Let $f,g\colon X\rightarrow Y$ be maps between spaces.

I am looking for criteria for the equalizer of $f$ and $g$ to be a homotopy equalizer and I am happy to get answers for whatever model category of spaces you prefer.

As the equalizer of $f$ and $g$ can be written as the pullback of $(f,g)\colon X\rightarrow Y\times Y$ along the diagonal $Y\rightarrow Y\times Y$, it is a homotopy equalizer if $(f,g)$ is a fibration, which unfortunately doesn't happen very often.

I feel there should be something weaker which is more likely to happen. An indication for this is the following thought. If $k\colon Z\rightarrow Y$ and $l\colon W\rightarrow Y$, the pullback of $k$ and $l$ is the equalizer of the maps from $Z\times W$ to $Y$ given by projecting to one factor first and then using $k$ respectively $l$. The map analogous to $(f,g)$ is then $k\times l$. But $k\times l$ is often not a fibration even if the pullback is a homotopy pullback, e.g. if only one of the maps is a fibration and the other is not.

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