I mention that the question of topologising $\pi_1X$ when $X$ has a universal cover is generalised in 10.5.8 of the book Topology and Groupoids.

The question as a whole is relevant to the history of topology, see for example the book with that title edited by I.M. James (Elsevier, 1999).

In the early 20th century, topologists such as Dehn were looking for generalisations to all dimensions of the fundamental group $\pi_1(X,x)$ of a pointed space. At the 1932 ICM in Zurich, E. Cech gave a seminar on higher homotopy groups $\pi_n(X,x)$, and explained that they were abelian for $n \geqslant 2$. On this ground, Alexandroff and Hopf argued that they were not the hoped for generalisation, and only a small paragraph appeared in the ICM Proceedings. The importance of higher homotopy groups became clear later with work of Hurewicz.

The argument for the abelian nature of higher homotopy groups is nowadays often expressed as "a group object in the category of groups is an abelian group".

The notion of groupoid was defined by Brandt in 1926, for applications in number theory, and many applications to differential geometry were developed by C. Ehresmann and his group. He also developed notions of $n$-fold groupoids as for instance for $n=2$ as groupoid objects in the category of groupoids. This led me in 1965 to look for "strict higher homotopy groupoids" and a higher Seifert-Van Kampen Theorem. This was found in dimension 2 in collaboration with P.J.Higgins in 1974, giving applications to second relative homotopy groups, relevant to J.H.C.Whitehead's 1949 paper "Combinatorial Homotopy II", and in all dimensions in 1981. That theory is developed not for pointed spaces but for filtered spaces. The argument for this is developed in this paper - essentially that to calculate something about a space you need information on it, and that information will have a certain structure which may be able to be formalised appropriately. The usual structure of a base point is rather minimalist for this end.

I mention that the question of topologising $\pi_1X$ when $X$ has a universal cover is generalised in 10.5.8 of the book Topology and Groupoids.

The question as a whole is relevant to the history of topology, see for example the book with that title edited by I.M. James (Elsevier, 1999).

In the early 20th century, topologists such as Dehn were looking for generalisations to all dimensions of the fundamental group $\pi_1(X,x)$ of a pointed space. At the 1932 ICM in Zurich, E. Cech gave a seminar on higher homotopy groups $\pi_n(X,x)$, and explained that they were abelian for $n \geqslant 2$. On this ground, Alexandroff and Hopf argued that they were not the hoped for generalisation, and only a small paragraph appeared in the ICM Proceedings. The importance of higher homotopy groups became clear later with work of Hurewicz. The historical side of this is discussed in this paper.

The argument for the abelian nature of higher homotopy groups is nowadays often expressed as "a group object in the category of groups is an abelian group".

The notion of groupoid was defined by Brandt in 1926, for applications in number theory, and many applications to differential geometry were developed by C. Ehresmann and his group. He also developed notions of $n$-fold groupoids as for instance for $n=2$ as groupoid objects in the category of groupoids. This led me in 1965 to look for "strict higher homotopy groupoids" and a higher Seifert-Van Kampen Theorem. This was found in dimension 2 in collaboration with P.J.Higgins in 1974, giving applications to second relative homotopy groups, relevant to J.H.C.Whitehead's 1949 paper "Combinatorial Homotopy II", and in all dimensions in 1981. That theory is developed not for pointed spaces but for filtered spaces. The argument for this is developed in this paper - essentially that to calculate something about a space you need information on it, and that information will have a certain structure which may be able to be formalised appropriately. The usual structure of a base point is rather minimalist for this end.

I mention that the question of topologising $\pi_1X$ when $X$ has a universal cover is generalised in 10.5.8 of the book Topology and Groupoids.

The question as a whole is relevant to the history of topology, see for example the book with that title edited by I.M. James (Elsevier, 1999).

In the early 20th century, topologists such as Dehn were looking for generalisations to all dimensions of the fundamental group $\pi_1(X,x)$ of a pointed space. At the 1932 ICM in Zurich, E. Cech gave a seminar on higher homotopy groups $\pi_n(X,x)$, and explained that they were abelian for $n \geqslant 2$. On this ground, Alexandroff and Hopf argued that they were not the hoped for generalisation, and only a small paragraph appeared in the ICM Proceedings. The importance of higher homotopy groups became clear later with work of Hurewicz.

The argument for the abelian nature of higher homotopy groups is nowadays often expressed as "a group object in the category of groups is an abelian group".

The notion of groupoid was defined by Brandt in 1926, for applications in number theory, and many applications to differential geometry were developed by C. Ehresmann and his group. He also developed notions of $n$-fold groupoids as for instance for $n=2$ as groupoid objects in the category of groupoids. This led me in 1965 to look for "strict higher homotopy groupoids" and a higher Seifert-Van Kampen Theorem. This was found in dimension 2 in collaboration with P.J.Higgins in 1974, giving applications to second relative homotopy groups, relevant to J.H.C.Whitehead's 1968 paper "Combinatorial Homotopy II", and in all dimensions in 1981. That theory is developed not for pointed spaces but for filtered spaces. The argument for this is developed in this paper - essentially that to calculate something about a space you need information on it, and that information will have a certain structure which may be able to be formalised appropriately. The usual structure of a base point is rather minimalist for this end.

I mention that the question of topologising $\pi_1X$ when $X$ has a universal cover is generalised in 10.5.8 of the book Topology and Groupoids.

The question as a whole is relevant to the history of topology, see for example the book with that title edited by I.M. James (Elsevier, 1999).

In the early 20th century, topologists such as Dehn were looking for generalisations to all dimensions of the fundamental group $\pi_1(X,x)$ of a pointed space. At the 1932 ICM in Zurich, E. Cech gave a seminar on higher homotopy groups $\pi_n(X,x)$, and explained that they were abelian for $n \geqslant 2$. On this ground, Alexandroff and Hopf argued that they were not the hoped for generalisation, and only a small paragraph appeared in the ICM Proceedings. The importance of higher homotopy groups became clear later with work of Hurewicz.

The argument for the abelian nature of higher homotopy groups is nowadays often expressed as "a group object in the category of groups is an abelian group".

The notion of groupoid was defined by Brandt in 1926, for applications in number theory, and many applications to differential geometry were developed by C. Ehresmann and his group. He also developed notions of $n$-fold groupoids as for instance for $n=2$ as groupoid objects in the category of groupoids. This led me in 1965 to look for "strict higher homotopy groupoids" and a higher Seifert-Van Kampen Theorem. This was found in dimension 2 in collaboration with P.J.Higgins in 1974, giving applications to second relative homotopy groups, relevant to J.H.C.Whitehead's 1949 paper "Combinatorial Homotopy II", and in all dimensions in 1981. That theory is developed not for pointed spaces but for filtered spaces. The argument for this is developed in this paper - essentially that to calculate something about a space you need information on it, and that information will have a certain structure which may be able to be formalised appropriately. The usual structure of a base point is rather minimalist for this end.