# Kerodon

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Definition 1.5.4.3. Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given a morphism $f: A \rightarrow B$ and $g: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$. We will say that $f$ is weakly left orthogonal to $g$ if, for every pair of morphisms $u: A \rightarrow X$ and $v: B \rightarrow Y$ satisfying $g \circ u= v \circ f$, the associated lifting problem

$\xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r]^-{u} & X \ar [d]^{g} \\ B \ar [r]^-{v} \ar@ {-->}[ur] & Y }$

admits a solution (that is, there exists a map $h: B \rightarrow X$ satisfying $g \circ h = v$ and $h \circ f = u$). In this case, we will also say that $g$ is weakly right orthogonal to $f$.

If $S$ and $T$ are collections of morphisms of $\operatorname{\mathcal{C}}$, we say that $S$ is weakly left orthogonal to $T$ if every morphism $f \in S$ is weakly left orthogonal to every morphism $g \in T$. In this case, we also say that $T$ is weakly right orthogonal to $S$. In the special case where $S = \{ f\}$ is a singleton, we abbreviate this condition by saying that $f$ is weakly left orthogonal to $T$, or $T$ is weakly right orthogonal to $f$. In the special case $T = \{ g\}$ is a singleton, we abbreviate this condition by saying that $g$ is weakly right orthogonal to $S$, or $S$ is weakly left orthogonal to $g$.