces that are the same simplex'.

James Deikun (Oct 16 2021 at 17:40):

Also, I think you have a good intuition for simplicial complexes, not semisimplicial sets, since you are having trouble with the numbering and orientation of faces, which are the main distinctions between simplicial complexes and semisimplicial sets.

Reid Barton (Oct 17 2021 at 11:41):

Leopold Schlicht said:

John Baez said:

... except with arrows on all the edges, etc.

Why do the arrows have a direction?

You can start by just thinking about semisimplicial sets with only 0- and 1-simplices and why they are directed graphs and not undirected graphs.

Reid Barton (Oct 17 2021 at 11:45):

(Unless you mean why do we want them to be there, which is a more complicated question.)

Patrick Nicodemus (Oct 17 2021 at 19:58):

@Leopold Schlicht I will respond to your question with my answer for why the representable functor $y([n])$ is the 'right' choice of representative for the $n$ simplex $\Delta^n$. I also answer your question "why are the degenerate simplices there" by explaining that they play a strong role wrt products - the product of semisimplicial sets does not represent the geometric Cartesian product of the associated complexes, but the product of simplicial sets does, and the difference comes down to degeneracies.

Patrick Nicodemus (Oct 17 2021 at 19:59):

This is a bit of an abstract answer so it might be too high level for this concrete situation but there's a general theorem of category theory that says the following.

Let $M$ be a small category, and let $C$ be a category with all small colimits. Let $F : M\to C$ be a functor. For the concrete case that follows, I am interested in the case where $M$ is the simplex category, and $C$ is the category of topological spaces, and $F$ sends the ordinal $[n]={0..n}$ to $\Delta^{n}$, which is the convex hull of $n+1$ points in a real vector space in general position.

Patrick Nicodemus (Oct 17 2021 at 19:59):

For any presheaf $P$ on $M$, we can construct the "weighted colimit"" of $F$ with weights in $P$ (also known as the "Indexed colimit" in some older literature. This is a gluing construction which is built out of taking $|P(m)|$ many copies of $F(m)$ for each object $m$ in $M$, and gluing these together along the maps of the diagram $F$ in a way specified by the maps of $P$. In this way we can interpret presheaves on $M$ as combinatorial schemas for gluing together the objects in the diagram $F$ along the maps of $F$; i.e. interpreting the objects $F(m)$ as basic building blocks, out of which we can construct more complex objects.

Patrick Nicodemus (Oct 17 2021 at 19:59):

If $F$ is regarded as fixed and $P$ is allowed to vary, then this defines a functor from presheaves on $M$ to $C$ which sends each presheaf $P$ to the weighted colimit of $F$ with weights in $P$; it sends each combinatorial schema to the thing it describes in $C$.

Patrick Nicodemus (Oct 17 2021 at 19:59):

This functor, which I call the "realization" $R$ (see the nlab page on nerve-realization for more on this) has some interesting theoretical properties. $R$ is in a precise sense a canonical choice of extension of $F: M\to C$ along the Yoneda embedding $y : M\to \hat{M}$, it is the "left Kan extension" of $F$ along $y$, and we have a natural isomorphism $R\circ y \cong F$. $R$ is the unique colimit-preserving functor such that $R\circ y\cong F$. We can interpret this as follows. Each presheaf $P$ on $M$ can be written in a canonical way as the colimit of representable presheaves. This is known as the Density theorem, and it is a corollary of the Yoneda lemma. Since we ask that for every representable presheaf $y(m)$, $R(y(m))\cong F(m)$ naturally in $m$, and $R$ preserves colimits, it follows that for an arbitrary presheaf $P$, $R(P)$ should be glued together out of the objects $F(m)$ in precisely the same way that $P$ itself is to be glued together out of the representable presheaves $y(m)$ as prescribed by the Density lemma.

Patrick Nicodemus (Oct 17 2021 at 19:59):

So, to answer your question "why do we let $y(n)$ represent $\Delta^{n+1}$, my answer to this is that the isomorphism $R\circ y\cong F$ tells us that there is a canonical choice of presheaf $P$ as combinatorial schema such that $R(P)\cong F(m)$; namely we should take $P=y(m)$. Thus the representable functor $y([n])$ is the canonical choice of presheaf whose realization is $\Delta^n$.

