14.22 Simplicial objects in abelian categories
Recall that an abelian category is defined in Homology, Section 12.5.
Lemma 14.22.1. Let $\mathcal{A}$ be an abelian category.
The categories $\text{Simp}(\mathcal{A})$ and $\text{CoSimp}(\mathcal{A})$ are abelian.
A morphism of (co)simplicial objects $f : A \to B$ is injective if and only if each $f_ n : A_ n \to B_ n$ is injective.
A morphism of (co)simplicial objects $f : A \to B$ is surjective if and only if each $f_ n : A_ n \to B_ n$ is surjective.
A sequence of (co)simplicial objects
\[ A \xrightarrow {f} B \xrightarrow {g} C \]
is exact at $B$ if and only if each sequence
\[ A_ i \xrightarrow {f_ i} B_ i \xrightarrow {g_ i} C_ i \]
is exact at $B_ i$.
Proof.
Pre-additivity is easy. A final object is given by $U_ n = 0$ in all degrees. Existence of direct products we saw in Lemmas 14.6.2 and 14.9.2. Kernels and cokernels are obtained by taking termwise kernels and cokernels.
$\square$
For an object $A$ of $\mathcal{A}$ and an integer $k$ consider the $k$-truncated simplicial object $U$ with
$U_ i = 0$ for $i < k$,
$U_ k = A$,
all morphisms $U(\varphi )$ equal to zero, except $U(\text{id}_{[k]}) = \text{id}_ A$.
Since $\mathcal{A}$ has both finite limits and finite colimits we see that both $\text{cosk}_ k U$ and $i_{k!}U$ exist. We will describe both of these and the canonical map $i_{k!}U \to \text{cosk}_ kU$.
Lemma 14.22.2. With $A$, $k$ and $U$ as above, so $U_ i = 0$, $i < k$ and $U_ k = A$.
Given a $k$-truncated simplicial object $V$ we have
\[ \mathop{\mathrm{Mor}}\nolimits (U, V) = \{ f : A \to V_ k \mid d^ k_ i \circ f = 0, \ i = 0, \ldots , k \} \]
and
\[ \mathop{\mathrm{Mor}}\nolimits (V, U) = \{ f : V_ k \to A \mid f \circ s^{k - 1}_ i = 0, \ i = 0, \ldots , k - 1 \} . \]
The object $i_{k!} U$ has $n$th term equal to $\bigoplus _\alpha A$ where $\alpha $ runs over all surjective morphisms $\alpha : [n] \to [k]$.
For any $\varphi : [m] \to [n]$ the map $i_{k!} U(\varphi )$ is described as the mapping $\bigoplus _\alpha A \to \bigoplus _{\alpha '} A$ which maps to component corresponding to $\alpha : [n] \to [k]$ to zero if $\alpha \circ \varphi $ is not surjective and by the identity to the component corresponding to $\alpha \circ \varphi $ if it is surjective.
The object $\text{cosk}_ k U$ has $n$th term equal to $\bigoplus _\beta A$, where $\beta $ runs over all injective morphisms $\beta : [k] \to [n]$.
For any $\varphi : [m] \to [n]$ the map $\text{cosk}_ k U(\varphi )$ is described as the mapping $\bigoplus _\beta A \to \bigoplus _{\beta '} A$ which maps to component corresponding to $\beta : [k] \to [n]$ to zero if $\beta $ does not factor through $\varphi $ and by the identity to each of the components corresponding to $\beta '$ such that $\beta = \varphi \circ \beta '$ if it does.
The canonical map $ c : i_{k !} U \to \text{cosk}_ k U $ in degree $n$ has $(\alpha , \beta )$ coefficient $A \to A$ equal to zero if $\alpha \circ \beta $ is not the identity and equal to $\text{id}_ A$ if it is.
The canonical map $ c : i_{k !} U \to \text{cosk}_ k U $ is injective.
Proof.
The proof of (1) is left to the reader.
