Pullback Lemma
Theorem
Let $\mathbf C$ be a metacategory.
Suppose the following is a commutative diagram in $\mathbf C$:
- $\begin{xy}\xymatrix@+0.5em{ F \ar[d]_*+{h''} \ar[r]^*+{f'} & E \ar[d]^*+{h'} \ar[r]^*+{g'} & D \ar[d]^*+{h} \\ A \ar[r]_*+{f} & B \ar[r]_*+{g} & C }\end{xy}$
Suppose furthermore that the right square is a pullback.
Then the left square is a pullback if and only if the outer rectangle is.
Proof
Necessary Condition
Suppose the left square is a pullback.
To verify the outer rectangle is a pullback, suppose we are given the following commutative diagram:
- $\begin{xy}\xymatrix@+0.5em{ P \ar@/^1em/[rrrd]^*+{p_1} \ar@/_1em/[ddr]_*+{p_2} \\ & & & D \ar[d]^*+{h} \\ & A \ar[r]_*+{f} & B \ar[r]_*+{g} & C }\end{xy}$
The right square being a pullback implies there is a unique $v: P \to E$ , making the following commute:
- $\begin{xy}\xymatrix@+0.5em{ P \ar@/^1em/[rrrd]^*+{p_1} \ar@/_1em/[ddr]_*+{p_2} \ar@{-->}@/^/[rrd]_*+{v} \\ & & E \ar[d]^*+{h'} \ar[r]^*+{g'} & D \ar[d]^*+{h} \\ & A \ar[r]_*+{f} & B \ar[r]_*+{g} & C }\end{xy}$
Since the left square is also a pullback, we subsequently find a unique $u: P \to F$ , making the following commute:
- $\begin{xy}\xymatrix@+0.5em{ P \ar@/^1em/[rrrd]^*+{p_1} \ar@/_1em/[ddr]_*+{p_2} \ar@/^/[rrd]_*+{v} \ar@{-->}[rd]_*+{u} \\ & F \ar[d]_*+{h''} \ar[r]^*+{f'} & E \ar[d]^*+{h'} \ar[r]^*+{g'} & D \ar[d]^*+{h} \\ & A \ar[r]_*+{f} & B \ar[r]_*+{g} & C }\end{xy}$
In conclusion, the outer rectangle is a pullback.
$\Box$
Sufficient Condition
Suppose the outer rectangle is a pullback.
To verify the left square is a pullback, suppose we are given the following commutative diagram:
- $\begin{xy}\xymatrix@+0.5em{ P \ar@/^/[rrd]^*+{p_1} \ar@/_1em/[ddr]_*+{p_2} \\ & & E \ar[d]^*+{h'} \\ & A \ar[r]_*+{f} & B }\end{xy}$
Evidently, we can amend this diagram as follows:
- $\begin{xy}\xymatrix@+0.5em{ P \ar@/^/[rrd]_*{p_1} \ar@/^1em/[rrrd]^*+{g' \circ p_1} \ar@/_1em/[ddr]_*+{p_2} \\ & & E \ar[d]^*+{h'} \ar[r]^*+{g'} & D \ar[d]^*+{h} \\ & A \ar[r]_*+{f} & B \ar[r]_*+{g} & C }\end{xy}$
Because the outer rectangle is a pullback, we thus obtain a unique $u: P \to F$, making the following commute:
- $\begin{xy}\xymatrix@+0.5em{ P \ar@/^1em/[rrrd]^*+{g' \circ p_1} \ar@/_1em/[ddr]_*+{p_2} \ar@{-->}[rd]_*+{u} \\ & F \ar[d]_*+{h''} \ar[r]^*+{f'} & E \ar[d]^*+{h'} \ar[r]^*+{g'} & D \ar[d]^*+{h} \\ & A \ar[r]_*+{f} & B \ar[r]_*+{g} & C }\end{xy}$
If we now can prove that $f' \circ u = p_1$, then we're done.
To this end, remark that they both complete the following commutative diagram:
- $\begin{xy}\xymatrix@+0.5em{ P \ar@{-->}[rrd] \ar@/^1em/[rrrd]^*+{g' \circ p_1} \ar@/_1em/[ddr]_*+{p_2} \\ & & E \ar[d]^*+{h'} \ar[r]^*+{g'} & D \ar[d]^*+{h} \\ & A \ar[r]_*+{f} & B \ar[r]_*+{g} & C }\end{xy}$
Since the right square is a pullback, there is only one such morphism.
Therefore, we conclude $f' \circ u = p_1$.
Hence the left square is also a pullback.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.3$: Lemma $5.8$