nLab
permuting limits and colimits
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Contents
Context
Limits and colimits
limits and colimits
1-Categorical
limit and colimit
limits and colimits by example
commutativity of limits and colimits
small limit
filtered colimit
sifted colimit
connected limit , wide pullback
preserved limit , reflected limit , created limit
product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum
finite limit
Kan extension
weighted limit
end and coend
fibered limit
2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
In category theory one may ask when an expression of the form “lim colim lim \, colim ” can be replaced by an expression of the form “colim lim colim \, lim ”, i.e., under which circumstances limits and colimits can be “permuted”.
There are (at least) two different types of such “permutation” of limits and colimits:
Commutativity of limits and colimits
Commutativity of limits and colimits refers to the situation when we have a diagram of the form
(1) X : I × J → C
X \,\colon\, I\times J\to C
and the canonical morphism
(2) colim i ∈ I lim j ∈ J X i , j ⟶ lim j ∈ J colim i ∈ I X i , j
\underset
{i\in I}
{colim}
\;
\underset
{j\in J}
{lim}
\;
X_{i,j}
\longrightarrow
\underset
{j\in J}
{lim}
\;
\underset
{i\in I}
{colim}
\;
X_{i,j}
is an isomorphism (or an equivalence in the case of ∞-categories ).
This situation is discussed at commutativity of limits and colimits
Distributivity of limits over colimits
Distributivity of limits over colimits , in its easiest possible formulation, is stated for X X as above (1) , and requires the canonical map
(3) colim i ∈ I J lim j ∈ J X i ( j ) , j ⟶ lim j ∈ J colim i ∈ I X i , j
\underset
{i\in I^J}
{colim}
\;
\underset
{j\in J}
{lim}
\;
X_{i(j),j}
\longrightarrow
\underset
{j\in J}
{lim}
\;
\underset
{i\in I}
{colim}
X_{i,j}
to be an isomorphism (or an equivalence in the case of ∞-categories ), where now – on the left of (3) – i i is an object of the functor category I J I^J , and i ( j ) i(j) denotes the value of this functor at j ∈ J j\in J .
More generally, one may allow the indexing category I I to depend on j ∈ J j\in J , in that the functor I × J → J I\times J\to J is replaced by some arbitrary Grothendieck fibration in categories and I J I^J by the category of its sections .
This situation is discussed at distributivity of limits over colimits .
Last revised on December 18, 2022 at 18:28:13.
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