In his wonderful article “On proof and progress in mathematics“, Bill Thurston describes (among many other topics) how one’s understanding of given concept in mathematics (such as that of the derivative) can be vastly enriched by viewing it simultaneously from many subtly different perspectives; in the case of the derivative, he gives seven standard such perspectives (infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic) and then mentions a much later perspective in the sequence (as describing a flat connection for a graph).

One can of course do something similar for many other fundamental notions in mathematics. For instance, the notion of a group can be thought of in a number of (closely related) ways, such as the following:

- (0) Motivating examples: A group is an abstraction of the operations of addition/subtraction or multiplication/division in arithmetic or linear algebra, or of composition/inversion of transformations.
- (1) Universal algebraic: A group is a set with an identity element , a unary inverse operation , and a binary multiplication operation obeying the relations (or axioms) , , for all .
- (2) Symmetric: A group is all the ways in which one can transform a space to itself while preserving some object or structure on this space.
- (3) Representation theoretic: A group is identifiable with a collection of transformations on a space which is closed under composition and inverse, and contains the identity transformation.
- (4) Presentation theoretic: A group can be generated by a collection of generators subject to some number of relations.
- (5) Topological: A group is the fundamental group of a connected topological space .
- (6) Dynamic: A group represents the passage of time (or of some other variable(s) of motion or action) on a (reversible) dynamical system.
- (7) Category theoretic: A group is a category with one object, in which all morphisms have inverses.
- (8) Quantum: A group is the classical limit of a quantum group.
- etc.

One can view a large part of group theory (and related subjects, such as representation theory) as exploring the interconnections between various of these perspectives. As one’s understanding of the subject matures, many of these formerly distinct perspectives slowly merge into a single unified perspective.

From a recent talk by Ezra Getzler, I learned a more sophisticated perspective on a group, somewhat analogous to Thurston’s example of a sophisticated perspective on a derivative (and coincidentally, flat connections play a central role in both):

- (37) Sheaf theoretic: A group is identifiable with a (set-valued) sheaf on the category of simplicial complexes such that the morphisms associated to collapses of -simplices are bijective for (and merely surjective for ).

This interpretation of the group concept is apparently due to Grothendieck, though it is motivated also by homotopy theory. One of the key advantages of this interpretation is that it generalises easily to the notion of an -group (simply by replacing with in (37)), whereas the other interpretations listed earlier require a certain amount of subtlety in order to generalise correctly (in particular, they usually themselves require higher-order notions, such as -categories).

The connection of (37) with any of the other perspectives of a group is elementary, but not immediately obvious; I enjoyed working out exactly what the connection was, and thought it might be of interest to some readers here, so I reproduce it below the fold.

[Note: my reconstruction of Grothendieck's perspective, and of the appropriate terminology, is likely to be somewhat inaccurate in places: corrections are of course very welcome.]

** — 1. Flat connections — **

To see the relationship between (37) and more traditional concepts of a group, such as (1), we will begin by recalling the machinery of flat connections.

Let be a group, be a topological space. A principal -connection on can be thought of as an assignment of a group element to every path in which obey the following four properties:

- Invariance under reparameterisation: if is a reparameterisation of , then .
- Identity: If is a constant path, then is the identity element.
- Inverse: If is the reversal of a path , then is the inverse of .
- Groupoid homomorphism: If starts where ends (so that one can define the concatenation ), then . (Depending on one’s conventions, one may wish to reverse the order of the group multiplication on the right-hand side.)

Intuitively, represents a way to use the group to connect (or “parallel transport”) the fibre at the initial point of to the fibre at the final point; see this previous blog post for more discussion. Note that the identity property is redundant, being implied by the other three properties.

We say that a connection is *flat* if is the identity element for every “short” closed loop , thus strengthening the identity property. One could define “short” rigorously (e.g. one could use “contractible” as a substitute), but we will prefer here to leave the concept intentionally vague.