Patrick Nicodemus (Oct 17 2021 at 20:00):

As for the degenerate simplices, surely others have explained this in the thread or you've seen this in the literature but the most immediately accessible technical reason why we care about degenerate simplices is that if $\sigma$ is degenerate in $X_n$ and $\tau$ is degenerate in $Y_n$, it is not necessarily the case that the pair $(\sigma,\tau)$ is degenerate in $(X\times Y)_n = X_n\times Y_n$. Here is an exercise in combinatorics to help you understand this fact: call a simplex maximal nondegenerate if it is nondegenerate and it is not the face of any other nondegenerate simplex. If you can explicitly describe the maximal nondegenerate simplices of a simplicial set then you have a good sense of what its geometric realization should look like. For example, it is easy to see that $y([n])$ has one and only one maximal nondegenerate
simplex, namely the identity map $[n]\to [n]$ in degree $n$. Prove that the maximal nondegenerate simplices of the product complex $y([p])\times y([q])$ are all concentrated in degree $p+q$ and are in one-to-one correspondence with $(p,q)$ shuffles. (There could be a minor typo here in that it should really be $(p+1,q+1)$ shuffles or something of this ilk.)

Patrick Nicodemus (Oct 17 2021 at 20:01):

The category of semisimplicial sets has categorical products. But these products are not geometrically meaningful. These products do not really represent the product of the complexes in Top they describe. Another way to say this is that the realization functor from semisimplicial sets to Top does not preserve products. It is easy to see why this is the case. The product of a $p$ dimensional thing and a $q$ dimensional thing is a $p+q$ dimensional thing. But for semisimplicial sets $X$ and $Y$ $\sigma$ is a $p$-simplex in $X$ and $\tau$ is a $q$ simplex in $Y$, there is no obvious element of the product semisimplicial set $X\times Y$ which represents the pair $(\sigma,\tau)$. Where in the product semisimplicial set $X\times Y$ can we see the product of these cells? On the other hand if $\sigma\in X_n$ and $\tau\in Y_n$ then the pair $(\sigma,\tau)$ is a well defined member of $(X\times Y)_n= X_n\times Y_n$. But this is not geometrically meaningful. The product of $\sigma,\tau$ should be $2n$ dimensional, not $n$ dimensional. It is not easy to see how to interpret the meaining of the pair $(\sigma,\tau)$ in $X\times Y$.

Patrick Nicodemus (Oct 17 2021 at 20:02):

But if $X$ and $Y$ are simplicial, rather than semi-simplicial sets, something incredible becomes possible. Degeneracies allow a simplex that, intuitively, is $p$ dimensional, to masquerade for formal purposes as a simplex of arbitrarily high dimension. Thus if $\sigma\in X_p$ and $\tau$ is in $Y_q$, then we may repeatedly apply degeneracy maps to $\sigma,\tau$ respectively until we get elements $\sigma',\tau'$ in $X_{p+q},Y_{p+q}$ respectively. Then the pair $(\sigma',\tau')$ in $(X\times Y)_{p+q}$ is a thing of the "right dimension". Actually there are many different possible choices of ways to write $x,y$ as degenerate $p+q$ cells $x',y'$; together all these degenerate cells represent a triangulation of the product $\Delta^p\times\Delta^q$. To see what I am saying, I encourage you to calculate by hand the geometric realization of $y([1])\times y([1])$ and see that it is a square, and $y([1])\times y([2])$ and see that it describes a triangular prism.

Patrick Nicodemus (Oct 17 2021 at 20:04):

As a consequence of this. the geometric realization functor from simplicial sets to topological spaces preserves products up to handwaving about the distinction between Top and CGHaus, CGWH or some other good category of well behaved topological spaces

Patrick Nicodemus (Oct 17 2021 at 20:04):

I see that you say you have read the literature but i continue to learn more about simplicial sets every day by reading the literature. Just today I came across Kan's paper which demonstrates that the homotopy groups of any space or simplicial set can be expressed as the homology groups of a certain chain complex of not-necessarily Abelian groups. This reduces homotopy to homology. Interestingly one also has the theorem of Dold-Thom which shows that homology of a space $X$ can be reduced to the homotopy groups of a space $S(X)$, so the two notions are intertranslateable.
There's been a lot written about simplicial sets. I recommend reading the first few sections of Manin and Gelfand's Methods of Homological Algebra for their treatment of simplicial sets and geometric realization.

John Baez (Oct 18 2021 at 13:50):

Leopold Schlicht said:

John Baez said:

... except with arrows on all the edges, etc.

Why do the arrows have a direction?

Arrows always have a direction, that's what arrows are: a direction.

But the reason edges in a simplicial set have a direction is that each edge $e$ has a specified "starting-point" $d_1(e)$ and "ending-point" $d_0(e)$

John Baez (Oct 18 2021 at 13:53):

Leopold Schlicht said:

John Baez said:

Here we aren't drawing the degenerate simplexes, but you have to remember that they are there!

But why are they there?

