


  • etc.







— 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 {G} be a group, {X} be a topological space. A principal {G}-connection {\omega} on {X} can be thought of as an assignment of a group element {\omega(\gamma) \in G} to every path {\gamma} in {X} which obey the following four properties:

  • Invariance under reparameterisation: if {\gamma'} is a reparameterisation of {\gamma}, then {\omega(\gamma)=\omega(\gamma')}.
  • Identity: If {\gamma} is a constant path, then {\omega(\gamma)} is the identity element.
  • Inverse: If {-\gamma} is the reversal of a path {\gamma}, then {\omega(-\gamma)} is the inverse of {\omega(\gamma)}.
  • Groupoid homomorphism: If {\gamma_2} starts where {\gamma_1} ends (so that one can define the concatenation {\gamma_1+\gamma_2}), then {\omega(\gamma_1 + \gamma_2) = \omega(\gamma_2) \omega(\gamma_1)}. (Depending on one’s conventions, one may wish to reverse the order of the group multiplication on the right-hand side.)

Intuitively, {\omega(\gamma)} represents a way to use the group {G} to connect (or “parallel transport”) the fibre at the initial point of {\gamma} 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 {\omega} is flat if {\omega(\gamma)} is the identity element for every “short” closed loop {\gamma}, 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 {G} and the base space {X} are continuous rather than discrete. However, there is a combinatorial model for connections which is suitable for discrete groups, in which the base space {X} is now an (abstract) simplicial complex {\Delta} – a vertex set {V}, together with a number of simplices in {V}, by which we mean ordered {d+1}-tuples {(x_0,\ldots,x_d)} of distinct vertices in {V} for various integers {d} (with {d} being the dimension of the simplex {(x_0,\ldots,x_d)}). 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 {d+1}-tuple {\{x_0,\ldots,x_d\}} is represented by at most one simplex (thus, for instance, a complex cannot contain both an edge {(0,1)} and its reversal {(1,0)}). Though it will not matter too much here, one can think of the vertex set {V} here as being restricted to be finite.

A path {\gamma} in a simplicial complex {\Delta} is then a sequence of {1}-simplices {(x_i,x_{i+1})} or their formal reverses {-(x_i,x_{i+1})}, with the final point of each {1}-simplex being the initial point of the next. If {G} is a (discrete) group, a principal {G}-connection {\omega} on {\Delta} is then an assignment of a group element {\omega(\gamma) \in G} to each such path {\gamma}, 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 {\omega(b \leftarrow a)} it assigns to each {1}-simplex {(a,b)}. (I have written the simplex {b \leftarrow a} from right to left, as this makes the composition law cleaner.)

So far, only the {1}-skeleton (i.e. the simplices of dimension at most {1}) of the complex have been used. But one can use the {2}-skeleton to define the notion of a flat connection: we say that a principal {G}-connection {\omega} on {\Delta} is flat if the boundary of every {2}-simplex {(a,b,c)}, oriented appropriately, is assigned the identity element, or more precisely that {\omega(c \leftarrow a)^{-1} \omega(c \leftarrow b) \omega(b \leftarrow a) = e}, or in other words that {\omega(c \leftarrow a) = \omega(c \leftarrow b) \omega(b \leftarrow a)}; thus, in this context, a “short loop” means a loop that is the boundary of a {2}-simplex. Note that this corresponds closely to the topological concept of a flat connection when applied to, say, a triangulated manifold.

Fix a group {G}. Given any simplicial complex {\Delta}, let {{\mathcal O}(\Delta)} be the set of flat connections on {\Delta}. One can get some feeling for this set by considering some basic examples:

  • If {\Delta} is a single {0}-dimensional simplex (i.e. a point), then there is only the trivial path, which must be assigned the identity element {e} of the group. Thus, in this case, {{\mathcal O}(\Delta)} can be identified with {\{e\}}.
  • If {\Delta} is a {1}-dimensional simplex, say {(0,1)}, then the path from {0} to {1} can be assigned an arbitrary group element {\omega(1 \leftarrow 0) \in G}, and this is the only degree of freedom in the connection. So in this case, {{\mathcal O}(\Delta)} can be identified with {G}.
  • Now suppose {\Delta} is a {2}-dimensional simplex, say {(0,1,2)}. Then the group elements {\omega(1 \leftarrow 0)} and {\omega(2 \leftarrow 1)} are arbitrary elements of {G}, but {\omega(2 \leftarrow 0)} is constrained to equal {\omega(2 \leftarrow 1) \omega(1 \leftarrow 0)}. This determines the entire flat connection, so {{\mathcal O}(\Delta)} can be identified with {G^2}.
  • Generalising this example, if {\Delta} is a {k}-dimensional simplex, then {{\mathcal O}(\Delta)} can be identified with {G^k}.

