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The oplax limit is the category of sections for the functor from the Grothendieck construction to the base category.

The strong limit is the category of cartesian sections (every arrow in the base category gets mapped to a cartesian one).

Notice how this goes along very well with the interpretation as dependent product and as $\forall$: The set theoretic product is just the set of sections into the disjoint union.

[edit]

Given a strong functor $F:X\to\mathrm{Cat}$ we denote the Grothendieck construction by $$\mathrm{Gr}(F).$$ There is a canonical functor $\pi:\mathrm{Gr}(F)\to X$. Sections of this functor are functors $$s:X\to\mathrm{Gr}(F)$$ such that $s\circ\pi=\mathrm{id}$.

So even more explicitely a sections amounts to

forall $x\in X$ an object $s_x\in F(x)$ and ...

The oplax limit is the category of sections for the functor from the Grothendieck construction to the base category.

The strong limit is the category of cartesian sections (every arrow in the base category gets mapped to a cartesian one).

Notice how this goes along very well with the interpretation as dependent product and as $\forall$: The set theoretic product is just the set of sections into the disjoint union.

[edit]

Given a strong functor $F:X\to\mathrm{Cat}$ we denote the Grothendieck construction by $$\mathrm{Gr}(F).$$ There is a canonical functor $\pi:\mathrm{Gr}(F)\to X$. Sections of this functor are functors $$s:X\to\mathrm{Gr}(F)$$ such that $s\circ\pi=\mathrm{id}$.

So even more explicitely a section amounts to

forall $x\in X$ an object $s_x\in F(x)$ and ...

The oplax limit is the category of sections for the functor from the Grothendieck construction to the base category.

The strong limit is the category of cartesian sections (every arrow in the base category gets mapped to a cartesian one).

Notice how this goes along very well with the interpretation as dependent product and as $\forall$: The set theoretic product is just the set of sections into the disjoint union.

[edit]

Given a strong functor $F:X\to\mathrm{Cat}$ we denote the Grothendieck construction by $$\mathrm{Gr}(F).$$ There is a canonical functor $\pi:\mathrm{Gr}(F)\to X$. Sections of this functor are functors $$s:X\to\mathrm{Gr}(F)$$ such that $s\circ\pi=\mathrm{id}$.

So even more explicitely a sections amounts to

forall $x\in X$ an object $s_x\in F(x)$ ...

(and: forall $f:x\to x'$ a morphism $s_f:s_x\to s_{x'}$ over $f$, [such that $s_{\mathrm{id}_x} =\mathrm{id}_{s_x}$ and $s_{f\circ g}=s_f\circ s_g$])

The oplax limit is the category of sections for the functor from the Grothendieck construction to the base category.

The strong limit is the category of cartesian sections (every arrow in the base category gets mapped to a cartesian one).

Notice how this goes along very well with the interpretation as dependent product and as $\forall$: The set theoretic product is just the set of sections into the disjoint union.

[edit]

Given a strong functor $F:X\to\mathrm{Cat}$ we denote the Grothendieck construction by $$\mathrm{Gr}(F).$$ There is a canonical functor $\pi:\mathrm{Gr}(F)\to X$. Sections of this functor are functors $$s:X\to\mathrm{Gr}(F)$$ such that $s\circ\pi=\mathrm{id}$.

So even more explicitely a sections amounts to

forall $x\in X$ an object $s_x\in F(x)$ and ...

The oplax limit is the category of sections for the functor from the Grothendieck construction to the base category.

The strong limit is the category of cartesian sections (every arrow in the base category gets mapped to a cartesian one).

Notice how this goes along very well with the interpretation as dependent product and as $\forall$: The set theoretic product is just the set of sections into the disjoint union.

[edit]

Given a strong functor $F:X\to\mathrm{Cat}$ we denote the Grothendieck construction by $$\mathrm{Gr}(F).$$ There is a canonical functor $\pi:\mathrm{Gr}(F)\to X$. Sections of this functor are functors $$s:X\to\mathrm{Gr}(F)$$ such that $s\circ\pi=\mathrm{id}$.

So even more explicitely a sections amounts to

forall $x\in X$ an object $s_x\in F(x)$ ...

(and: forall $f:x\to x'$ a morphism $s_f:s_x\to s_{x'}$, [such that $s_{\mathrm{id}_x} =\mathrm{id}_{s_x}$ and $s_{f\circ g}=s_f\circ s_g$])

The oplax limit is the category of sections for the functor from the Grothendieck construction to the base category.

The strong limit is the category of cartesian sections (every arrow in the base category gets mapped to a cartesian one).

Notice how this goes along very well with the interpretation as dependent product and as $\forall$: The set theoretic product is just the set of sections into the disjoint union.

[edit]

Given a strong functor $F:X\to\mathrm{Cat}$ we denote the Grothendieck construction by $$\mathrm{Gr}(F).$$ There is a canonical functor $\pi:\mathrm{Gr}(F)\to X$. Sections of this functor are functors $$s:X\to\mathrm{Gr}(F)$$ such that $s\circ\pi=\mathrm{id}$.

So even more explicitely a sections amounts to

forall $x\in X$ an object $s_x\in F(x)$ ...

(and: forall $f:x\to x'$ a morphism $s_f:s_x\to s_{x'}$ over $f$, [such that $s_{\mathrm{id}_x} =\mathrm{id}_{s_x}$ and $s_{f\circ g}=s_f\circ s_g$])