Because for a simplicial set we have $n+1$ maps called "degeneracies" from its set of $n$-simplices to its set of $(n+1)$-simplices. This is just part of the definition. So, for example, each 1-simplex (or "edge") gives rise to two 2-simplexes (or "triangles"), and each 2-simplex (or "triangle") gives rise to three 3-simplexes (or "tetrahedra"), and so on.

John Baez (Oct 18 2021 at 13:57):

As a result, every simplicial set except the empty simplicial set has, for all $n$, a nonempty set of $n$-simplexes. But we in examples don't draw most of these, because we don't draw the degenerate ones.

John Baez (Oct 18 2021 at 14:00):

You can draw the degenerate ones, and you need to learn how to understand this stuff. (Alas, it's too tiresome to explain how without a blackboard; it would take me an hour instead of the 2 minutes I'd need with a blackboard.) But we don't need to draw the degenerate simplices, which is good because there are always infinitely many, of arbitrarily high dimension, unless your simplicial set is empty.

Leopold Schlicht (Oct 18 2021 at 15:18):

James Deikun said:

There is a nice consistent numbering for faces where the facet that glues the i-th face to the j-the face is the i-th face of the j-th face and the j-1-th face of the i-th face. This is a good thing to work out on the triangle and the tetrahedron.

This comment is really helpful, thanks!

Leopold Schlicht (Oct 18 2021 at 15:19):

Reid Barton said:

(Unless you mean why do we want them to be there, which is a more complicated question.)

Yeah, that's what I really wanted to ask. :grinning_face_with_smiling_eyes:

Leopold Schlicht (Oct 18 2021 at 16:38):

@Patrick Nicodemus and @John Baez Thanks for your very helpful explanations!

I think this clears up the vast majority of my original questions and confusions!

Leopold Schlicht (Oct 18 2021 at 17:04):

John Baez said:

But the reason edges in a simplicial set have a direction is that each edge $e$ has a specified "starting-point" $d_1(e)$ and "ending-point" $d_0(e)$

Is there some specific reason for $d_1(e)$ being called the source whereas $d_0(e)$ is called the target? Im wondering because Lurie uses the same convention.

Patrick Nicodemus (Oct 18 2021 at 18:02):

Leopold Schlicht said:

Is there some specific reason for $d_1(e)$ being called the source whereas $d_0(e)$ is called the target? Im wondering because Lurie uses the same convention.

The totally ordered set $[n] = \left\{0 < 1 < 2 \dots < n\right\}$ represents the $n$-simplex which is the convex hull of $n+1$ points $v_0,\dots, v_n$ in general position in $n+1$ dimensional space. In general when we talk about these simplices, we understand that the simplices carry an ordering, $v_0 < v_1 < \dots < v_n$. This is useful for a few reasons. In homology theory, simplices carry an orientation, which is defined in terms of the ordering of the vertices, so this is one motivation for why we would insist on an ordering of the vertices. In turn let me motivate the concept of orientation - when we glue two triangles together in the plane along a common edge to form a rhombus, we need some way of saying that the two glued faces "cancel each other out", so that the boundary of the rhombus only consists of the four outer edges, and does not include some kind of superposition of the two edges that pass through the center. The way we do this is by specifying that these edges have opposite "orientation". Also, some deep theorems in topology, like Poincare duality, rely on a space being orientable, and we define the orientability of the space ultimately in terms of coherent orientations of simplices.

Patrick Nicodemus (Oct 18 2021 at 18:05):

Also, there is a close connection between simplicial objects and the theory of monoids. If you have some monoid $M$, then there is a cosimplicial object $X$, where $X_n$ is the set of all (ordered!) strings of $n+1$ elements $m_0,m_1,\dots, m_n$. The face maps $\delta_i: n\to n+1$ corresponding to inserting a copy of the monoid identity element $e$ in the $i$-th position, and the degeneracies $s_i : n+2\to n+1$ correspond to multiplying together two adjacent elements. Here, the ordering of the elements is important because if the monoid is not commutative, then permuting the elements may give a different answer for the multiplication.

Patrick Nicodemus (Oct 18 2021 at 18:06):

All I am trying to stress here is that there are very good reasons to think of simplices of a simplicial set as having their vertices fixed in a certain order.