Typically, one studies connections when the structure group and the base space are continuous rather than discrete. However, there is a combinatorial model for connections which is suitable for discrete groups, in which the base space is now an (abstract) simplicial complex – a vertex set , together with a number of *simplices* in , by which we mean ordered -tuples of distinct vertices in for various integers (with being the *dimension* of the simplex ). In our definition of a simplicial complex, we add the requirement that if a simplex lies in the complex, then all faces of that simplex (formed by removing one of the vertices, but leaving the order of the remaining vertices unchanged) also lie in the complex. We also assume a well defined *orientation*, in the sense that every -tuple is represented by at most one simplex (thus, for instance, a complex cannot contain both an edge and its reversal ). Though it will not matter too much here, one can think of the vertex set here as being restricted to be finite.

A *path* in a simplicial complex is then a sequence of -simplices or their formal reverses , with the final point of each -simplex being the initial point of the next. If is a (discrete) group, a *principal -connection* on is then an assignment of a group element to each such path , obeying the groupoid homomorphism property and the inverse property (and hence the identity property). Note that the reparameterisation property is no longer needed in this abstract combinatorial model. Note that a connection can be determined by the group elements it assigns to each -simplex . (I have written the simplex from right to left, as this makes the composition law cleaner.)

So far, only the -skeleton (i.e. the simplices of dimension at most ) of the complex have been used. But one can use the -skeleton to define the notion of a *flat* connection: we say that a principal -connection on is flat if the boundary of every -simplex , oriented appropriately, is assigned the identity element, or more precisely that , or in other words that ; thus, in this context, a “short loop” means a loop that is the boundary of a -simplex. Note that this corresponds closely to the topological concept of a flat connection when applied to, say, a triangulated manifold.

Fix a group . Given any simplicial complex , let be the set of flat connections on . One can get some feeling for this set by considering some basic examples:

- If is a single -dimensional simplex (i.e. a point), then there is only the trivial path, which must be assigned the identity element of the group. Thus, in this case, can be identified with .
- If is a -dimensional simplex, say , then the path from to can be assigned an arbitrary group element , and this is the only degree of freedom in the connection. So in this case, can be identified with .
- Now suppose is a -dimensional simplex, say . Then the group elements and are arbitrary elements of , but is constrained to equal . This determines the entire flat connection, so can be identified with .
- Generalising this example, if is a -dimensional simplex, then can be identified with .

An important operation one can do on flat connections is that of *pullback*. Let be a *morphism* from one simplicial complex to another ; by this, we mean a map from the vertex set of to the vertex set of such that every simplex in maps to a simplex in in an order preserving manner (though note that is allowed to be non-injective, so that the dimension of the simplex can decrease by mapping adjacent vertices to the same vertex). Given such a morphism, and given a flat connection on , one can then pull back that connection to yield a flat connection on , defined by the formula

for any -simplex in , with the convention that is the identity for any . It is easy to see that this is still a flat connection. Also, if and are morphisms, then the operations of pullback by and then by compose to equal the operation of pullback by : . In the language of category theory, pullback is a contravariant functor from the category of simplicial complexes to the category of sets (with each simplicial complex being mapped to its set of flat connections).

A special case of a morphism is an *inclusion morphism* to a simplicial complex from a subcomplex . The associated pullback operation is the *restriction* operation, that maps a flat connection on to its restriction to .

** — 2. Sheaves — **

We currently have a set of flat connections assigned to each simplicial complex , together with pullback maps (and in particular, restriction maps) connecting these sets to each other. One can easily observe that this system of structures obeys the following axioms:

- (Identity) There is only one flat connection on a point.
- (Locality) If is the union of two simplicial complexes, then a flat connection on is determined by its restrictions to and . In other words, the map is an injection from to .
- (Gluing) If , and are flat connections on which agree when restricted to , (and if the orientations of on the intersection agree) then there exists a flat connection on which agrees with on . (Note that this gluing of and is unique, by the previous axiom. It is important that the orientations match; we cannot glue to , for instance.)

