# Kerodon

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Construction 8.1.0.1 (The Twisted Arrow Category). Let $\operatorname{\mathcal{C}}$ be a category. We define a new category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as follows:

• An object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is a morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$.

• Let $f: X \rightarrow Y$ and $f': X' \rightarrow Y'$ be objects of $\operatorname{Tw}(\operatorname{\mathcal{C}})$. A morphism from $f$ to $f'$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is a pair of morphisms $u: X' \rightarrow X$, $v: Y \rightarrow Y'$ in $\operatorname{\mathcal{C}}$ satisfying $f' = v \circ f \circ u$, so that we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [d]^-{f} & X' \ar [l]_-{u} \ar [d]^{f'} \\ Y \ar [r]^-{v} & Y'.}$
• Let $f: X \rightarrow Y$, $f': X' \rightarrow Y'$, and $f'': X'' \rightarrow Y''$ be objects of $\operatorname{Tw}(\operatorname{\mathcal{C}})$. If $(u,v)$ is a morphism from $f$ to $f'$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ and $(u',v')$ is a morphism from $f'$ to $f''$ in $\operatorname{\mathcal{C}}$, then the composition $(u',v') \circ (u, v)$ in $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is the pair $(u \circ u', v' \circ v)$.

We will refer to $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as the twisted arrow category of $\operatorname{\mathcal{C}}$.