An important operation one can do on flat connections is that of pullback. Let {\phi: \Delta \rightarrow \Delta'} be a morphism from one simplicial complex {\Delta} to another {\Delta'}; by this, we mean a map from the vertex set of {\Delta} to the vertex set of {\Delta'} such that every simplex in {\Delta} maps to a simplex in {\Delta'} in an order preserving manner (though note that {\phi} 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 {\omega'} on {\Delta'}, one can then pull back that connection to yield a flat connection {\phi^* \omega'} on {\Delta}, defined by the formula

\displaystyle  \phi^* \omega'( w \leftarrow v ) := \omega'( \phi(w) \leftarrow \phi(v) )

for any {1}-simplex {(v,w)} in {\Delta}, with the convention that {\omega'(u \leftarrow u)} is the identity for any {u}. It is easy to see that this is still a flat connection. Also, if {\phi: \Delta \rightarrow \Delta'} and {\psi: \Delta' \rightarrow \Delta''} are morphisms, then the operations of pullback by {\psi} and then by {\phi} compose to equal the operation of pullback by {\psi \circ \phi}: {\phi^* \psi^* = (\psi \circ \phi)^*}. 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 {\iota: \Delta \rightarrow \Delta'} to a simplicial complex {\Delta'} from a subcomplex {\Delta}. The associated pullback operation is the restriction operation, that maps a flat connection {\omega'} on {\Delta'} to its restriction {\omega'\downharpoonright_\Delta} to {\Delta}.

— 2. Sheaves —

We currently have a set {{\mathcal O}(\Delta)} of flat connections assigned to each simplicial complex {\Delta}, 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 {\Delta = \Delta_1 \cup \Delta_2} is the union of two simplicial complexes, then a flat connection on {\Delta} is determined by its restrictions to {\Delta_1} and {\Delta_2}. In other words, the map {\omega \mapsto (\omega\downharpoonright_{\Delta_1}, \omega\downharpoonright_{\Delta_2})} is an injection from {{\mathcal O}(\Delta)} to {{\mathcal O}(\Delta_1) \times {\mathcal O}(\Delta_2)}.
  • (Gluing) If {\Delta = \Delta_1 \cup \Delta_2}, and {\omega_1, \omega_2} are flat connections on {\Delta_1, \Delta_2} which agree when restricted to {\Delta_1 \cap \Delta_2}, (and if the orientations of {\Delta_1, \Delta_2} on the intersection {\Delta_1 \cap \Delta_2} agree) then there exists a flat connection {\omega} on {\Delta} which agrees with {\omega_1,\omega_2} on {\Delta_1, \Delta_2}. (Note that this gluing of {\omega_1} and {\omega_2} is unique, by the previous axiom. It is important that the orientations match; we cannot glue {(0,1)} to {(1,0)}, 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 {G} 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 {\Delta = (0,1,2)} is the standard {2}-simplex (together with subsimplices), and {\Delta'} is the subcomplex formed by removing the {2}-face {(0,1,2)} and the {1}-face {(0,2)}, leaving only the {1}-faces {(0,1), (1,2)} and the {0}-faces {0,1,2}. Then of course every flat connection on {\Delta} restricts to a flat connection on {\Delta'}. But the flatness property ensures that this restriction is invertible: given a flat connection on {\Delta'}, there exists a unique extension of this connection back to {\Delta}. This is nothing more than the property, remarked earlier, that to specify a flat connection on the {2}-simplex {(0,1,2)}, it suffices to know what the connection is doing on {(0,1)} and {(1,2)}, as the behaviour on the remaining edge can then be deduced from the group law; conversely, any specification of the connection on those two {1}-simplices determines a connection on the remainder of the {2}-simplex.

This observation can be generalised. Given any simplicial complex {\Delta}, define a {k}-dimensional collapse {\Delta'} of {\Delta} to be a simplicial complex obtained from {\Delta} by removing the interior of a {k}-simplex, together with one of its faces; thus for instance the complex consisting of {(0,1), (1,2)} (and subsimplices) is a {2}-dimensional collapse of the {2}-simplex {(0,1,2)} (and subsimplices). We then see that the sheaf of flat connections obeys an additional axiom:

  • (Grothendieck’s axiom) If {\Delta'} is a {k}-dimensional collapse of {\Delta}, then the restriction map from {{\mathcal O}(\Delta)} to {{\mathcal O}(\Delta')} is surjective for all {k}, and bijective for {k \geq 2}.

This axiom is trivial for {k=0}. For {k=1}, 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 {2}-simplex, and the value of a flat connection on that edge is thus completely unconstrained. (In any event, the {k=1} case of this axiom can be deduced from the sheaf axioms.) For {k=2}, it follows because if one can remove a {2}-simplex and one of its edges from a complex, then the edge is not the boundary of any other {2}-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 {2}-simplex. For {k=3}, it follows because if oen removes a {3}-simplex and one of its {2}-simplex faces, the constraint associated to that {2}-simplex is implied by the constraints coming from the other three faces of the {3}-simplex (I recommend drawing a tetrahedron and chasing some loops around to see this), and so one retains bijectivity. For {k \geq 4}, the axiom becomes trivial again because the {k}-simplices and {k-1}-simplices have no impact on the definition of a flat connection.