Let us take the rules of (2) and (3) as the definition of a simplicial object, call it $\tilde U$. We will show that it is an incarnation of $i_{k!}U$. This will prove (2), (3) at the same time. We have to show that given a morphism $f : U \to \text{sk}_ kV$ there exists a unique morphism $\tilde f : \tilde U \to V$ which recovers $f$ upon taking the $k$-skeleton. From (1) we see that $f$ corresponds with a morphism $f_ k : A \to V_ k$ which maps into the kernel of $d^ k_ i$ for all $i$. For any surjective $\alpha : [n] \to [k]$ we set $\tilde f_\alpha : A \to V_ n$ equal to the composition $\tilde f_\alpha = V(\alpha ) \circ f_ k : A \to V_ n$. We define $\tilde f_ n : \tilde U_ n \to V_ n$ as the sum of the $\tilde f_\alpha $ over $\alpha : [n] \to [k]$ surjective. Such a collection of $\tilde f_\alpha $ defines a morphism of simplicial objects if and only if for any $\varphi : [m] \to [n]$ the diagram
\[ \xymatrix{ \bigoplus _{\alpha : [n] \to [k]\text{ surjective}} A \ar[r]_-{\tilde f_ n} \ar[d]_{(3)} & V_ n \ar[d]^{V(\varphi )} \\ \bigoplus _{\alpha ' : [m] \to [k]\text{ surjective}} A \ar[r]^-{\tilde f_ m} & V_ m } \]
is commutative. Choosing $\varphi = \alpha $ shows our choice of $\tilde f_\alpha $ is uniquely determined by $f_ k$. The commutativity in general may be checked for each summand of the left upper corner separately. It is clear for the summands corresponding to $\alpha $ where $\alpha \circ \varphi $ is surjective, because those get mapped by $\text{id}_ A$ to the summand with $\alpha ' = \alpha \circ \varphi $, and we have $\tilde f_{\alpha '} = V(\alpha ') \circ f_ k = V(\alpha \circ \varphi ) \circ f_ k = V(\varphi ) \circ \tilde f_\alpha $. For those where $\alpha \circ \varphi $ is not surjective, we have to show that $V(\varphi ) \circ \tilde f_\alpha = 0$. By definition this is equal to $V(\varphi ) \circ V(\alpha ) \circ f_ k = V(\alpha \circ \varphi ) \circ f_ k$. Since $\alpha \circ \varphi $ is not surjective we can write it as $\delta ^ k_ i \circ \psi $, and we deduce that $V(\varphi ) \circ V(\alpha ) \circ f_ k = V(\psi ) \circ d^ k_ i \circ f_ k = 0$ see above.
Let us take the rules of (4) and (5) as the definition of a simplicial object, call it $\tilde U$. We will show that it is an incarnation of $\text{cosk}_ k U$. This will prove (4), (5) at the same time. The argument is completely dual to the proof of (2), (3) above, but we give it anyway. We have to show that given a morphism $f : \text{sk}_ kV \to U$ there exists a unique morphism $\tilde f : V \to \tilde U$ which recovers $f$ upon taking the $k$-skeleton. From (1) we see that $f$ corresponds with a morphism $f_ k : V_ k \to A$ which is zero on the image of $s^{k - 1}_ i$ for all $i$. For any injective $\beta : [k] \to [n]$ we set $\tilde f_\beta : V_ n \to A$ equal to the composition $\tilde f_\beta = f_ k \circ V(\beta ) : V_ n \to A$. We define $\tilde f_ n : V_ n \to \tilde U_ n$ as the sum of the $\tilde f_\beta $ over $\beta : [k] \to [n]$ injective. Such a collection of $\tilde f_\beta $ defines a morphism of simplicial objects if and only if for any $\varphi : [m] \to [n]$ the diagram
\[ \xymatrix{ V_ n \ar[d]_{V(\varphi )} \ar[r]_-{\tilde f_ n} & \bigoplus _{\beta : [k] \to [n]\text{ injective}} A \ar[d]^{(5)} \\ V_ m \ar[r]^-{\tilde f_ m} & \bigoplus _{\beta ' : [k] \to [m]\text{ injective}} A } \]
is commutative. Choosing $\varphi = \beta $ shows our choice of $\tilde f_\beta $ is uniquely determined by $f_ k$. The commutativity in general may be checked for each summand of the right lower corner separately. It is clear for the summands corresponding to $\beta '$ where $\varphi \circ \beta '$ is injective, because these summands get mapped into by exactly the summand with $\beta = \varphi \circ \beta '$ and we have in that case $\tilde f_{\beta '} \circ V(\varphi ) = f_ k \circ V(\beta ') \circ V(\varphi ) = f_ k \circ V(\beta ) = \tilde f_\beta $. For those where $\varphi \circ \beta '$ is not injective, we have to show that $\tilde f_{\beta '} \circ V(\varphi ) = 0$. By definition this is equal to $f_ k \circ V(\beta ') \circ V(\varphi ) = f_ k \circ V(\varphi \circ \beta ')$. Since $\varphi \circ \beta '$ is not injective we can write it as $\psi \circ \sigma ^{k - 1}_ i$, and we deduce that $f_ k \circ V(\beta ') \circ V(\varphi ) = f_ k \circ s^{k - 1}_ i \circ V(\psi ) = 0$ see above.
