2.1. Definable cohomology for simplicial complexes

A simplicial complex K is a family of finite sets that is closed downwards (i.e., $\sigma \subseteq \tau \in K\implies \sigma \in K$ ). A simplex or face of K is any element $\sigma \in K$ . A vertex of K is any element v of $\mathrm {dom}(K):=\bigcup K$ . Let $K,L$ be two simplicial complexes. A simplicial map $f\colon K\rightarrow L$ is then any function $ f\colon \mathrm {dom}(K)\rightarrow \mathrm {dom}(L)$ such that $ \{f(v_{0}),\ldots ,f(v_{n})\}\in L$ for all $\{v_{0},\ldots ,v_{n}\}\in K$ . The dimension $\mathrm {dim}(\sigma )$ of a face $\sigma $ of K is simply the number $|\sigma |-1$ . For example, $\mathrm {dim}(\emptyset )=(-1)$ and $\mathrm {dim}(\{v\})=0$ for every $v\in \mathrm {dom}(K)$ . The dimension $\mathrm {dim}(K)$ of $ K$ is the supremum over $\{\mathrm {dim}(\sigma )\mid \sigma \in K\}$ . A simplicial complex is finite if it has finitely many vertices, and countable if it has countably many vertices. It is locally finite if each vertex belongs to finitely many faces. For each $n\in \mathbb {N}$ , the singular n-faces Footnote ¹ of K comprise the set

$$\begin{align*}K^{(n)}:=\{(v_0,\ldots,v_{n})\in \mathrm{dom}(K)^{n+1}\colon \{v_0,\ldots,v_{n}\}\in K \}.\end{align*}$$

We fix an abelian Polish group G and consider, for every $n\in \mathbb {N}$ , the collection

$$\begin{align*}C^{n}(K;G):= C(K^{(n)},G)\end{align*}$$

of all maps from the countable set $K^{(n)}$ to G. Endowed with the group operation $(\zeta ,\eta )\mapsto (\zeta +\eta )$ of pointwise addition $(\zeta +\eta )(\bar {v})=\zeta (\bar {v})+\eta (\bar {v})$ , and the product topology of countably many copies of G, the collection $C^{n}(K;G)$ forms the abelian Polish group of all G-valued cochains of K. For every $n\geq 0$ , we have the coboundary map

(3) $$ \begin{align} \begin{aligned} \delta^{n}:C^{n}(K;G) &\rightarrow C^{n+1}(K;G), \text{ with}\\ \left( \delta^{n}( \zeta) \right) \big((v_{0},\ldots ,v_{n+1})\big) &=\sum_{i=0}^{n}\left( -1\right) ^{i}\zeta (v_{0},\ldots ,\hat{v}_{i},\ldots ,v_{n+1}), \end{aligned} \end{align} $$

where $\hat {v}_{i}$ denotes the omission of $v_i$ from $\bar {v}$ . Notice that $\delta ^{n}$ is a continuous homomorphism. This gives rise to the Polish cochain complex $C^{\bullet }(K;G)$ of G-valued cochains of K:

$$\begin{align*}0\longrightarrow C^{0}(K;G)\overset{\delta^0} \longrightarrow C^{1}(K;G)\longrightarrow\cdots\longrightarrow C^{n}(K;G) \overset{\delta^{n+1}}\longrightarrow C^{n}(K;G) \longrightarrow \cdots\end{align*}$$

Definition 2.1. Let K be a countable simplicial complex, and let G be an abelian Polish group. For every $n\in \mathbb {N}$ , the n-dimensional definable cohomology group $\mathrm {H}^{n}_{\mathrm {def}}(K;G)$ of K with coefficients in G is the n-dimensional cohomology group $\mathrm {H}^{n}(K;G)$ of the Polish cochain complex $C^{\bullet }(K;G)$ , viewed as the group with a Polish cover

$$\begin{align*}0\longrightarrow \mathrm{B}^{n}(K;G)\longrightarrow \mathrm{Z}^{n}(K;G) \longrightarrow \mathrm{Z}^{n}(K;G)/ \mathrm{B}^{n}(K;G)\longrightarrow 0\end{align*}$$

where $\mathrm {Z}^{n}(K;G)= \mathrm {ker}(\delta ^{n})$ is the Polish group of n-dimensional G-valued cocycles of K and $\mathrm {B}^{n}(K;G)=\mathrm {im}(\delta ^{n-1})$ is the Polishable group of n-dimensional G-valued coboundaries of K.

Of course, if we forget about the quotient Borel structure on $\mathrm {H}^{n}_{\mathrm {def}}(K;G)$ and instead treat it as an abstract group, then we recover the classical n-dimensional simplicial cohomology group of K with coefficients in G.