Patrick Nicodemus (Oct 18 2021 at 18:07):

In particular for a $1$ simplex $\sigma = [v_0,v_1]$, because we have the ordering $0<1$, in some contexts it is natural to think of the edge as going from $v_0$ to $v_1$, and not the symmetric relationship of being an edge between $v_0$ and $v_1$

Patrick Nicodemus (Oct 18 2021 at 18:10):

Perhaps I should have started with this example, but since Lurie works in higher category theory there is a very specific reason why he would care about the orientations of edges. If $C$ is any small category then there is a simplicial set associated called the nerve of $C$. The $0$ cells (vertices) of $C$ are the objects. The $1$ cells are the morphisms $f : c\to c'$, where we set $d_1(f)=c$, $d_0(f)=c'$. The $n$ cells for higher $n$ are strings of $n$ successive consecutive morphisms, and face and degeneracy maps are given by composition of maps and inserting copies of the identity map. In this case a $1$ cell in the nerve literally denotes a morphism from its source, $d_1(f)$, to its target, $c_0$. Higher category theory studies simplicial sets that have similar formal properties to the nerve of a category (weak Kan complexes) and so the name "source" and "target" are borrowed from this important case which is being generalized.

John Baez (Oct 19 2021 at 02:54):

Leopold Schlicht said:

John Baez said:

But the reason edges in a simplicial set have a direction is that each edge $e$ has a specified "starting-point" $d_1(e)$ and "ending-point" $d_0(e)$

Is there some specific reason for $d_1(e)$ being called the source whereas $d_0(e)$ is called the target? I'm wondering because Lurie uses the same convention.

Patrick already answered this, but: each face of an simplex contains all but one of the vertices of that simplex. So, the only reasonable definition of the ith face of a simplex is that it's the face missing the ith vertex.

John Baez (Oct 19 2021 at 02:55):

In the case of a 1-simplex this leads to the superficially annoying fact that the 1-simplex should be drawn as arrow going from its 1st face to its 0th face, not the other way around.

John Baez (Oct 19 2021 at 02:56):

But this actually works out nicely: it means that in homology, where any path defines a 1-chain, the boundary of this 1-chain is its target minus its source.

John Baez (Oct 19 2021 at 02:58):

And this fits in nicely with other conventions; for example when we integrate a function $f$ over an interval we get

$\int_a^b f(x) dx = F(b) - F(a)$

not the other way around. Here $a$ is the source of the path from $a$ to $b$, and $b$ is the target!

John Baez (Oct 19 2021 at 02:58):

This example becomes important in deRham cohomology.

John Baez (Oct 19 2021 at 02:59):

Or, for a simpler example: when we think of an arrow from a point $a$ in $\mathbb{R}^n$ to a point $b$ as a vector, this vector is

$b - a$

Leopold Schlicht (Oct 19 2021 at 11:02):

Thanks!

Leopold Schlicht (Oct 25 2021 at 15:34):

Again to the very definition of a simplicial set. Let $C_\bullet$ be a simplicial set. Consider the degeneracy map $s_0\colon C_0\to C_1$ and the face maps $d_0, d_1\colon C_1\to C_0$. Which simplicial identity gives us $d_0\circ s_0=d_1\circ s_0 = \mathrm{id}_{C_0}$?

At first I thought it is the simplicial identity that says "$d_i\circ s_j=\mathrm{id}_{C_n}$ if $i=j$ or $i = j+1$", but then I noticed that $i$ and $j$ are required to satisfy $0\leq i,j\leq n$. In this case, $n=0$, so we can't plug in $i=1$ and $j=0$ to get $d_1\circ s_0=\mathrm{id}_{C_0}$!

Is there maybe a typo in these identities? What is very confusing is that for fixed $i$, there is not "the map $s_i$", but there is such a map for each $n\geq i$: $s_i\colon C_n\to C_{n+1}$. And it isn't always clear what the domain and codomain of $s_i$ is supposed to be when it isn't explicitly mentioned.

Morgan Rogers (he/him) (Oct 25 2021 at 15:38):

looks like a typo to me?

Leopold Schlicht (Oct 25 2021 at 15:39):

Thanks. How would one state the simplicial identities without typos? I can't find any reference that lists the simplicial identities with explicitly saying what the domain and codomain of $d_i$ and $s_i$ are.

Reid Barton (Oct 25 2021 at 15:48):

It's just all the values of $i$, $j$, $n$ that make sense. I agree the bounds on Kerodon are wrong (post a comment!), this sort of thing is why nobody writes them, I guess.

Reid Barton (Oct 25 2021 at 15:48):

Even in the Lean formalization of these identities (https://github.com/leanprover-community/mathlib/blob/07f1235dbf45b6c74fa8f0695213a2bf2b7130a0/src/algebraic_topology/simplicial_object.lean#L89-L103) we don't write out explicitly what the domains/codomains are.

Morgan Rogers (he/him) (Oct 25 2021 at 15:49):

It looks like the domains and codomains are given in Notation 1.1.1.8 and 1.1.1.9 on Kerodon.

