Could not include topos theory - contents
By the logic of space, one may think of a topos as being (the category of sheaves on) some generalization of a topological space. The construction of the first fundamental group of a topological space happens to generalize to an arbitrary topos, even without a predefined notion of path or interval object. Instead, by the logic of Grothendieck's Galois theory, the homotopy groups can be detected from the behaviour of locally constant sheaves.
(At present the following is just lifted/adapted from another article, (see references below) and needs some editing and possibly correcting. T.P.)
Let $\mathbb{E}$ be a Grothendieck topos (think of $\mathbb{E}$ as the category, $Sh(X)$, of set valued sheaves on a space $X$). Within $\mathbb{E}$, we can pick out a subcategory, $\mathbb{C}$, of locally finite, locally constant objects in $\mathbb{E}$. (If $X$ is a space with $\mathbb{E}= Sh(X)$, $\mathbb{C}$ corresponds to those sheaves whose ‘espace étalé’ is a finite covering space of $X$.) Picking a base point in $X$ generalises to picking a ‘fiber functor’ $F :\mathbb{C} \to \mathbf{Sets_{fin}}$, a functor satisfying various conditions implying that it is pro-representable. (If $x_0 \in X$ is a base point $\{x_0\}\to X$ induces a ‘fibre functor’ $Sh(X)\to Sh\{x_0\} \cong \mathbf{Sets}$, by pullback.)
Mike: I presume that more generally any “point” of $\mathbb{E}$, meaning a geometric morphism $Set\to \mathbb{E}$, supplies a fibre functor (its inverse image)? Of course, in general $\mathbb{E}$ might not have a point. Are there other examples of fibre functors when $\mathbb{E}=Sh(X)$?
If $F$ is ‘pro-representable’ by $P$, then $\pi_1(\mathbb{E},F)$ is defined to be $Aut(P)$, which is a profinite group. (Usually we will simply write $\pi_1(\mathbb{E})$, for this.) Grothendieck proves there is an equivalence of categories
the category of finite $\pi_1(\mathbb{E})$-sets. (This is what is called Grothendieck's Galois theory.)
If $X$ is a locally nicely behaved space such as a CW complex and $\mathbb{E} = Sh(X)$, then $\pi_1(\mathbb{E})$ is the profinite completion of $\pi_1(X)$. This profinite completion occurs only because Grothendieck considers locally finite objects. Without this restriction, a covering space $Y$ of $X$ would correspond to a $\pi_1(X)$-set, $Y^\prime$, but if $Y$ is a finite covering of $X$ then the homomorphism from $\pi_1(X)$ to the finite group of transformations of $Y$ factors through the profinite completion of $\pi_1(X)$.
This idea of using covering spaces or their analogue in $\mathbb{E}$ raises several important points:
these are homotopy theoretic results, but no paths (no interval object) are used. The argument involving sheaf theory, the theory of pro-representable functors, etc., is of a purely categorical nature. This means it is applicable to spaces where the use of paths, and other homotopies is impossible because of bad (or unknown) local properties. Such spaces have been studied within shape theory and strong shape theory, although not by using exactly Grothendieck’s fundamental group, nor using sheaf theory. (See below for more on this connection and such sources as Lisica and Mardešić, Edwards and Hastings, Cordier and Porter, Mardešić and Segal for more information on Shape and Strong Shape).
As no paths are used, these methods can also be applied to ‘non-spaces’, e.g. locales and possibly to their non-commutative analogues, quantales.
For instance, classically one could consider a field $k$ and an algebraic closure $K$ of $k$ and then choose $\mathbb{C}$ to be a category of étale algebras over $k$, in such a way that $\pi_1(\mathbb{E}) \cong Gal(K/k)$, the Galois group of $k$. A beautiful treatment of this can be found in Douady and Douady, (see below), and the link with locales (which is very strong) is explored in Joyal and Tierney. It, in fact, leads to a classification theorem for Grothendieck toposes. From this viewpoint, low dimensional homotopy theory is seen as being part of Galois theory, or vice versa. Of course, the really interesting question is how to fit higher dimensional homotopy theory into a higher dimensional Galois theory, and, again, -vice versa-. See A. Grothendieck, (1975?), Letter to L. Breen. NOT to Quillen as is sometimes claimed. More on that at homotopy group of an ∞-stack.
Mike: From a higher-categorical perspective, the reason $\pi_1(X)$ classifies covering spaces is that covering spaces are fibrations with discrete fibers, and so are classified by functors $\Pi_\infty(X)\to Set$. But since $Set$ is a 1-category, any such functors factors through the 1-categorical reflection of $\Pi_\infty(X)$, which is the ordinary fundamental groupoid $\Pi_1(X)$. Thus, to ask what higher homotopy groups classify, one should consider not $\pi_i(X)$ but $\Pi_i(X)$, which one might expect to classify fibrations over $X$ whose fibers are homotopy $(i-1)$-types.
Does a topos have a fundamental groupoid? A fundamental $i$-groupoid? A fundamental $\infty$-groupoid?
Tim: I am not an expert on the ‘not enough points’ case, but do know that a long time ago people generalised to a fundamental groupoid. (I think it was a preprint from Montpellier in which I saw this, but I know it was also taken up by others in (?) the 1970s. More recently Marta Bunge and Eduardo Dubuc have published on this and look at at Eduardo’s:
http://arxiv.org/abs/0706.1771
David: The paper Higher Monodromy shows what the fundamental 2-groupoid classifies.
Mike: Just looking at the abstract of Higher Monodromy, what I was saying above (for $i=2$) looks like the special case of their theory for locally constant stacks with values in the 2-category $Gpd$ of 1-types. They don’t seem to treat the topos case, though.
One may consider the vertical categorification of this situation from the notion of topos to that of (∞,1)-topos. The discussion of the construction of homotopy groups of objects in such a higher topos is at
The above was taken from an article:
The general construction of the first geometric homotopy group of objects in a Grothendieck topos is in section 8.4 of
A discussion of the fundamental groupoid for a general topos is in
A discussion for “locally simply connected” toposes is in
Other references are
D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag.
J.T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glasnik Mat. 19(39) (1984) 335-399.
J.-M. Cordier and Tim Porter, (1989), Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications, Ellis Horwood. Reprinted Dover (2008).
S. Mardešić and J. Segal, (1982) Shape Theory, North Holland.
S. Mardešić, Strong Shape and Homology, Springer monographs in mathematics, Springer-Verlag.
On categorical Galois theory
and for a more traditional approach:
For the link with locale theory
When we have no paths, in internal case, one may find the article of Pataraia useful (beware many typoses, but the article is dense with content):
Luis Javier Hernández-Paricio, Fundamental pro-groupoids and covering projections(pdf)