4.17 Cofinal and initial categories
In the literature sometimes the word “final” is used instead of cofinal in the following definition.
Definition 4.17.1. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say $\mathcal{I}$ is cofinal in $\mathcal{J}$ or that $H$ is cofinal if
for all $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$ there exist an $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a morphism $y \to H(x)$, and
given $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$, $x, x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and morphisms $y \to H(x)$ and $y \to H(x')$ there exist a sequence of morphisms
\[ x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x' \]
in $\mathcal{I}$ and morphisms $y \to H(x_ i)$ in $\mathcal{J}$ such that the diagrams
\[ \xymatrix{ & y \ar[ld] \ar[d] \ar[rd] \\ H(x_{2k}) & H(x_{2k + 1}) \ar[l] \ar[r] & H(x_{2k + 2}) } \]
commute for $k = 0, \ldots , n - 1$.
Lemma 4.17.2. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is cofinal in $\mathcal{J}$. Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism
\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \circ H = \mathop{\mathrm{colim}}\nolimits _\mathcal {J} M \]
if either side exists.
Proof.
Omitted.
$\square$
Definition 4.17.3. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say $\mathcal{I}$ is initial in $\mathcal{J}$ or that $H$ is initial if
for all $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$ there exist an $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a morphism $H(x) \to y$,
for any $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$, $x , x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and morphisms $H(x) \to y$, $H(x') \to y$ in $\mathcal{J}$ there exist a sequence of morphisms
\[ x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x' \]
in $\mathcal{I}$ and morphisms $H(x_ i) \to y$ in $\mathcal{J}$ such that the diagrams
\[ \xymatrix{ H(x_{2k}) \ar[rd] & H(x_{2k + 1}) \ar[l] \ar[r] \ar[d] & H(x_{2k + 2}) \ar[ld] \\ & y } \]
commute for $k = 0, \ldots , n - 1$.
This is just the dual notion to “cofinal” functors.
Lemma 4.17.4. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is initial in $\mathcal{J}$. Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism
\[ \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \circ H = \mathop{\mathrm{lim}}\nolimits _\mathcal {J} M \]
if either side exists.
Proof.
Omitted.
$\square$
Lemma 4.17.5. Let $F : \mathcal{I} \to \mathcal{I}'$ be a functor. Assume
the fibre categories (see Definition 4.32.2) of $\mathcal{I}$ over $\mathcal{I}'$ are all connected, and
for every morphism $\alpha ' : x' \to y'$ in $\mathcal{I}'$ there exists a morphism $\alpha : x \to y$ in $\mathcal{I}$ such that $F(\alpha ) = \alpha '$.
Then for every diagram $M : \mathcal{I}' \to \mathcal{C}$ the colimit $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \circ F$ exists if and only if $\mathop{\mathrm{colim}}\nolimits _{\mathcal{I}'} M$ exists and if so these colimits agree.
Proof.
One can prove this by showing that $\mathcal{I}$ is cofinal in $\mathcal{I}'$ and applying Lemma 4.17.2. But we can also prove it directly as follows. It suffices to show that for any object $T$ of $\mathcal{C}$ we have
\[ \mathop{\mathrm{lim}}\nolimits _{\mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_{F(i)}, T) = \mathop{\mathrm{lim}}\nolimits _{(\mathcal{I}')^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_{i'}, T) \]
If $(g_{i'})_{i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')}$ is an element of the right hand side, then setting $f_ i = g_{F(i)}$ we obtain an element $(f_ i)_{i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})}$ of the left hand side. Conversely, let $(f_ i)_{i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})}$ be an element of the left hand side. Note that on each (connected) fibre category $\mathcal{I}_{i'}$ the functor $M \circ F$ is constant with value $M_{i'}$. Hence the morphisms $f_ i$ for $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ with $F(i) = i'$ are all the same and determine a well defined morphism $g_{i'} : M_{i'} \to T$. By assumption (2) the collection $(g_{i'})_{i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')}$ defines an element of the right hand side.
$\square$
Lemma 4.17.6. Let $\mathcal{I}$ and $\mathcal{J}$ be a categories and denote $p : \mathcal{I} \times \mathcal{J} \to \mathcal{J}$ the projection. If $\mathcal{I}$ is connected, then for a diagram $M : \mathcal{J} \to \mathcal{C}$ the colimit $\mathop{\mathrm{colim}}\nolimits _\mathcal {J} M$ exists if and only if $\mathop{\mathrm{colim}}\nolimits _{\mathcal{I} \times \mathcal{J}} M \circ p$ exists and if so these colimits are equal.
Proof.
This is a special case of Lemma 4.17.5.
$\square$
Comments (3)
Comment #7095 by Elías Guisado on
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Comment #7098 by Johan on