One can consider more abstract assignments of sets to simplicial complexes, together with pullback maps, which obey these three axioms. A system which obeys the first two axioms is known as a *pre-sheaf*, while a system that obeys all three is known as a sheaf. (One can also consider pre-sheaves and sheaves on more general topological spaces than simplicial complexes, for instance the spaces of smooth (or continuous, or holomorphic, etc.) functions (or forms, sections, etc.) on open subsets of a manifold form a sheaf.)

Thus, flat connections associated to a group form a sheaf. But flat connections form a special type of sheaf that obeys an additional property (listed above as (37)). To explain this property, we first consider a key example when is the standard -simplex (together with subsimplices), and is the subcomplex formed by removing the -face and the -face , leaving only the -faces and the -faces . Then of course every flat connection on restricts to a flat connection on . But the flatness property ensures that this restriction is invertible: given a flat connection on , there exists a unique extension of this connection back to . This is nothing more than the property, remarked earlier, that to specify a flat connection on the -simplex , it suffices to know what the connection is doing on and , as the behaviour on the remaining edge can then be deduced from the group law; conversely, any specification of the connection on those two -simplices determines a connection on the remainder of the -simplex.

This observation can be generalised. Given any simplicial complex , define a -dimensional *collapse* of to be a simplicial complex obtained from by removing the interior of a -simplex, together with one of its faces; thus for instance the complex consisting of (and subsimplices) is a -dimensional collapse of the -simplex (and subsimplices). We then see that the sheaf of flat connections obeys an additional axiom:

- (Grothendieck’s axiom) If is a -dimensional collapse of , then the restriction map from to is surjective for all , and bijective for .

This axiom is trivial for . For , it is true because if an edge (and one of its vertices) can be removed from a complex, then it is not the boundary of any -simplex, and the value of a flat connection on that edge is thus completely unconstrained. (In any event, the case of this axiom can be deduced from the sheaf axioms.) For , it follows because if one can remove a -simplex and one of its edges from a complex, then the edge is not the boundary of any other -simplex and thus the connection on that edge only constrained to precisely be the product of the connection on the other two edges of the -simplex. For , it follows because if oen removes a -simplex and one of its -simplex faces, the constraint associated to that -simplex is implied by the constraints coming from the other three faces of the -simplex (I recommend drawing a tetrahedron and chasing some loops around to see this), and so one retains bijectivity. For , the axiom becomes trivial again because the -simplices and -simplices have no impact on the definition of a flat connection.

Grothendieck’s beautiful observation is that the converse holds: if a (concrete) sheaf obeys Grothendieck’s axiom, then it is equivalent to the sheaf of flat connections of some group defined canonically from the sheaf. Let’s see how this works. Suppose we have a sheaf , which is concrete in the sense that is a set, and the morphisms between these sets are given by functions. In analogy with the preceding discussion, we’ll refer to elements of as (abstract) flat connections, though *a priori* we do not assume there is a group structure behind these connections.

By the sheaf axioms, there is only one flat connection on a point, which we will call the trivial connection. Now consider the space flat connections on the standard -simplex . If the sheaf was indeed the sheaf of flat connections on a group , then is canonically identifiable with . Inspired by this, we will *define* to equal the space of flat connections on . The flat connections on any other -simplex can then be placed in one-to-one correspondence with elements of by the morphism , so flat connections on can be viewed as being *equivalent* to an element of .

At present, is merely a set, not a group. To make it into a group, we need to introduce an identity element, an inverse operation, and a multiplication operation, and verify the group axioms.

To obtain an identity element, we look at the morphism from to a point, and pull back the trivial connection on that point to obtain a flat connection on , which we will declare to be the identity element. (Note from the functorial nature of pullback that it does not matter which point we choose for this.)

Now we define the multiplication operation. Let , then and are flat connections on . By using the morphism from to , we can pull back to to create a flat connection on that is equivalent to . The restriction of and to the point is trivial, so by the gluing axiom we can glue and to a flat connection on the complex . By Grothendieck’s axiom, one can then uniquely extend this connection to the -simplex , which can then be restricted to the edge . Mapping this edge back to , we obtain an element of , which we will define to be .

