nLab dagger equalizer

Idea

The analogue of equalizers in dagger categories.

Definition

For two parallel morphisms $f \colon x \to y$ and $g \colon x \to y$ in a dagger category $(\mathcal{C}, \dagger)$, a dagger equalizer is a dagger monomorphism $e \colon \mathrm{eq} \to x$ from an object $\mathrm{eq} \in \mathcal{C}$ such that

is a fork (i.e., $f \circ e = g \circ e$), and every morphism $h \colon z \to x$ so that $f \circ h = g \circ h$ factors through $e$:

If $(\mathcal{C}, \dagger)$ also has a zero object $0$, then when $g=0_!: x \to 0 \to y$ is the unique zero morphism, the dagger equalizer $e \colon \mathrm{eq} \to x$ is called the dagger kernel. This is the analogue of kernels in ordinary category theory.

References

An exposition of dagger categories that discusses dagger equalizers and dagger kernels in their relation to linear structure and quantum measurement:

• Chris Heunen, Jamie Vicary: Categories for Quantum Theory, Oxford University Press (2019) [ISBN:9780198739616]

based on:

• Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics (2012) [pdf, pdf]

Dagger equalizers and dagger kernels used to axiomatize the category Hilb of Hilbert spaces:

• Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces (arXiv:2109.07418)