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Hi all,

It is known (cf. Lurie's book "Higher Topos Theory", for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan complexes, aka quasi-categories. Let me recall that a simplicial space $X_\bullet$ is weak Kan if every map $\Lambda_i^n\longrightarrow X$ (where $\Lambda_i^n$ is the horn) can be extended to a map $\Delta^n\longrightarrow X$ for all $0< i < n$.

My problem is the following. Let us consider the fundamental $(\infty,0)$-groupoid $\pi_{\leq \infty}X$ of a nice space $X$: $X$ is the set of objects, morphisms between objects are maps $f:[0,1]\longrightarrow X$, $2$-morphisms are homotopies between morphisms, and so on. I am quite confused about how does one, "geometrically", see $\pi_{\leq \infty}X$ as a simplicial space; for an element of $\pi_2X$, for instance, has four edges, and hence four different face maps.

Many thanks!

It is known (cf. Lurie's book Higher Topos Theory for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan complexes, aka quasi-categories. Let me recall that a simplicial space $X_\bullet$ is weak Kan if every map $\Lambda_i^n\longrightarrow X$ (where $\Lambda_i^n$ is the horn) can be extended to a map $\Delta^n\longrightarrow X$ for all $0< i < n$.

My problem is the following. Let us consider the fundamental $(\infty,0)$-groupoid $\pi_{\leq \infty}X$ of a nice space $X$: $X$ is the set of objects, morphisms between objects are maps $f:[0,1]\longrightarrow X$, $2$-morphisms are homotopies between morphisms, and so on. I am quite confused about how does one, "geometrically", see $\pi_{\leq \infty}X$ as a simplicial space; for an element of $\pi_2X$, for instance, has four edges, and hence four different face maps.

Hi all,

It is known (cf. Lurie's book "Higher Topos Theory", for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" viewed as Kan complexes. Let me recall that a simplicial space $X_\bullet$ is Kan if every map $\Lambda_i^n\longrightarrow X$ (where $\Lambda_i^n$ is the horn) can be extended to a map $\Delta^n\longrightarrow X$ for all $0< i < n$.

My problem is the following. Let us consider the fundamental $(\infty,0)$-groupoid $\pi_{\leq \infty}X$ of a nice space $X$: $X$ is the set of objects, morphisms between objects are maps $f:[0,1]\longrightarrow X$, $2$-morphisms are homotopies between morphisms, and so on. I am quite confused about how does one, "geometrically", see $\pi_{\leq \infty}X$ as a simplicial space; for an element of $\pi_2X$, for instance, has four edges, and hence four different face maps.

Many thanks!

Hi all,

It is known (cf. Lurie's book "Higher Topos Theory", for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as weak Kan complexes, aka quasi-categories. Let me recall that a simplicial space $X_\bullet$ is weak Kan if every map $\Lambda_i^n\longrightarrow X$ (where $\Lambda_i^n$ is the horn) can be extended to a map $\Delta^n\longrightarrow X$ for all $0< i < n$.

My problem is the following. Let us consider the fundamental $(\infty,0)$-groupoid $\pi_{\leq \infty}X$ of a nice space $X$: $X$ is the set of objects, morphisms between objects are maps $f:[0,1]\longrightarrow X$, $2$-morphisms are homotopies between morphisms, and so on. I am quite confused about how does one, "geometrically", see $\pi_{\leq \infty}X$ as a simplicial space; for an element of $\pi_2X$, for instance, has four edges, and hence four different face maps.

Many thanks!