This operation is closed. To verify the identity property, observe that if , then by starting with the simplex and pulling back under the morphism that sends to but is the identity on , we obtain a flat connection on which is equal to on , equivalent to the identity on , and is equivalent to on (after identifying with ). From the definition of group multiplication, this shows that ; a similar argument (using a slightly different morphism from to ) gives .

Now we establish associativity. Let . Using the definition of multiplication, we can create a flat connection on the -simplex which equals on , is equivalent to on , and is equivalent to on . We then glue on the edge and extend the flat connection to be equivalent to on . Using Grothendieck’s axiom and the definition of multiplication, we can then extend the flat connection to the -simplex to be equivalent to on . By another use of that axiom, we can also extend the flat connection to the -simplex , to be equivalent to on . Now that we have three of the four faces of the -simplex , we can now apply the case of Grothendieck’s axiom and extend the flat connection to the entire -simplex, and in particular to the -simplex . Using the definition of multiplication again, we thus see that , giving associativity.

Next, we establish the inverse property. It will suffice to establish the existence of a left-inverse and a right-inverse for each group element, since the associativity property will then guarantee that these two inverses equal each other. We shall just establish the left-inverse property, as the right-inverse is analogous. Let be arbitrary. By the gluing axiom, one can form a flat connection on the complex which equals on and is equivalent to the identity on . By Grothendieck’s axiom, this can be extended to a flat connection on ; the restriction of this connection to is equivalent to some element of , which we define to be . By construction, as required.

We have just shown that is a group. The last thing to do is to show that this abstract sheaf can be indeed identified with the sheaf of -flat connections. This is fairly straightforward: given an abstract flat connection on a complex, the restriction of that connection to any edge is equivalent to an element of . To verify that this genuinely determines a -connection on that complex, we need to verify that if and are both in the complex, that the group elements assigned to these edges invert each other. But we can pullback to the -simplex by mapping to and to , creating a flat connection that is equal to on , equivalent to on , and equivalent to the identity on ; by definition of multiplication or inverse we conclude that invert each other as desired.

Thus the abstract connection defines a -connection. From the definition of multiplication it is also clear that every -simplex in the complex imposes the right relation on the three elements of associated to the edges of that simplex that makes the -connection flat. Thus we have a canonical way to associate a -flat connection to each abstract flat connection; the only remaining things to do are verify that this association is bijective.

We induct on the size of the complex. If the complex is not a single simplex, the claim follows from the induction hypothesis by viewing the complex as the union of two (possibly overlapping) smaller complexes, and using the gluing and locality axioms. So we may assume that the complex consists of a single simplex. If the simplex is or -dimensional the claim is easy; for the claim follows from Grothendieck’s axiom (which applies both for the abstract flat connections (by hypothesis) and for -flat connections (as verified earlier)) and the induction hypothesis.

## 19 comments

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19 October, 2009 at 5:31 pm

AnonymousWonder if this post is missing an expository tag.

[Added, thanks. (It seems my readers are more familiar with my organisational system than I am.) - T.]19 October, 2009 at 5:50 pm

LiorIn the definition of a flat connection, should begin where

ends.[Corrected, thanks - T.]19 October, 2009 at 8:37 pm

LiorYou also have “At present, G is merely a group, not a set” rather than the reverse.

[Corrected, thanks - T.]19 October, 2009 at 11:36 pm

anonymousIn the interpretation (37) the wording is a little bit garbled: “such that the morphisms associated to collapse of {d}-simplices for are bijective” should probably be “such that the morphisms associated to collapses of {d}-simplices are bijective”.

[Fixed, thanks - T.]20 October, 2009 at 6:49 am

timurDoes numbering (37) mean you had a list up to (36) before coming across this interpretation?

20 October, 2009 at 6:44 pm

Terence TaoWell, no, but certainly there are more ways to view a group than are listed here. The precise number (37) is also used in Thurston’s article, referenced above.

20 October, 2009 at 9:28 am

A semana nos arXivs… « Ars Physica[...] Grothendieck’s definition of a group (alguém disse Langlands Duality? ) [...]