Remark 2.2. Several alternative definitions of the simplicial cohomology groups $\mathrm {H}^{n}(K;G)$ may be found in the literature. Each involves variants of the cochain complex $C^{\bullet }(K;G)$ described above. For example, one may work with the chain complex $C^{\bullet }_{\mathrm {alt}}(K;G)$ of alternating cochains instead. An alternating n-cochain is any n-cochain $\zeta \in C^{\bullet }(K;G)$ with the property that

$$\begin{align*}\zeta(v_0,\ldots,v_n)=\mathrm{sgn}(\pi)\zeta(v_{\pi(0)},\ldots,v_{\pi(n)}), \end{align*}$$

for any permutation $\pi $ of the set $\{0,\ldots ,n\}$ . The inclusion $C^{\bullet }_{\mathrm {alt}}(K;G)\hookrightarrow C^{\bullet }(K;G)$ yields an isomorphism between $\mathrm {H}^{n}(K;G)$ and $\mathrm {H}^{n}_{\mathrm {alt}}(K;G)$ , as is well known (see, for example, [Reference Serre78]). It is clear that $C^{\bullet }_{\mathrm {alt}}(K;G)$ inherits a Polish structure from $C^{\bullet }(K;G)$ and that $C^{\bullet }_{\mathrm {alt}}(K;G)\hookrightarrow C^{\bullet }(K;G)$ is continuous. Hence, $\mathrm {H}^{n}(K;G)$ and $\mathrm {H}^{n}_{\mathrm {alt}}(K;G)$ are definably isomorphic as groups with a Polish cover. As this isomorphism commutes with direct limits, similar remarks will apply to the Čech cohomology groups of locally compact Polish spaces which we define below.

2.2. An indexing poset

Below, we identify n with the set $\{0,\ldots ,n-1\}$ . Let $\mathbb {N}^n$ be the space of all functions from n to $\mathbb {N}$ , and set $\mathbb {N}^{<\mathbb {N}}=\bigcup _{n\in \mathbb {N}}\mathbb {N}^n$ . The Baire space $\mathcal {N}:=\mathbb {N}^{\mathbb {N}}$ is the space of all functions from $\mathbb {N}$ to $\mathbb {N}$ , which we view as a Polish space equipped with the product topology on discrete copies of $\mathbb {N}$ . For every $\alpha \in \mathcal {N}$ and all $n\in \mathbb {N}$ , we let $\alpha |n\in \mathbb {N}^n$ be the finite sequence $(\alpha (0),\ldots ,\alpha (n-1))$ . For every $s=(s(0),\ldots ,s(n-1))\in \mathbb {N}^{<\mathbb {N}}$ , we denote by $\mathcal {N}_s$ the clopen subset $\{\alpha \in \mathcal {N}\colon \alpha |n=s \}$ of $\mathcal {N}$ . The collection $\{\mathcal {N}_s\colon s\in \mathbb {N}^{<\mathbb {N}}\}$ of all such sets forms a basis for the topology on $\mathcal {N}$ . Let $(X_s\colon s\in \mathbb {N}^{<\mathbb {N}})$ be a family of sets parametrized by $\mathbb {N}^{<\mathbb {N}}$ . To each such family we may apply Suslin’s $\mathcal {A}$ -operation, producing the set

$$\begin{align*}\mathcal{A}(X_{s}):=\bigcup_{\alpha\in \mathcal{N}}\bigcap_{n\in \mathbb{N}}X_{\alpha|n}. \end{align*}$$

Notice that while the family $(X_s)$ is countable, the operation $\mathcal {A}$ involves an uncountable union. Historically, this operation was used to explicitly define subsets of the real line which are analytic but not Borel. Indeed, every Borel subset of a Polish space X is the result of the $\mathcal {A}$ -operation applied to some system $(X_s)$ of closed subsets of X – but not all sets derived from the $\mathcal {A}$ -operation applied to a system of closed subsets is Borel; see [Reference Kechris44, Theorem 25.7].

