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Stream: theory: category theory

Topic: a certain 2-limit

Morgan Rogers (he/him) (Apr 09 2020 at 09:29):

In a 2-category (I won't be especially careful about how strict I want my 2-categories to be unless someone tells me it's important) we have enough structure to talk about internal adjunctions. There is a particular form of diagram that I came across some time ago in this context, and I wonder if it has a name, or looks like something familiar to anyone: a coreflexive pair, which is to say $f,g: A \rightrightarrows B$ with $r: B \to A$ satisfying $r \circ f = r \circ g = \mathrm{id}_A$, but such that $(f \dashv r \dashv g)$.
There are plenty of ways to describe this diagram, but has anyone encountered limits over diagrams of this form before? Is there a neat way to describe the equalizer of this pair as a weighted colimit, for example?

Mike Shulman (Apr 09 2020 at 13:56):

Not an answer to the question about limits, but diagrams of this sort do come up in the theory of lax-idempotent 2-monads, where they in fact extend to the entire "simplex 2-category" (an enhancement of the simplex category to a 2-category making various face and degeneracy operators adjoint).