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Stream: learning: questions

Topic: Double semicategory?

Adittya Chaudhuri (Mar 20 2026 at 07:44):

There is a notion called semicategory, which is exactly like a category but without the requiremnent for the existence of the identity morphisms.

Recently, I came across the following situation:

I was able to construct a structure like a double category $\mathbb{D}=[\mathsf{D}_{1} \rightrightarrows \mathsf{D}_{0} ]$, where $\mathsf{D}_{1}$ is the category of arrows and $\mathsf{D}_{0}$ is the category of objects, where I could not find a natural way to define the identity assigning functor $\mathsf{U} \colon \mathsf{D}_{0} \to \mathsf{D}_{1}$. But, the composition functor $\circ \colon \mathsf{D}_{1} \times_{\mathsf{D}_{0} } \mathsf{D}_{1} \to \mathsf{D}_{1}$ makes the composition of horizontal 1-cells associative.

The above situation naturally led me to ask whether there already exists a notion of double semicategory in the literature with the same spirit as a semicategory is to a category.