Commutative diagram shapes

Definition 1.4.1 (Diagram shapes). A commutative square in an \(\infty \)-category \(C\) is a functor \([1] \times [1] \to C\). We will often display such a square as

Throughout, we read the first coordinate of \([1] \times [1]\) as the row and the second as the column, so that the four objects \((0,0)\), \((0,1)\), \((1,0)\) and \((1,1)\) are displayed as \(x\), \(y\), \(z\) and \(w\), respectively. In particular \(\{0\} \times [1]\) and \(\{1\} \times [1]\) are the top and bottom rows, while \([1] \times \{0\}\) and \([1] \times \{1\}\) are the left and right columns.

Let \(\pushout \subseteq [1] \times [1]\) be the subposet spanned by the three objects \((0,0)\), \((0,1)\) and \((1,0)\), and let \(\pullback \subseteq [1] \times [1]\) be the subposet spanned by the three objects \((1,0)\), \((0,1)\) and \((1,1)\). A span in \(C\) is a functor \(\pushoutto C\), and a cospan in \(C\) is a functor \(\pullbackto C\). We display them as

Every commutative square gives rise to both a span and a cospan via restriction along the two inclusions \(\pushout \hookrightarrow [1] \times [1]\) and \(\pullback \hookrightarrow [1] \times [1]\).

Finally, let \([\omega ] := \{0 \leq 1 \leq 2 \leq 3 \leq \cdots \}\) be the poset of natural numbers. An infinite sequence of morphisms in \(C\) is a functor \(x_{\bullet }\colon [\omega ] \to C\), which we display as \[ x_0 \xrightarrow {f_0} x_1 \xrightarrow {f_1} x_2 \xrightarrow {f_2} \dots , \] where \(x_n\) is the evaluation of \(x_{\bullet }\) at \(n \in [\omega ]\) and \(f_n\) is obtained from the inclusion \([1] \cong \{n \leq n+1\} \hookrightarrow [\omega ]\).