Here, we will be interested in the closed subset $\mathcal {N}^{*}$ of $\mathcal {N}$ consisting of all non-decreasing such functions (i.e., all $\alpha =(\alpha (0),\alpha (1),\ldots )\in \mathcal {N}$ such that $\alpha (n)\leq \alpha (n+1)$ for all $n\in \mathbb {N}$ ). It is easy to see that $\mathcal {N}^{*}$ and $\mathcal {N}$ are homeomorphic. Similarly, we denote by $(\mathbb {N}^n)^{*}$ and $(\mathbb {N}^{<\mathbb {N}})^{*}$ the sets of all non-decreasing functions in $\mathbb {N}^n$ and $\mathbb {N}^{<\mathbb {N}}$ , respectively. We set $\mathcal {N}^{*}_s:=\mathcal {N}_s\cap \mathcal {N}^{*}$ for all $s\in (\mathbb {N}^{<\mathbb {N}})^{*}$ . We endow $\mathcal {N}^{*}$ with the pointwise partial ordering $\leq $ and the associated meet operation $(\alpha ,\beta )\mapsto \alpha \wedge \beta $ and join operation $(\alpha ,\beta )\mapsto \alpha \vee \beta $ , where

$$\begin{align*}\alpha\leq \beta \iff \forall n\in \mathbb{N} \; (\alpha(n)\leq \beta(n)),\end{align*}$$

$$\begin{align*}(\alpha\wedge \beta)(n):= \min\{\alpha(n),\beta(n)\} \text{ and } (\alpha\vee \beta)(n):= \max\{\alpha(n),\beta(n)\}.\end{align*}$$

Occasionally, we will also employ the lexicographic linear ordering $\leq _{\mathrm {lex}}$ of $\mathcal {N}^{*}$ with

$$\begin{align*}\alpha\leq_{\mathrm{lex}}\beta \iff \big( \alpha=\beta \text{ or } \exists n\in \mathbb{N} \; (\alpha|n=\beta|n \text{ and } \alpha(n)<\beta(n)\big). \end{align*}$$

Lastly, observe that to every family of sets $\big {(}X_s\colon s\in (\mathbb {N}^{<\mathbb {N}})^{*}\big {)}$ , parametrized by elements of $(\mathbb {N}^{<\mathbb {N}})^{*}$ , we may still apply Suslin’s $\mathcal {A}$ -operation by first setting $X_s:=\emptyset $ for all $s\in \mathbb {N}^{<\mathbb {N}}\setminus (\mathbb {N}^{<\mathbb {N}})^{*}$ .

2.3. Covering systems

Let X be a locally compact Polish space. A cofiltration or exhaustion of X is an increasing sequence $X_0\subseteq X_1\subseteq \cdots $ of compact subsets of X, with $X_0=\emptyset $ and $\bigcup _{n\in \mathbb {N}}X_n=X$ . An open cover of X is any family $\mathcal {U}$ of open sets so that $\bigcup \mathcal {U}=X$ . An open cover $\mathcal {U}$ is locally finite if for every $x\in X$ , there is an open neighborhood $O\subseteq X$ of x which intersects only finitely many elements of $\mathcal {U}$ . If $K\subseteq X$ , then we let $\mathcal {U}\upharpoonright K:=\{U\in \mathcal {U}\colon U\cap K \neq \emptyset \}$ . Notice that if $\mathcal {U}$ is locally finite and K is compact, then $\mathcal {U}\upharpoonright K$ is finite. Let $\mathcal {U},\mathcal {V}$ be open covers of X. We write $\mathcal {U}\preceq \mathcal {V}$ if $\mathcal {V}$ refines $\mathcal {U}$ (i.e., if there exists a function $r\colon \mathcal {V}\to \mathcal {U}$ so that $V\subseteq r(V)$ for all $V\in \mathcal {V}$ ). We term such an r a refinement map.

Notice that if $\boldsymbol {\mathcal {U}}$ is a covering system for X, then for every open cover $\mathcal {U}$ of X and every $m\in \mathbb {N}$ , there exists $\beta \in \mathcal {N}^{*}$ so that

$$\begin{align*}\mathcal{U}\upharpoonright X_m\preceq \mathcal{U}_{\beta}\upharpoonright X_m. \end{align*}$$

This follows simply from (E1), since $\{X\}=\mathcal {U}\upharpoonright X_0\preceq \mathcal {U}_{\alpha }\upharpoonright X_0$ for all $\alpha \in \mathcal {N}^{*}$ .

While the definition of a covering system $\boldsymbol {\mathcal {U}}=\big {(}(X_n),(\mathcal {U}_{\alpha }),(r^{\beta }_{\alpha })\big {)}$ for X involves an uncountable family of open covers and an uncountable family of refinement maps, $\boldsymbol {\mathcal {U}}$ can still be fully recovered from a certain countable family of ‘finitary approximations’, via a procedure which resembles Suslin’s operation $\mathcal {A}$ . The next definition and the remark following it make this precise. Notice that by (L1) and (L2), the definition of $\mathcal {U}_s$ and $r^{t}_{s}$ below does not depend on the choice of $\alpha $ and $\beta $ .

