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Stream: deprecated: id my structure

Topic: weak functor?

Joshua Meyers (Jan 07 2022 at 09:36):

Consider a functor $U:D\to C$. If the comma category $d/U$ has an initial object for each $d\in D$, then we can construct a left adjoint $F:C\to D$ by sending $c$ to the object $d'$, where $(d',f)$ is initial in $d/U$ . If $d/U$ merely has a weakly initial object for each $d\in D$, then we can construct some sort of "weak functor" $F:C\to D$ by sending $c$ to the family of objects $d'\in D$ such that $(d',f)$ is weakly initial in $d/U$ for some $f:d\to d'$. Anyone know a name for this sort of "weak functor"?

Joshua Meyers (Jan 07 2022 at 09:47):

This seems relevant: https://www.mscs.dal.ca/~pare/OctFest.pdf
Except it is about the case where $d/U$ has a jointly initial family, rather than at least one weakly initial object

Joshua Meyers (Jan 07 2022 at 22:06):

OK in the case where $U$ is an inclusion of a subcategory, there's a name for this: weakly reflective

There are some references for this