The composition $i_{k!}U \to \text{cosk}_ kU$ is the unique map of simplicial objects which is the identity on $A = U_ k = (i_{k!}U)_ k = (\text{cosk}_ kU)_ k$. Hence it suffices to check that the proposed rule defines a morphism of simplicial objects. To see this we have to show that for any $\varphi : [m] \to [n]$ the diagram
\[ \xymatrix{ \bigoplus _{\alpha : [n] \to [k]\text{ surjective}} A \ar[d]_{(3)} \ar[r]_{(6)} & \bigoplus _{\beta : [k] \to [n]\text{ injective}} A \ar[d]^{(5)} \\ \bigoplus _{\alpha ' : [m] \to [k]\text{ surjective}} A \ar[r]^{(6)} & \bigoplus _{\beta ' : [k] \to [m]\text{ injective}} A } \]
is commutative. Now we can think of this in terms of matrices filled with only $0$'s and $1$'s as follows: The matrix of (3) has a nonzero $(\alpha ', \alpha )$ entry if and only if $\alpha ' = \alpha \circ \varphi $. Likewise the matrix of (5) has a nonzero $(\beta ', \beta )$ entry if and only if $\beta = \varphi \circ \beta '$. The upper matrix of (6) has a nonzero $(\alpha , \beta )$ entry if and only if $\alpha \circ \beta = \text{id}_{[k]}$. Similarly for the lower matrix of (6). The commutativity of the diagram then comes down to computing the $(\alpha , \beta ')$ entry for both compositions and seeing they are equal. This comes down to the following equality
\[ \# \left\{ \beta \mid \beta = \varphi \circ \beta ' \text{ and } \alpha \circ \beta = \text{id}_{[k]} \right\} = \# \left\{ \alpha ' \mid \alpha ' = \alpha \circ \varphi \text{ and } \alpha ' \circ \beta ' = \text{id}_{[k]} \right\} \]
whose proof may safely be left to the reader.
Finally, we prove (7). This follows directly from Lemmas 14.18.7, 14.19.4, 14.21.3 and 14.21.9.
$\square$
Definition 14.22.3. Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer $\geq 0$. The Eilenberg-Maclane object $K(A, k)$ is given by the object $K(A, k) = i_{k!}U$ which is described in Lemma 14.22.2 above.
Lemma 14.22.4. Let $\mathcal{A}$ be an abelian category. Let $A$ be an object of $\mathcal{A}$ and let $k$ be an integer $\geq 0$. Consider the simplicial object $E$ defined by the following rules
$E_ n = \bigoplus _\alpha A$, where the sum is over $\alpha : [n] \to [k + 1]$ whose image is either $[k]$ or $[k + 1]$.
Given $\varphi : [m] \to [n]$ the map $E_ n \to E_ m$ maps the summand corresponding to $\alpha $ via $\text{id}_ A$ to the summand corresponding to $\alpha \circ \varphi $, provided $\mathop{\mathrm{Im}}(\alpha \circ \varphi )$ is equal to $[k]$ or $[k + 1]$.
Then there exists a short exact sequence
\[ 0 \to K(A, k) \to E \to K(A, k + 1) \to 0 \]
which is term by term split exact.
Proof.
The maps $K(A, k)_ n \to E_ n$ resp. $E_ n \to K(A, k + 1)_ n$ are given by the inclusion of direct sums, resp. projection of direct sums which is obvious from the inclusions of index sets. It is clear that these are maps of simplicial objects.
$\square$
Lemma 14.22.5. Let $\mathcal{A}$ be an abelian category. For any simplicial object $V$ of $\mathcal{A}$ we have
\[ V = \mathop{\mathrm{colim}}\nolimits _ n i_{n!}\text{sk}_ n V \]
where all the transition maps are injections.
Proof.
This is true simply because each $V_ m$ is equal to $(i_{n!}\text{sk}_ n V)_ m$ as soon as $n \geq m$. See also Lemma 14.21.10 for the transition maps.
$\square$
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