Reid Barton (Oct 25 2021 at 15:52):

What I mean is that $d_i$ depends on $n$, but this is not reflected in the notation.

Reid Barton (Oct 25 2021 at 15:52):

Much like how people often write $d^2 = 0$, even if it really means something like $d_n \circ d_{n+1} = 0$.

Nick Hu (Oct 25 2021 at 18:06):

The simplicial identities make much more sense if you think of an $n$ simplex as a length $n$ list of points. In that case, each face map corresponds to deleting some point in the list, and each degeneracy map corresponds to duplicating a point. Then the simplicial identities merely say stuff like 'it doesn't matter in which order points are deleted/duplicated', and 'duplicating a point and then deleting it is equivalent to doing nothing'.

Nick Hu (Oct 25 2021 at 18:09):

I would actually like to see an explicit derivation that $s_0$ and $s_1$ for the 1-simplex in the terminal simplicial set are equal from the simplicial identities. It seems quite tricky to do, but clear if you abstract away to the presheaf view of simplicial sets.

John Baez (Oct 25 2021 at 22:09):

Doesn't that equation follow from the simplicial identities using the fact that this 1-simplex is itself $s_0$ of the unique 0-simplex?

John Baez (Oct 25 2021 at 22:11):

These identities say $s_1 \circ s_0 = s_0 \circ s_0$.

Nick Hu (Oct 25 2021 at 22:54):

I believe that equation does follow pretty directly from the simplicial identities (at least the ones at [[simplicial identities]]), but I got stuck showing that $[*] \xrightarrow{s_0} [*, *]$ is an epimorphism

John Baez (Oct 25 2021 at 23:18):

What do things like $[\ast, \ast]$ mean?

For a general simplicial set $s_0$ is not is an isomorphism from n-simplices to (n+1)-simplices, of course. It is for the terminal simplical set, but here we need more than the simplicial identies per se: we need the concept of terminal object. One can show that the terminal simplicial set has one n-simplex for each n, and thus all the maps $s_i, \partial_i$ are isomorphisms. I'm not quite sure what rules you want to impose on how we show this.

Nick Hu (Oct 26 2021 at 00:03):

I'm using $[*, *]$ to denote the 1-simplex in the terminal simplicial set, i.e. the simplex with two points. I agree that all the face and degeneracy maps are isomorphisms, but I had thought that the implication was the (only) two degeneracy maps from the 1-simplex to the 2-simplex are bona-fide equal.

Nick Hu (Oct 26 2021 at 00:17):

Actually, it's not even clear to me from the simplicial identities that $[*] \xrightarrow{s_0} [*, *]$ is an isomorphism; the simplicial identities only say (immediately) that the face maps $[*, *] \to [*]$ are retractions of $s_0$

James Deikun (Oct 26 2021 at 00:24):

The simplicial identities are true in every simplicial set, not just the terminal one. If from the simplicial identities alone you can derive something that is only true of the terminal simplicial set, there is a serious problem somewhere.

Nick Hu (Oct 26 2021 at 00:25):

Seems like this is what people thought might be a typo from before, unless simplicial sets are trying to encode something very subtle. My intuition says that in all the face/degeneracy maps in the terminal simplicial set should be unique. Furthermore, the simplicial identities do seem to be encoding something like a 'one-step coherence' (e.g. duplicating a point then deleting it is equivalent to not having done anything), so extrapolating from this, it could be the case that what they are really trying to say is something to the effect of 'every parallel map consisting entirely of face and degeneracy maps is equal'. At least, from the definition as presented in [[simplicial identities]] this does not seem to be the case. I can think of reasons for why you would, or would not, want this to be true though, so I can't say with confidence if it's supposed to be the case.

James Deikun (Oct 26 2021 at 00:26):

It is definitely not the case in general that all parallel maps you can make that way are equal.

Nick Hu (Oct 26 2021 at 00:27):

Right, so then I guess the answer is that the two face maps $[*, *] \to [*]$ cannot be identified

Nick Hu (Oct 26 2021 at 00:28):

But this is a bit perplexing because taking the view of the terminal simplicial set as a presheaf on the simplex category which maps everything to $\{*\}$ (I believe this is what it should be), suggests that there is a unique map between every simplex in this simplicial set.

James Deikun (Oct 26 2021 at 00:29):

They can't be identified by the simplicial identities, any more than all elements in the trivial group are identified by the group axioms ...

James Deikun (Oct 26 2021 at 00:30):

However, in fact in the terminal simplicial set, they are equal.

