2019-02-09 13:50:59 8 Comments

Suppose we are given a functor

$F:(A\times B)^{\operatorname{op}}\to \operatorname{Set}$.

It's well-known that the Grothendieck construction in this case evaluates as

$\int_{A\times B}F = (A\times B)/F$.

We could also apply this construction pointwise to obtain a functor

$\int_A F:B^{op}\to \operatorname{Cat}$

sending $b\mapsto A/F(b)$

and similarly

$\int_B F:A^{op}\to \operatorname{Cat}$

We can apply the Grothendieck construction again to each of these functors to obtain categories

$\int_A\int_B F$

and

$\int_B\int_A F$

Is it the case that $\int_A \int_B F\cong \int_B \int_A F\cong \int_{A\times B} F$?

### Related Questions

#### Sponsored Content

#### 1 Answered Questions

### [SOLVED] Factorization of colimits through slices?

**2018-09-17 00:56:45****Harry Gindi****158**View**5**Score**1**Answer- Tags: ct.category-theory

#### 0 Answered Questions

### What if adjoints are just too big to give unit and counit?

**2018-06-15 17:45:30****Fosco Loregian****146**View**4**Score**0**Answer- Tags: ct.category-theory

#### 2 Answered Questions

### [SOLVED] Twisted-arrow construction for 2-categories

**2017-08-24 19:31:48****Harry Gindi****393**View**4**Score**2**Answer- Tags: ct.category-theory homotopy-theory simplicial-stuff 2-categories

#### 1 Answered Questions

### [SOLVED] Does Dyer's Thesis prove that the sheaf/fibration equivalence fails in dimension n>2?

**2017-08-06 10:53:26****Harry Gindi****336**View**11**Score**1**Answer- Tags: ct.category-theory

#### 1 Answered Questions

### [SOLVED] Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck construction?

**2017-01-01 21:01:44****Mathemologist****223**View**2**Score**1**Answer- Tags: ct.category-theory higher-category-theory grothendieck-construction stable-category

#### 2 Answered Questions

### [SOLVED] Does the functor Sch to Top have a right adjoint?

**2016-02-19 18:33:50****Minseon Shin****1050**View**17**Score**2**Answer- Tags: ag.algebraic-geometry reference-request ct.category-theory

#### 0 Answered Questions

### A generalization of the Spanier-Whitehead construction

**2015-05-19 10:44:39****Fosco Loregian****104**View**2**Score**0**Answer- Tags: ct.category-theory

#### 0 Answered Questions

### Composition of Cat-valued distributors - compatible with grothendieck construction?

**2013-04-22 19:56:35****Gerrit Begher****158**View**1**Score**0**Answer- Tags: grothendieck-construction ct.category-theory profunctors limits-and-colimits

#### 0 Answered Questions

### Fullness of Internal Yoneda "Embeddings"

**2012-08-29 00:20:08****user17165****252**View**4**Score**0**Answer- Tags: ct.category-theory

#### 0 Answered Questions

### Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

**2010-12-20 12:22:50****Harry Gindi****553**View**8**Score**0**Answer- Tags: ho.history-overview simplicial-stuff terminology at.algebraic-topology combinatorial-geometry

## 1 comments

## @David White 2019-02-09 15:19:54

Yes. See section 4.2 of this paper by Harpaz and Prasma, which also extends the result to the setting of model categories.

Another good source on these topics is Emily Riehl's book. The Fubini result is extended to the setting of $n$-categories by this thesis.

## @Harry Gindi 2019-02-09 15:27:51

Great, thanks a bunch!