Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




Monodromy is the name for the action of the the homotopy groups of a space XX on fibers of covering spaces or locally constant ∞-stacks on XX.

In point-set topology

We discuss monodromy of covering spaces in elementary point-set topology.



(monodromy of a covering space)

Let XX be a topological space and EpXE \overset{p}{\to} X a covering space. Write Π 1(X)\Pi_1(X) for the fundamental groupoid of XX.

Define a functor

Fib E:Π 1(X)Set Fib_E \;\colon\; \Pi_1(X) \longrightarrow Set

to the category Set of sets as follows:

  1. to a point xXx \in X assign the fiber p 1({x})Setp^{-1}(\{x\}) \in Set;

  2. to the homotopy class of a path γ\gamma connecting xγ(0)x \coloneqq \gamma(0) with yγ(1)y \coloneqq \gamma(1) in XX assign the function p 1({x})p 1({y})p^{-1}(\{x\}) \longrightarrow p^{-1}(\{y\}) which takes x^p 1({x})\hat x \in p^{-1}(\{x\}) to the endpoint of a path γ^\hat \gamma in EE which lifts γ\gamma through pp with starting point γ^(0)=x^\hat \gamma(0) = \hat x

    p 1(x) p 1(y) (x^=γ^(0)) γ^(1). \array{ p^{-1}(x) &\overset{}{\longrightarrow}& p^{-1}(y) \\ (\hat x = \hat \gamma(0)) &\mapsto& \hat \gamma(1) } \,.

This construction is well defined for a given representative γ\gamma due to the unique path-lifting property of covering spaces (this lemma) and it is independent of the choice of γ\gamma in the given homotopy class of paths due to the homotopy-lifting property (this lemma). Similarly, these two lifting properties give that this construction respects composition in Π 1(X)\Pi_1(X) and hence is indeed a functor.

Hence this defines a “permutation groupoid representation” of Π 1(X)\Pi_1(X).


Given a homomorphism between two covering spaces E ip iXE_i \overset{p_i}{\to} X, hence a continuous function f:E 1E 2f \colon E_1 \to E_2 which respects fibers in that the diagram

E 1 f E 2 p 1 p 2 X \array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X }

commutes, then the component functions

f| {x}:p 1 1({x})p 2 1({x}) f\vert_{\{x\}} \;\colon\; p_1^{-1}(\{x\}) \longrightarrow p_2^{-1}(\{x\})

are compatible with the monodromy Fib EFib_{E} (def. 1) along any path γ\gamma between points xx and yy from def. 1 in that the following diagrams of sets commute

p 1 1(x) f| {x} p 2 1(x) Fib E 1([γ]) Fib E 2([γ]) p 1 1(y) f| {y} p 2 1({y}). \array{ p_1^{-1}(x) &\overset{f\vert_{\{x\}}}{\longrightarrow}& p_2^{-1}(x) \\ {}^{\mathllap{Fib_{E_1}([\gamma])}}\downarrow && \downarrow^{\mathrlap{ Fib_{E_2}([\gamma]) }} \\ p_1^{-1}(y) &\underset{f\vert_{\{y\}}}{\longrightarrow}& p_2^{-1}(\{y\}) } \,.

This means that ff induces a natural transformation between the monodromy functors of E 1E_1 and E 2E_2, respectively, and hence that constructing monodromy is itself a functor from the category of covering spaces of XX to that of permutation representations of the fundamental groupoid of XX:

Fib:Cov(X)Set Π 1(X). Fib \;\colon\; Cov(X) \longrightarrow Set^{\Pi_1(X)} \,.

For any x^p 1 1(x)\hat x \in p_1^{-1}(x) let γ^\hat \gamma be the unique path in EE with γ^(0)=x^\hat \gamma(0) = \hat x and pγ^=γp \circ \hat \gamma = \gamma. By definition 1 we have

Fib E 1([γ(f)])(x^)=γ^(1) Fib_{E_1}([\gamma(f)])(\hat x) = \hat \gamma(1)

and hence

f(Fib E 1([γ(f)])(x^))=f(γ^(1)) f(Fib_{E_1}([\gamma(f)])(\hat x)) = f(\hat \gamma(1))

Now fγ^f \circ \hat \gamma satisfies fγ^(0)=f(x^)f \circ \hat \gamma(0) = f(\hat x) and pfγ^=γp \circ f \circ \hat \gamma = \gamma by the fact that ff preserves fibers. Hence by uniqueness of path lifting (this lemma), fγ^f \circ \hat \gamma is the unique lift of γ\gamma with starting point f(x^)f(\hat x). By def. 1 this means that

Fib E 2([γ])(f(x^))=f(γ^(1)). Fib_{E_2}([\gamma])(f(\hat x)) = f (\hat \gamma(1)) \,.

This is the equality to be shown.



(fundamental theorem of covering spaces)

The reconstruction of covering spaces from monodromy is an inverse functor to the monodromy functor. The resulting equivalence of categories

Cov(X)FibRecSet Π 1(X) Cov(X) \underoverset {\underset{Fib}{\longrightarrow}} {\overset{Rec}{\longleftarrow}} {} Set^{\Pi_1(X)}

between the category of covering spaces and the permutation groupoid representations of the fundamental groupoid is known as the fundamental theorem of covering spaces.


(fundamental groupoid of covering space)

Let EpXE \overset{p}{\longrightarrow} X be a covering space.

Then the fundamental groupoid Π 1(E)\Pi_1(E) of the total space EE is equivalently the Grothendieck construction of the monodromy functor Fib E:Π 1(X)SetFib_E \;\colon\; \Pi_1(X) \to Set

Π 1(E) Π 1(X)Fib E \Pi_1(E) \;\simeq\; \int_{\Pi_1(X)} Fib_E


  • objects are pairs (x,x^)(x,\hat x) consisting of a point xXx \in X and en element x^Fib E(x)\hat x \in Fib_E(x);

  • morphisms [γ^]:(x,x^)(x,x^)[\hat \gamma] \colon (x,\hat x) \to (x', \hat x') are morphisms [γ]:xx[\gamma] \colon x \to x' in Π 1(X)\Pi_1(X) such that Fib E([γ])(x^)=x^Fib_E([\gamma])(\hat x) = \hat x'.


By the uniqueness of the path-lifting (this lemma) and the very definition of the monodromy functor.


Let XX be a path-connected topological space and let EpXE \overset{p}{\to} X be a covering space. Then the total space EE is

  1. path-connected precisely if the monodromy Fib EFib_E is a transitive action;

  2. simply connected precisely if the monodromy Fib EFib_E is free action.


By example 1.

In cohesive \infty-Toposes

Let H\mathbf{H} be a cohesive (∞,1)-topos and XHX \in \mathbf{H} any object. Then the locally constant ∞-stacks on XX are represented by morphisms XLConstCore(Grpd)X \to LConst Core(\infty Grpd). By adjunction such morphisms are equivalent to (∞,1)-functors Π(X)Core(Grpd)\Pi(X) \to Core(\infty Grpd) This morphism exhibits the monodromy of the locally constant ∞-stack.

Specifically, the restriction BΩ xΠ(X)Π(X)Grpd\mathbf{B}\Omega_x \Pi(X) \hookrightarrow \Pi(X) \to \infty Grpd to the delooping BΩ xΠ(X)\mathbf{B}\Omega_x \Pi(X) of the loop space object Ω xΠ(X)\Omega_x \Pi(X) at a chosen baspoint x:*Xx : {*} \to X is the monodromy action of loops based at xXx \in X on the fiber of the locally constant \infty-stack over xx.


Revised on July 10, 2017 13:53:58 by Urs Schreiber (