20 October, 2009 at 11:45 am

RajI got stuck for a little bit on this: “pullback is a contravariant functor from the category of simplicial complexes to the category of flat connections.” I think what you mean to say is that pullback gives us a contravariant functor from the category of simplicial complexes to the category of sets, where at the level of objects a simplicial complex gets sent to the set of all flat connections on it.

Is my interpretation right, or am I missing something?

[Yes, thanks, this is what was intended. I've corrected the text. -T]20 October, 2009 at 12:38 pm

RajAlso, this reminded me of another point of view which is closely related to the category-theoretic definition (7). Namely, if you think of a group as a category with one object and invertible morphisms, then you can take its nerve, which is a simplicial complex with only one 0-simplex. This simplicial complex is special in two other ways: a) it is Kan, which means that any (k-1)-dimensional horn can be filled by a k-simplex, and b) this horn-filling is unique for k > 1. Another way to state these conditions is by defining various “horn maps” that take k-simplices to (k-1)-dimensional horns. Then the Kan property says that these horn maps are bijective, and the uniqueness property says that the maps are bijective for k > 1.

Superficially, this looks very similar to Grothendieck’s axiom, and I’m guessing it’s not just a coincidence. Do you know how to relate the two?

20 October, 2009 at 12:39 pm

Rajsorry, I meant to say “the Kan property says that these horn maps are surjective”.

20 October, 2009 at 6:45 pm

Terence TaoOne would have to ask a category theorist for a proper answer, but I do recall Ezra mentioning a close connection of Grothendieck’s interpretation of a group with nerves in his talk. Unfortunately I do not recall the precise nature of the connection…

21 October, 2009 at 4:14 pm

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26 October, 2009 at 4:14 pm

Ben WielandI think Raj is right that these two definitions are pretty much the same. You can replace simplicial complexes with pretty much any geometric category, like simplicial sets. The n-simplex is not symmetric in the category of simplicial sets, so you have several different horn conditions, rather than the single one for simplicial sets. If you only impose some of them, you can expand the definition to monoids, dropping inverses.

The Kan complex version doesn’t mention sheaves, because a sheaf on the category of simplicial sets is the same thing as a simplicial set. This is a general principle that a sheaf on the category of sheaves is represented by a sheaf in the original category of sheaves. In particular, if we have no sheaf condition, that is, if we take a diagram (category) D and look at all functors from D to sets, as in the case of simplicial sets, a sheaf on this category is determined by its restriction to the sheaves represented by objects of D, and thus it essentially a functor from D to sets.

In fact, the category of sheaves on the category of simplicial complexes is not a lot larger than the category of simplicial complexes. By locality, a sheaf is determined by its restriction to the full subcategory of standard simplices. I claim that any functor yields a sheaf, just like a simplicial set is any functor on the full subcategory (of simplicial sets) spanned by the standard simplices. The category of functors here is a bit bigger than the category of simplicial complexes, but not a lot bigger. It just adds all colimits which contains things like an interval with the two ends glued together, or two intervals with the same pair of endpoints. Depending on your definition of morphism of simplicial complex, objects like these are all the extra objects, that is, simplices glued along faces of the same dimension. (whereas simplicial sets are simplices glued along linear maps more general than inclusions)

27 October, 2009 at 3:12 pm

Ben WielandIn my first paragraph, that should have been “the single [horn condition] for simplicial

complexes,” in contrast to several for simplicialsets.In my last paragraph, I imply that simplicial sets allow more general gluing maps than simplicial complexes. That is not true: neither condition is weaker. What is true is that simplical sets allow gluing along non-injective maps, such as gluing the boundary of a 2-simplex to a point to get a 2-sphere (with a single top cell). But simplicial sets are restricted in their morphisms and gluing maps by the order of the vertices of each simplex.

31 October, 2009 at 6:41 pm

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6 July, 2010 at 9:10 pm

LMSchmittIt should, perhaps, be noticed that a set G with associative operation o and left-neutral element 1=c , i.e., 1 o x =x, as-well-as left inverse I(x) in G for every x in G, i.e., I(x) o x = 1, is already a group in the above-mentioned universal-algebra sense.

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