Horn Filling Condition in Quasicategories

Question

A simplicial set $X$ is called a Kan complex if every horn $\Lambda^k [n]$, $0 \leq k \leq n$ has a filling:

Here $\Lambda^k [n]$ is the union of the faces $\Delta[n]_i$ with $1 \leq i \leq n$, $i \neq k$, inside $\Delta [n]$.

If this filling condition holds for $0 < k < n$, then we say that the simplicial set $X$ is a quasi-category.

For $n = 2$, this is saying that we can compose $1$-morphisms up to homotopy, but I can't visualize/comprehend what this is saying for higher $n$. Could someone provide some examples with $n = 3$, or somehting to help me comprehend this inner horn filling condition?

Answer 1

So having for example $f:\Lambda^3_2 \to X$ is having the data of

• 4 0-simplicies $x_0,x_1,x_2,x_3$,
• 6 1-simplicies $f_{0,1},f_{1,2},f_{2,3},f_{0,2},f_{0,3},f_{1,3}$ that we can interpret as 'morphisms' $f_{i,j}: x_i \to x_j$ for $i<j$ in $\{0<1<2<3\}$, meaning the two 0-faces of $f_{i,j}$ are $x_i$ and $x_j$,
• 3 2-simplicies $f_{0,2,3}, f_{0,1,2}, f_{1,2,3}$, ($f_{0,1,3}$ beeing the one missing), that we can interpret as homotopies witnessing compositions : for example $f_{0,2,3} : f_{2,3}\circ f_{0,2} \sim f_{0,3}$,
• no 3-simplex.

So the horn filling condition states that given that data you can find the missing 2-simplex $f_{0,1,3}$, and the missing 3-simplex $f_{0,1,2,3}$, this 3-simplex, i.e. a homotopy of compositions of homotopies : $$f_{0,1,2,3} : f_{0,2,3} \circ f_{0,1,2} \sim f_{0,1,3} \circ f_{1,2,3}$$ The first composition of homotopies, $f_{0,2,3} \circ f_{0,1,2}$, says that we can decompose $f_{0,3}$ as $f_{2,3} \circ f_{0,2}$ first and then decomposite further to get $f_{2,3} \circ (f_{1,2} \circ f_{0,1})$.

The second composition of homotopies, $f_{0,1,3} \circ f_{1,2,3}$ says we can decompose $f_{0,3}$ as $f_{1,3} \circ f_{0,1}$ first and then as $(f_{2,3} \circ f_{1,3}) \circ f_{0,1}$.

So this 3-simplex encodes assosciativiy of the compositions of 3 morphisms up to homotopy, a homotopy between $f_{2,3} \circ (f_{1,2} \circ f_{0,1})$ and $(f_{2,3} \circ f_{1,3}) \circ f_{0,1}$.

I like to see 1-simplicies $f_{i,i+1}$ as the morphisms I want to compose, and $f_{i,j}$ with $|i-j| > 1$ as a candidate for compositions, so here for example let's say $f_{0,1} = f$, $f_{1,2} = g$, $f_{1,3} = h$, then $f_{0,2}$ will be a candidate for the composition $g \circ f$, $f_{1,3}$ will be a candidate of the composition $h \circ g$, and $f_{0,3}$ a candidate for composition $h\circ g \circ f$, but this $f_{0,3}$ can a candidate for $h \circ g \circ f$ in two ways : $f_{1,3} \circ f_{0,1}$ i.e. $(h\circ g) \circ f$ or $f_{2,3} \circ f_{0,2}$ i.e. $h\circ (g \circ f)$. 2 simplices when they exist are ways to compose and 3 simplicies are ways to compose those compositions and so on.

4-horns can be interpreted in a similar way but it is tedious to write down and you need to have good vision in 4-dimensions.