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Stream: learning: questions

Topic: Grothendieck construction variants of poset-valued presheaf

Naso (May 04 2023 at 02:45):

Let $F : \mathcal{T}^{op} \to \mathcal{Pos}$ be a poset-valued presheaf on a topological space.

There are four related ways we could put a poset structure on $\coprod_{A \in \mathcal{T}} FA$.

$(A, a) \preceq (B, b) \iff A \subseteq B, \quad a \leq_{FA} {b \mid_A}$
or
$(A, a) \preceq (B, b) \iff B \subseteq A, \quad {a \mid_B} \leq_{FB} b$
or
$(A, a) \preceq (B, b) \iff A \subseteq B, \quad {b \mid_A} \leq_{FA} a$
or
$(A, a) \preceq (B, b) \iff B \subseteq A, \quad b \leq_{FB} {a \mid_B}$

according to nlab , the first one is the Grothendieck construction of $F$. Where do the other ones come from? I am getting confused about where opposites are coming in. I think (iii) is the Grothendieck construction of F composed with the contravariant functor $(-)^{op} : \mathcal{Pos} \to \mathcal{Pos}$, so in this case the covariant Grothendieck construction applies . What about (ii) and (iv)?

I have a practical application in mind for which the 'right' order seems to be (ii), and I'm trying to motivate and also understand myself the construction better.

Mike Shulman (May 04 2023 at 18:52):

I think (i) is the ordinary contravariant GC, (ii) is the ordinary covariant GC, and (iii) and (iv) are the two GCs of $(-)^{\rm op} \circ F$ as you suggest.

Mike Shulman (May 04 2023 at 18:53):

(Of course nothing here depends on $\mathcal{T}$ being a topology, it could be any poset.)

Naso (May 04 2023 at 23:53):

Mike Shulman said:

I think (i) is the ordinary contravariant GC, (ii) is the ordinary covariant GC, and (iii) and (iv) are the two GCs of $(-)^{\rm op} \circ F$ as you suggest.

Thanks Mike. I see now that we can apply the contravariant GC to $F$ or the covariant GC to $F$ and we get different results. That explains it!