Remark 2.5. Notice that we can fully recover the covering system $\boldsymbol {\mathcal {U}}$ from its approximation $\boldsymbol {\mathcal {U}}_{\mathrm {fin}}$ . Indeed, for each $n\in \mathbb {N}$ and any sequence $(s_k)$ in $(\mathbb {N}^n)^{*}$ with $s_0(i)<s_1(i)<\cdots $ for all $i<n$ , we have that

$$\begin{align*}X_n=\bigcap_{k\in\mathbb{N}} \bigcup_{U\in\, \mathcal{U}_{s_k}}\mathrm{cl}(U).\end{align*}$$

The above equality is easily established using (1) and (3) of Lemma 2.6 below. If for $s\in (\mathbb {N}^n)^{*}$ we momentarily denote by $\widehat {\mathcal {U}}_{s}$ the set of all open covers $\mathcal {U}$ of X with $\mathcal {U}\upharpoonright X_{|s|}= \mathcal {U}_s$ , then

$$\begin{align*}\mathcal{A}\big(\widehat{\mathcal{U}}_{s}\big)=\bigcup_{\alpha\in\mathcal{N}^{*}}\bigcap_{n\in\mathbb{N}}\widehat{\mathcal{U}}_{\alpha|n} =\{\mathcal{U}_{\alpha}\colon \alpha\in\mathcal{N}^{*}\},\end{align*}$$

and similarly for the collection $(r^{\beta }_\alpha )_{\alpha \leq \beta }$ . This motivates the notation $\boldsymbol {\mathcal {U}}=\mathcal {A}(\boldsymbol {\mathcal {U}}_{\mathrm {fin}})$ in Definition 2.4 for indicating the relationship between a covering system $\boldsymbol {\mathcal {U}}$ and its approximation $\boldsymbol {\mathcal {U}}_{\mathrm {fin}}$ .

Proof. (1) follows from the local finiteness of $\mathcal {U}_{\alpha }$ , and (4) is a direct consequence of (3), since X is locally compact.

For (2), we may assume that $\emptyset \not \in \bigcup _{\alpha \in \mathcal {N}} \mathcal {U}_{\alpha }$ . The rest follows from (1), since by (L1) we have

$$\begin{align*}\bigcup_{\alpha\in\mathcal{N}^{*}} \mathcal{U}_{\alpha}=\bigcup_{\alpha\in\mathcal{N}^{*}}\bigcup_{n\in \mathbb{N}} (\mathcal{U}_{\alpha}\upharpoonright X_n)=\bigcup_{n\in \mathbb{N}} \bigcup_{s\in\mathbb{N}^{n}} \mathcal{U}_s. \end{align*}$$

For (3), we will find $\alpha $ as the limit of a sequence $(\alpha _n)$ in $\mathcal {N}$ . We inductively define $\alpha _n$ as follows: let $\alpha _0$ be any element of $\mathcal {N}$ ; assuming that $\alpha _{n-1}$ has been defined, use (E1) to get $\alpha _{n}\in \mathcal {N}$ with $\alpha _{n}|(n-1)=\alpha _{n-1}|(n-1)$ , so that $\mathcal {U}\upharpoonright X_{n}\preceq \mathcal {U}_{\alpha |n}\upharpoonright X_n$ . Notice that $(\alpha _n)$ converges to the unique $\alpha \in \mathcal {N}$ with $ \alpha |n=\alpha _n|n$ . By (L1), we have that $\mathcal {U}\upharpoonright X_n\preceq \mathcal {U}_{\alpha }\upharpoonright X_n$ for all $n\in \mathbb {N}$ . Since $\bigcup _n X_n=X$ , it follows that $\mathcal {U}\preceq \mathcal {U}_{\alpha }$ .

Proof. Let d be a metric on X that is compatible with the topology, and let $(X_n)$ be any exhaustion of X with $X_{n}\subseteq \mathrm {int}(X_{n+1})$ . We will attain the desired covering system $\boldsymbol {\mathcal {U}}=\big {(}(X_n),(\mathcal {U}_{\alpha }),(r^{\beta }_{\alpha })\big {)}$ in the form of $\mathcal {A}(\boldsymbol {\mathcal {U}}_{\mathrm {fin}})$ where $\boldsymbol {\mathcal {U}}_{\mathrm {fin}}=((\mathcal {U}_s),(r^{t}_s))$ will be defined by a double induction.

$\underline {\mbox {Induction on the length of } s,t}$ .

