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Stream: learning: questions

Topic: equalisers/pullbacks of "global elements", but for mon. cats

Emily (Mar 16 2021 at 02:13):

In $\mathsf{Sets}$, given maps $x,y\colon\mathrm{pt}\rightrightarrows X$ with $x\neq y$, we have $\mathrm{Eq}(x,y)\cong\mathrm{pt}\times_{x,X,y}\mathrm{pt}\cong\emptyset$, where the pullback agrees with the equaliser because $\mathrm{pt}$ is terminal in $\mathsf{Sets}$.

When one passes however to a bicomplete monoidal category $(\mathcal{V},\otimes,\mathbf{1}_\mathcal{V})$ and replaces $\mathrm{pt}$ with $\mathbf{1}_{\mathcal{V}}$ and $\emptyset$ with the initial object $\varnothing_{\mathcal{V}}$ of $\mathcal{V}$, this may not be the case anymore.

Question: Are there nice conditions on $\mathcal{V}$ forcing this to be the case? I.e., such that, for any pair of maps $x,y\colon\mathbf{1}_{\mathcal{V}}\rightrightarrows V$ of $\mathcal{V}$, we have $\mathbf{1}_{\mathcal{V}}\times_{x,V,y}\mathbf{1}_\mathcal{V}\cong\varnothing_{\mathcal{V}}$?