Nick Hu (Oct 26 2021 at 00:32):

My point is that it should be possible to derive that all these maps are equal, in the terminal simplicial set, working in the presentation of simplicial sets as sets-of-simplices+simplicial identities (but I don't really see how)

James Deikun (Oct 26 2021 at 00:34):

It's a general fact about presheaf categories that all the objects of the base category are mapped to $\lbrace*\rbrace$ and all the arrows to $id_{\lbrace*\rbrace}$ ...

Nick Hu (Oct 26 2021 at 00:36):

Or actually, I guess the answer I'm looking for is that the terminal simplicial set is the simplicial set where all these face/degeneracy maps are identified. There is a distinct simplicial set, where there is only one simplex in each dimension, in which the two face maps $[*, *] \to [*]$ are not identified.

James Deikun (Oct 26 2021 at 00:40):

Oh, so you're trying to use the extra axiom of 'there is only one simplex in each dimension' to derive everything else about the terminal simplicial set. It seems this should actually be possible ... there is only one 'simplicial algebra structure' with only one single simplex in each dimension.

Martti Karvonen (Oct 26 2021 at 00:43):

Nick Hu said:

There is a distinct simplicial set, where there is only one simplex in each dimension, in which the two face maps $[*, *] \to [*]$ are not identified.

How could this be? Surely there's only one function from any singleton to any other singleton.

James Deikun (Oct 26 2021 at 00:44):

The proofs are pretty boring though. They basically all take the form of "i write an equation between two terms of 'simplicial algebra', and observe that both sides are simplices of the same dimension, therefore equal".

John Baez (Oct 26 2021 at 01:15):

Nick Hu said:

My intuition says that in all the face/degeneracy maps in the terminal simplicial set should be unique.

As I mentioned, the terminal simplicial set has one n-simplex for each n. This follows from a general fact about presheaf categories: the terminal presheaf has "one thing of each kind".

As a consequence, there's no choice about what the face and degeneracy maps can be in the terminal simplicial set: they're all maps from a one-element set to a one-element set .

As a further consequence, all these maps are isomorphisms. Indeed, they obey all possible well-formed equations, e.g. all degeneracy maps from n-simplices to (n+1)-simplices are equal.

John Baez (Oct 26 2021 at 01:17):

So yes: in the terminal simplicial set "all parallel maps consisting entirely of face and degeneracy maps are equal"... because all parallel maps are equal.

John Baez (Oct 26 2021 at 01:18):

But it's certainly not true in a general simplicial set that "all parallel maps consisting entirely of face and degeneracy maps are equal". That's why we have different names for $\partial_0$ and $\partial_1$, for example: they're parallel, but not equal.

Joachim Kock (Oct 26 2021 at 09:00):

Regarding the simplicial identities:

A good way to get to learn, grasp, and love them is to write them as commutative squares. (They are all squares, except those that state that a degeneracy map is a section to its adjacent face maps.)

The immediate benefit is that the ambiguity is eliminated and one can clearly see what are the domain and codomain of each face or degeneracy map. (It also helps figuring out what are the valid indices for a given simplicial identity -- and it helps figuring out if $d_i d_j$ means 'first $d_i$ then $d_j$' or the other way around!)

Furthermore, squares can be pasted! This shows how the simplicial identities can be combined to yield further identities. Often these further identities are important in applications. The ones called 'simplicial identities' are only a minimal generating set of identities.

Another benefit is that in applications to category theory it is important to demand some of these squares to be pullback squares -- a condition not so easy to express with algebraic identities. Most importantly, the Segal condition, which characterises categories among all simplicial sets, states that simplicial-identity squares with top face maps against bottom face maps are pullbacks. (There are actually several other classes of simplicial-identity squares that are important to know are pullbacks, characterising various properties of simplicial sets, e.g. split, stiff, complete, and the decomposition-space property. Or one can demand only a weak pullback property, as done in the recent paper Weak cartesian properties of simplicial sets by Carme Constantin, Tobias Fritz, Paolo Perrone, and Brandon Shapiro, motivated by probability theory.)

Finally, when dealing with simplicial groupoids or simplicial categories, it is often necessary to consider pseudo-simplicial objects. Here the simplicial identities hold only up to specified isomorphims, and these have to be specified as part of the data and are required to satisfy coherence conditions. The isomorphisms are 2-cells occupying the simplicial-identity squares, and the coherence constraints are expressed in terms of pasting of squares. (A simplicial-identity style presentation of these coherence laws were worked out by Rick Jardine 30 years ago in a paper called Supercoherence. There are 17 laws! But once you regard the simplicial identities as squares, the coherence laws are just cubes.)