Grothendieck’s beautiful observation is that the converse holds: if a (concrete) sheaf {\Delta \mapsto {\mathcal O}(\Delta)} obeys Grothendieck’s axiom, then it is equivalent to the sheaf of flat connections of some group {G}defined canonically from the sheaf. Let’s see how this works. Suppose we have a sheaf {\Delta \mapsto {\mathcal O}(\Delta)}, which is concrete in the sense that {{\mathcal O}(\Delta)} 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 {{\mathcal O}(\Delta)} 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 {{\mathcal O}(0,1)} flat connections on the standard {1}-simplex {(0,1)}. If the sheaf was indeed the sheaf of flat connections on a group {G}, then {{\mathcal O}(0,1)} is canonically identifiable with {G}. Inspired by this, we will define {G} to equal the space {{\mathcal O}(0,1)} of flat connections on {(0,1)}. The flat connections on any other {1}-simplex {(u,v)} can then be placed in one-to-one correspondence with elements of {G} by the morphism {u \mapsto 0, v \mapsto 1}, so flat connections on {(u,v)} can be viewed as being equivalent to an element of {G}.

At present, {G} 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 {(0,1)} to a point, and pull back the trivial connection on that point to obtain a flat connection {e} on {(0,1)}, 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 {g, h \in G}, then {g} and {h} are flat connections on {(0,1)}. By using the morphism {i \mapsto i-1} from {(1,2)} to {(0,1)}, we can pull back {h} to {(1,2)} to create a flat connection {\tilde h} on {(1,2)} that is equivalent to {h}. The restriction of {g} and {\tilde h} to the point {1} is trivial, so by the gluing axiom we can glue {g} and {\tilde h} to a flat connection on the complex {(0,1), (1,2)}. By Grothendieck’s axiom, one can then uniquely extend this connection to the {2}-simplex {(0,1,2)}, which can then be restricted to the edge {(0,2)}. Mapping this edge back to {(0,1)}, we obtain an element of {G}, which we will define to be {hg}.

This operation is closed. To verify the identity property, observe that if {g \in G}, then by starting with the simplex {(0,1,2)} and pulling back {g} under the morphism that sends {2} to {1} but is the identity on {0,1}, we obtain a flat connection on {(0,1,2)} which is equal to {g} on {(0,1)}, equivalent to the identity on {(1,2)}, and is equivalent to {g} on {(0,2)} (after identifying {(0,2)} with {(0,1)}). From the definition of group multiplication, this shows that {eg = g}; a similar argument (using a slightly different morphism from {(0,1,2)} to {(0,1)}) gives {ge = g}.

Now we establish associativity. Let {f,g,h \in G}. Using the definition of multiplication, we can create a flat connection on the {2}-simplex {(0,1,2)} which equals {h} on {(0,1)}, is equivalent to {g} on {(1,2)}, and is equivalent to {gh} on {(0,2)}. We then glue on the edge {(2,3)} and extend the flat connection to be equivalent to {f} on {(2,3)}. Using Grothendieck’s axiom and the definition of multiplication, we can then extend the flat connection to the {2}-simplex {(0,2,3)} to be equivalent to {f(gh)} on {(0,3)}. By another use of that axiom, we can also extend the flat connection to the {2}-simplex {(1,2,3)}, to be equivalent to {fg} on {(1,3)}. Now that we have three of the four faces of the {3}-simplex {(0,1,2,3)}, we can now apply the {k=3} case of Grothendieck’s axiom and extend the flat connection to the entire {3}-simplex, and in particular to the {2}-simplex {(0,1,3)}. Using the definition of multiplication again, we thus see that {f(gh) = (fg)h}, 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 {g \in G} be arbitrary. By the gluing axiom, one can form a flat connection on the complex {(0,1), (0,2)} which equals {g} on {(0,1)} and is equivalent to the identity on {(0,2)}. By Grothendieck’s axiom, this can be extended to a flat connection on {(0,1,2)}; the restriction of this connection to {(1,2)} is equivalent to some element of {G}, which we define to be {g^{-1}}. By construction, {g^{-1} g = e} as required.

We have just shown that {G} is a group. The last thing to do is to show that this abstract sheaf {{\mathcal O}} can be indeed identified with the sheaf of {G}-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 {G}. To verify that this genuinely determines a {G}-connection on that complex, we need to verify that if {(u,v)} and {(v,u)} are both in the complex, that the group elements {g, h} assigned to these edges invert each other. But we can pullback {(u,v),(v,u)} to the {2}-simplex {(0,1,2)} by mapping {0,2} to {u} and {1} to {v}, creating a flat connection that is equal to {g} on {(0,1)}, equivalent to {h} on {(1,2)}, and equivalent to the identity on {(0,2)}; by definition of multiplication or inverse we conclude that {g, h} invert each other as desired.

Thus the abstract connection defines a {G}-connection. From the definition of multiplication it is also clear that every {2}-simplex in the complex imposes the right relation on the three elements of {G} associated to the edges of that simplex that makes the {G}-connection flat. Thus we have a canonical way to associate a {G}-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 {0} or {1}-dimensional the claim is easy; for {k \geq 2} the claim follows from Grothendieck’s axiom (which applies both for the abstract flat connections (by hypothesis) and for {G}-flat connections (as verified earlier)) and the induction hypothesis.