## 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

1. 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

2. 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

1. for all $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$ there exist an $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a morphism $H(x) \to y$,

2. 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

1. the fibre categories (see Definition 4.32.2) of $\mathcal{I}$ over $\mathcal{I}'$ are all connected, and

2. 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. $\square$

Comment #7095 by Elías Guisado on

I think there is some imprecision in the definition of a (co)final functor. I guess it would be nice if 4.17.1 clarified that in the zig-zag of morphisms it does not actually matter whether the first or last arrow go to the right or to the left. If not, I think the definition would be wrong. Let me explain: I'm going to spell out the definition for a cofinal functor from Kashiwara-Schapira, https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf (in Proposition 2.5.2 of this text they prove 4.17.2). To explain it, some previous definitions must be introduced:

According to the definition on p. 13 of the linked text, a category $\mathcal{C}$ is connected if it is non-empty and for every pair of objects $x,y\in\mathcal{C}$, there is a sequence of objects $x=x_1,x_2,\dots,x_n=y$ in $\mathcal{C}$ such that at least one of $\operatorname{Mor}(x_i,x_{i+1})$ or $\operatorname{Mor}(x_{i+1},x_{i})$ is non-empty for all $1\leq i< n$.

As it is written on Definition 1.2.16 of this same text, given a functor $F:\mathcal{C}\to\mathcal{C}'$ and an object $A\in\mathcal{C}$, we denote the category $\mathcal{C}^A$ where: objects are pairs $(X,f)$, where $X\in\mathcal{C}$ is an object in $\mathcal{C}$ and $f:A\to F(X)$ is a morphism in $\mathcal{C}'$, and a morphism $(X,f)\to(Y,g)$ is a morphism $h:X\to Y$ in $\mathcal{C}$ such that $g=F(h)\circ f$.

Now, according to Definition 2.5.1 of same text, a functor $F:\mathcal{I}\to \mathcal{J}$ is said to be cofinal if the category $\mathcal{I}^j$ is connected for every object $j\in \mathcal{J}$.

I think the definition of Kashiwara-Schapira for a cofinal functor is different from 4.17.1. Indeed: let $\mathcal{J}$ be the free category generated by the quiver modded out by the relations $g_i\circ f_i=f$, for $i=1,2$, and let $\mathcal{I}$ be the full subcategory $b\xrightarrow{g_1} c\xleftarrow{g_2} d$. Then the inclusion $\mathcal{I}\to\mathcal{J}$ is cofinal according to definition 2.5.1 of Kashiwara-Schapira but is is not according to Definition 4.17.1, if we were to interpet literally that the only allowable zig-zag of morphisms in 4.17.1 must be of the form $(1)$, with the first arrow to the left and the last one to the right.

Comment #7096 by Elías Guisado on

And the analogous correction should be made in 4.17.3: on same definition of Kashiwara-Schapira, Definition 2.5.1, this concept is given as "A functor $F:\mathcal{I}\to\mathcal{J}$ is co-cofinal if $F^\mathrm{op}:\mathcal{I}^\mathrm{op}\to\mathcal{J}^\mathrm{op}$ is cofinal". So again, in the zig-zag $(1)$, it should not matter the orientation of the first and last morphisms.

Comment #7098 by on

@#7095 and #7096. Everything is fine, see the discussion on Definition 4.17.1.

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