Joachim Kock (Oct 26 2021 at 09:09):

Regarding deleting and duplicating points:

The simplicial identities make much more sense if you think of an $n$ simplex as a length $n$ list of points. In that case, each face map corresponds to deleting some point in the list, and each degeneracy map corresponds to duplicating a point. Then the simplicial identities merely say stuff like 'it doesn't matter in which order points are deleted/duplicated', and 'duplicating a point and then deleting it is equivalent to doing nothing'.

This viewpoint of deleting and duplicating points is valid, of course, but it is a little bit backwards in the sense that it is actually a description of the category $\Delta^\mathrm{op}$. I think it is better to describe $\Delta$ itself and safer to work here, because here the arrows are actual functions, and it is very easy to make calculations.

If somebody asks you to verify if some identity $d_i d_j s_{k-1} d_{i+1} = s_{k+1} d_{i-1} d_i d_j$ (or whatever) holds, the best is usually to translate it back into an identity in $\Delta$ involving the corresponding coface maps (injective monotone map omitting some point) and codegeneracy maps (surjective monotone maps where some point has double preimage), and then just compose the functions and check that they are the same. This does not even involve the '(co)simplicial identities'. It is purely a question of composing functions.

Actually, this is a possible answer to the original question of how to understand the simplicial identities: translate them back into $\Delta$ where they are the cosimplicial identities -- now they are just obvious statements about composition of monotone functions.

Alexander Campbell (Oct 26 2021 at 10:54):

Ross Street once told me that MacLane carried the simplicial identities around with him on a piece of cardboard.

Leopold Schlicht (Oct 28 2021 at 17:23):

Thanks very much!

Joachim Kock said:

Actually, this is a possible answer to the original question of how to understand the simplicial identities: translate them back into $\Delta$ where they are the cosimplicial identities -- now they are just obvious statements about composition of monotone functions.

But why does one use the category $\Delta$ at all? Why is it that presheaves on $\Delta^\mathrm{op}$ are the best candidate for a sort of mathematical object that implements the idea of "points, triangles, tetrahedrons, ... glued together"?

John Baez (Oct 28 2021 at 18:52):

Leopold Schlicht said:

But why does one use the category $\Delta$ at all? Why is it that presheaves on $\Delta^\mathrm{op}$ are the best candidate for a sort of mathematical object that implements the idea of "points, triangles, tetrahedrons, ... glued together"?

There are many interesting answers to these questions. What are your best answers to these questions, currently? That will help us say something new to you.

Patrick Nicodemus (Nov 03 2021 at 16:56):

Leopold Schlicht said:

But why does one use the category $\Delta$ at all? Why is it that presheaves on $\Delta^\mathrm{op}$ are the best candidate for a sort of mathematical object that implements the idea of "points, triangles, tetrahedrons, ... glued together"?

There's a notion in category theory of a "weighted colimit". If $C$ is a category, $F : C\to D$ is a functor, and $$P : C^{\rm op}\to \mathbf{Sets}$ is a presheaf, then one can define the "weighted colimit of $F$ with weights in $P$". The idea is that the weighted colimit should be built by gluing together copies of $F(c)$ with multiplicity given by $|P(c)|$. All gluings are along the maps of the diagram $F$, but the question of which copies of $F(c)$ are glued to which copies of $F(d)$ along the map $F(f)$ is answered by the presheaf map $P(f): P(d)\to P(c)$. These two argument limits are pretty interesting and expressive and a very useful general notion of gluing.

Imo pretty much all notions of "simplicial complex" I know of are adequately treated by this framework. Take $D$ to be the category $\mathbf{Top}$. Take $C$ to be a category whose objects are the various simplices $\Delta^0, \Delta^1$ and so on, and whose maps are some interesting class of linear maps between them along which we want to allow simplices to be glued. The class of linear maps could be all linear maps sending vertices to vertices, or only the ones that respect the ordering of the vertices, or only the injective ones, and so on. There are various possible choices and each will give rise to a slightly different notion of simplicial complex - if $F: C\to D$ is the inclusion functor into Top, simplicial complexes are weighted colimits of $F$ with coefficients in some presheaf.

Patrick Nicodemus (Nov 03 2021 at 16:57):

There is a notion of "abstract simplicial complex" which isn't obviously of this form, but it's not hard to see that abstract simplicial complexes are a reflexive subcategory of the symmetric simplicial sets.

Leopold Schlicht (Nov 05 2021 at 12:28):

Thanks!

Leopold Schlicht (Nov 05 2021 at 12:35):

Reid Barton said:

Leopold Schlicht said:

Why do the arrows have a direction?

You can start by just thinking about semisimplicial sets with only 0- and 1-simplices and why they are directed graphs and not undirected graphs.
(Unless you mean why do we want them to be there, which is a more complicated question.)

