equivalences in/of $(\infty,1)$-categories
The fundamental $\infty$-groupoid $\Pi_\infty(X)$ of a topological space $X$ is the ∞-groupoid whose k-morphisms are the $k$-dimensional paths in $X$. This is the higher refinement of the fundamental groupoid $\Pi_1(X)$.
It is also sometimes called the $\infty$-Poincaré groupoid of the space, in analogy to the term Poincaré groupoid for the fundamental groupoid.
The following definition is appropriate if we take a Kan complex as the definition of $\infty$-groupoid.
The fundamental $\infty$-groupoid $\Pi(X)$ of a topological space $X$ is given by the Kan complex
which is the singular simplicial complex of $X$.
This construction is right adjoint to geometric realization.
By choosing horn-fillers this becomes an algebraic Kan complex. In terms of these the homotopy hypothesis has a direct proof, exhibited by a Quillen equivalence
due to (Nikolaus).
One may regard the singular simplicial complex functor as the instance of the general abstract notion of fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos by regarding Top as a cohesive (∞,1)-topos. See discrete ∞-groupoid for more on this.
For other models of ∞Grpd there are correspondingly other constructions:
One can consider strict ∞-groupoid versions of the fundamental $\infty$-groupoid. These lose information about the homotopy type of the space, though, but are more tractable and may give in some applications all the information that one is interested in.
The study of strict fundamental $\infty$-groupoids have been pursued by Ronnie Brown and his school.
There is a strict homotopy 2-groupoid for a Hausdorff space defined by Hardie, K. A.; Kamps, K. H.; Kieboom, R. W. (MR1785844), and a weak homotopy 2-groupoid for a general space (by the same authors). They later introduced a homotopy double groupoid. There is no $n$-dimensional version of these ideas on offer.
A strict cubical omega-groupoid $\rho X_*$ for a filtered space $X_*$ was defined by Brown and Higgins in 1981. Form the filtered cubical complex $R X_*$ which in dimension $n$ consists of filtered maps $I^n_* \to X_*$ and take filter homotopy classes of these relative to the vertices. The proof that the compositions in $RX_*$ are inherited by $\rho X_*$ is one of the key points of the development.
It turns out that $\rho X_*$ is equivalent in a clear sense to the crossed complex $\Pi X_*$ defined using relative homotopy groups by Blakers in 1948 (with other terminology) and that the homotopy types modelled by crossed complexes, or by the corresponding globular or cubical gadget, are restricted, essentially to the linear homotopy types, with no quadratic information. Nonetheless, it is well known in mathematics that linear approximations can be useful.
Loday’s paper of 1982 on Spaces with finitely many homotopy groups introduced the entirely new idea of a cubical resolution of a space. Some details were completed by Richard Steiner. Loday also introduced the fundamental cat-n-group of an $n$-cube of spaces. In this way we get a model of a space $X$ by a multiple groupoid in which the $r$-dimensional homotopy of $X$ occurs in the right place in the model. Also you can calculate something with this model, and it has led to new algebraic constructions, such as a nonabelian tensor product of groups, with homotopical applications.
These strict groupoid models do satisfy the dimension condition.
The 1-truncation of $\Pi X$ is the fundamental groupoid of $X$:
For $X$ a locally contractible topological space, the fundamental $\infty$-groupoid $Sing X$ is equivalent to the fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos of the (∞,1)-sheaf (∞,1)-topos $(\infty,1)Sh(X)$.
Details on this are at geometric homotopy groups in an (∞,1)-topos.
This perspective suggests that when $X$ is not locally contractible, a better replacement for its fundamental $\infty$-groupoid (as usually defined) is the shape of $(\infty,1)Sh(X)$. As discussed there, this coincides with the traditional shape theory of $X$.
fundamental groupoid, fundamental ∞-groupoid
fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos
K.A. Hardie, K. H. Kamps, R. Kieboom, A homotopy 2-groupoid of a Hausdorff space. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures 8 (2000), no. 1-2, 209–234.
K.A. Hardie, K. H. Kamps, R. Kieboom, A Homotopy Bigroupoid of a Topological Space. Applied Categorical Structures 9 (2001) 311–327.
J.-L. Loday, Spaces with finitely many homotopy groups, J.Pure Appl. Alg., 24 (1982) 179–202.
J. M. Casas, G. Ellis, M. Ladra, T. Pirashvili, Derived functors and the homology of $n$-types, J. Algebra 256, 583–598 (2002).
The direct proof of the homotopy hypothesis for the algebraic version of the fundamental $\infty$-groupoid is in
Strict versions of fundamental $\infty$-groupoids are discussed in
See also
Ronnie Brown and Philip Higgins, Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra 22 (1981) 11-41.
Ronnie Brown, A new higher homotopy groupoid: the fundamental globular $\omega$-groupoid of a filtered space, Homotopy, Homology and Applications, 10 (2008), No. 1, pp.327-343.
Richard Steiner, Resolutions of spaces by $n$-cubes of fibrations, J. London Math. Soc.(2), 34, 169-176, 1986.