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.
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.