What's the answer to the question why we want that the faces of a simplicial sets are ordered (and, in particular, that the edges have a direction)? I understand that this is useful (or even necessary) for encoding categories as simplicial sets (which leads to $\infty$-categories), but is there a purely topological motivation for that? After all, simplicial sets probably haven't been invented in order to define $\infty$-categories, but to model a notion of "nice space" or something like that. And if I think about a triangle, even as an abstract space, I usually don't consider the order of the vertices of the triangle to be part of the data defining that space.

Leopold Schlicht (Nov 05 2021 at 12:46):

John Baez said:

Leopold Schlicht said:

But why does one use the category $\Delta$ at all? Why is it that presheaves on $\Delta^\mathrm{op}$ are the best candidate for a sort of mathematical object that implements the idea of "points, triangles, tetrahedrons, ... glued together"?

There are many interesting answers to these questions. What are your best answers to these questions, currently? That will help us say something new to you.

I thought about this question a lot in the last week. By now I think I obtained a quite good intuition about the simplicial identities, mostly using Nick Hu's mental trick of imagining $d_i$ as "delete the $i$-th vertex" and $s_i$ as "duplicate the $i$-th vertex" and drawing a lot of pictures. I found it useful to use that interpretation to come up with the simplicial identities myself, rather than staring at them and trying to "understand" them from that. But I am still not able to give a compelling answer to the question why presheaves on $\Delta^\mathrm{op}$ are the best candidate for a sort of mathematical object that implements the idea of "points, triangles, tetrahedrons, ... glued together". It's more like I'm just getting used to it, so I can't really answer your question about what my best answer is. :sweat_smile:

Zhen Lin Low (Nov 05 2021 at 12:51):

Leopold Schlicht said:

What's the answer to the question why we want that the faces of a simplicial sets are ordered (and, in particular, that the edges have a direction)? I understand that this is useful (or even necessary) for encoding categories as simplicial sets (which leads to $\infty$-categories), but is there a purely topological motivation for that? After all, simplicial sets probably haven't been invented in order to define $\infty$-categories, but to model a notion of "nice space" or something like that. And if I think about a triangle, even as an abstract space, I usually don't consider the order of the vertices of the triangle to be part of the data defining that space.

You can try to work with "symmetric" simplicial sets (i.e. presheaves on the category of non-empty finite sets) if you like. The miraculous fact is that the cartesian product of simplicial sets, defined in the completely general category theoretic way, is preserved by geometric realisation; or put it another way, gives us a functorial way of triangulating the geometric cartesian product. Symmetric simplicial sets do not have this property, semisimplicial sets do not have this property, etc.

James Deikun (Nov 05 2021 at 12:52):

Leopold Schlicht said:

What's the answer to the question why we want that the faces of a simplicial sets are ordered (and, in particular, that the edges have a direction)? I understand that this is useful (or even necessary) for encoding categories as simplicial sets (which leads to $\infty$-categories), but is there a purely topological motivation for that? After all, simplicial sets probably haven't been invented in order to define $\infty$-categories, but to model a notion of "nice space" or something like that. And if I think about a triangle, even as an abstract space, I usually don't consider the order of the vertices of the triangle to be part of the data defining that space.

My best answer to this is that to glue together simplices to make all kinds of nice spaces you have to care about their orientation, and when gluing along their faces you have to care about orientation of the faces, etc. And ordering the vertices induces an orientation on a simplex and all its facets in a nice consistent way.

James Deikun (Nov 05 2021 at 12:56):

There's also this "Goldilocks effect" with the Cartesian product, mentioned above, but tbh I don't like miracles and would like to understand the reason it happens better myself.

John Baez (Nov 05 2021 at 15:16):

Leopold Schlicht said:

I am still not able to give a compelling answer to the question why presheaves on $\Delta^\mathrm{op}$ are the best candidate for a sort of mathematical object that implements the idea of "points, edges, triangles, tetrahedrons, ... glued together". It's more like I'm just getting used to it, so I can't really answer your question about what my best answer is. :sweat_smile:

You should try to formulate your own guesses as to why simplicial sets are good. But to do this you should compare them with the alternatives. A [[symmetric simplicial set]] is like a simplicial set but where you can 'turn around' any edge, triangle, tetrahedron, etc. A [[semi-simplicial set]] is like a set but with only face maps, not degeneracies. There is also a concept of 'symmetric semi-simplicial set', which combines these ideas. (I don't know a reference, but you can make it up yourself.)

It's good to think about what you can with simplicial sets that you can't do with these other things... and vice versa. Each has its own advantages.