# Horn Filling Condition in Quasicategories

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?

• The n=3 condition can be used, for example, to prove that composition in the homotopy category of a quasicategory is well-defined. In general the n>2 conditions encode the homotopy commutativity of diagrams. Jun 29, 2019 at 6:14

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 $$i in $$\{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.