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Definitions

An object $d \in Obj(\mathcal D)$ is in the essential image of $F$ if there exists some $c \in Obj(\mathcal C)$ such that $d \cong F c$.

A sieve in $\mathcal D$ is a full subcategory of $\mathcal D$ such that if $y$ is in the sieve and there is a morphism $\varphi : x \to y$, then $x$ is in the sieve. Since sieves are required to be full, they can be regarded as subsets of $Obj(\mathcal{D})$.

A cosieve is the dual: if $x$ is in the cosieve and there is a morphism $\varphi : x \to y$, then $y$ is in the cosieve.

Question

Is there a word for a functor $F : \mathcal C \to \mathcal D$ satisfying the following three equivalent conditions:

  • the objects in the essential image of $F$ constitute a cosieve in $\mathcal D$,

  • the objects outside the essential image of $F$ constitute a sieve in $\mathcal D$,

  • if there is a morphism $Fc \to d$, then $d$ is in the essential image of $F$.

Of course a word for the dual is also fine.

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    $\begingroup$ Can you perhaps explain the meaning of "constitute a cosieve" here? Thanks! $\endgroup$ Commented Feb 7, 2023 at 21:36

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  • A discrete opfibration $F : \mathcal C \to \mathcal D$ has the property that every morphism $Fc \to d'$ is the image of a unique morphism $c \to c'$ on the nose (we need $Fc' = d'$).

  • A Grothendieck opfibration $F : \mathcal C \to \mathcal D$ has the property that every morphism $Fc \to d'$ is the image of a universal ("cartesian") morphism $c \to c'$ on the nose (we need $Fc' = d'$).

    Thus, every discrete opfibration is a Grothendieck opfibration.

    If the opfibration is cloven, meaning that a choice of cartesian morphisms has been made, then for every morphism $\varphi : d \to d'$ in $\mathcal D$, we get a functor $E^{-1}(\varphi) : E^{-1}(d) \to E^{-1}(d')$. Universality of the lifting of the arrow is what makes $E^{-1}$ a pseudofunctor.

  • A Street/weak opfibration $F : \mathcal C \to \mathcal D$ has the property that every morphism $Fc \to d$ is the image of a universal morphism $c \to c'$ up to isomorphism (we need $Fc' \cong d'$).

    Thus, every Grothendieck opfibration is a Street opfibration.

  • A functor $F : \mathcal C \to \mathcal D$ as described in the question, has the property that every morphism $Fc \to d$ is the image of some morphism $c \to c'$ up to isomorphism (we need $Fc' \cong d'$). Lacking universality, even if this were cloven (in the sense that a choice of liftings of morphisms to $\mathcal C$ has been made), we cannot expect $E^{-1}$ to satisfy the functor laws up to isomorphism.

    Edit: The question did not require the morphism to be in the image of $F$. Of course this will be automatic if $F$ is full.

So I guess a reasonable name (when morphisms are also lifted) would be weak ad hoc opfibration:

  • Weak because it lifts arrows up to isomorphism,
  • Ad hoc because we can choose liftings for arrows in an ad hoc manner, not caring about the bigger picture, since we need not satisfy pseudofunctoriality.
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    $\begingroup$ Since the question did not require the morphism to be in the image of $F$, we could go for weak objectwise opfibration, but does this still warrant using fibration terminology? $\endgroup$ Commented Feb 9, 2023 at 7:37

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