How is your notion related to SWOLs (small categories without loops) which occur in the theory of Haefliger on Complexes of groups. (There isn introduction here but you should really look at Bridson and Haefliger’s book (you only need the later part.) These are also cropping up in work on directed homotopy as loop free categories.

]]>Every isomorphism in the preorder reflection is an identity: is that equivalent to the condition you’re after?

Or equivalently, that the preorder reflection is isomorphic to the poset reflection?

I don’t know how this condition is referred to in the literature, if it is.

]]>Is there any notion and technical term in category theory for the concept of *acirclic category* described below ?

The most strict definition of the term, i.e. *category $\mathsf{C}$ such that the quiver obtained from $\mathsf{C}$ by applying the forgetful functor $Cat\rightarrow Quiv$ does not contain any directed-cycle-of-length-at- least-two-without-repeated-vertices* is probably much too combinatorial and strict (use of equality, use of negations…), but what about

*acirclic category* $:=$ category $\mathsf{C}$ such that there does **not** exist any finite sequence of objects $O_0,...,O_{\ell-1}$ such that

- no two of the $O_i$ are isomorphic in $\mathsf{C}$

- the hom-set $\mathsf{C}(O_i,O_{i+1})$ is non-empty for each $i\in\ell$, with index-calculations modulo $\ell$.

This at least does not speak of equality of objects.

I never saw this being an issue in category theory anywhere. In this case I am not saying this should be studied.

At least according to some definitions of importance, it *evidently* is unimportant from a category-theoretical perspective. Possible this has to do with the definition making too much use of negatives. Another aspect is that acirclic categories are intuitively very far from groupoids.

One instructive aspect of this might be to try to characterize acirclicity in categorical terms.

Terminological note: chose “acirclic” since it is more distinctive and different from the many standard uses of “acyclic”

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