Set $U_{\emptyset }=\emptyset $ , and assume that for some $n\in \mathbb {N}$ , we have defined for all $s,t,q\in (\mathbb {N}^n)^{*}$ with $s\leq t\leq q$

1. (1) a finite family $\mathcal {U}_s$ of open subsets of X covering $X_n$ , so that $\mathcal {U}_s\upharpoonright X_m= \mathcal {U}_{s|m}$ if $m<n$ , and

   $$\begin{align*}\mathrm{diam}(U)<\frac{1}{\min\{s(i)\colon i<n\}+1}, \text{ for all } U\in\mathcal{U}_s.\end{align*}$$

2. (2) a surjective refinement map $r^t_s \colon \mathcal {U}_t\to \mathcal {U}_s$ with $r^{t}_t=\mathrm {id}$ and $r^t_s\circ r^q_t=r^q_s$ .

We will extend this system to a system indexed by $(\mathbb {N}^{n+1})^{*}$ , which satisfies properties (1), (2) above with $n+1$ in place of n. For that, set $\boldsymbol {\bar {\ell }}:=(\ell ,\ldots ,\ell )\in \mathbb {N}^n$ and $O_{\ell }:=\bigcup \mathcal {U}_{\boldsymbol {\bar {\ell }}}$ for every $\ell \in \mathbb {N}$ . Notice that by our inductive assumption, we have a decreasing sequence of open sets

$$\begin{align*}O_{0}\supseteq O_{1}\supseteq\cdots\supseteq O_{\ell}\supseteq \cdots\supseteq X_n.\end{align*}$$

$\underline {\mbox {Induction on } \ell \in \mathbb {N}}$ .

By induction on $\ell \in \mathbb {N}$ , we choose a sequence $\mathcal {V}_{0}, \mathcal {V}_{1},\ldots , \mathcal {V}_{\ell },\ldots $ of finite families of open subsets of X so that $\mathcal {V}_{\ell }$ covers $X_{n+1}\setminus O_{\ell }$ with sets of diameter less than $1/(\ell +1)$ , which do not intersect $X_n$ . We can also make sure that $ \mathcal {V}_{\ell }\cup \mathcal {U_{\boldsymbol {\bar {\ell }}}}\preceq \mathcal {V}_{\ell +1}$ . We may now set

$$\begin{align*}\mathcal{U}_{s^{\frown}\ell}:=\mathcal{U}_s\cup \mathcal{V}_{\ell},\end{align*}$$

for every $s^{\frown }\ell \in (\mathbb {N}^{n+1})^{*}$ and observe that the analogue of point (1) above is satisfied by the new system. For example, notice that if $s^{\frown }\ell \in (\mathbb {N}^{n+1})^{*}$ , then $s\leq \boldsymbol {\bar {\ell }}$ , and hence, $\mathcal {U}_s\preceq \mathcal {U}_{\boldsymbol {\bar {\ell }}}$ . As a consequence, $O_{\ell }\subseteq \bigcup \mathcal {U}_s$ , and therefore, $\mathcal {U}_{s^{\frown }\ell }$ covers $X_{n+1}$ .

We now turn to the definition of the refining maps. Fix $\ell \in \mathbb {N}$ , and assume inductively that for every pair $s^{\frown }k, t^{\frown } m\in (\mathbb {N}^{n+1})^{*}$ with $s^{\frown }k\leq t^{\frown } m$ and $m\leq \ell $ , we have defined a surjective refinement map

$$\begin{align*}r^{ t^{\frown} m}_{s^{\frown}k}\colon \mathcal{U}_{t^{\frown} m} \to \mathcal{U}_{s^{\frown}k}, \end{align*}$$

which extends $r^{ t}_{s}$ , and assume that these maps altogether cohere with respect to composition as in point (2) above. Fix now any refinement map

$$\begin{align*}p^{\ell+1}_{\boldsymbol{\bar{\ell}}^{\frown}\ell}: \mathcal{V}_{\ell+1}\to \mathcal{U}_{\boldsymbol{\bar{\ell}}^{\frown}\ell}. \end{align*}$$

Notice that in the definition of $\mathcal {V}_{\ell }$ , we could have arranged, by removing superfluous elements, that $\mathcal {V}_{\ell }$ is ‘minimal’ (i.e., if we remove any V from $\mathcal {V}_{\ell }$ , then the resulting family is not going to be a cover of $X_{n+1}\setminus O_{\ell }$ ). By imposing this minimality assumption on $\mathcal {V}_{\ell }$ , we have that $p^{\ell +1}_{\boldsymbol {\bar {\ell }}^{\frown }\ell }$ is surjective on $\mathcal {V}_{\ell }$ . For every $s^\frown k \in (\mathbb {N}^{n+1})^{*}$ with $k\leq \ell +1$ , we define a refinement map

$$\begin{align*}p^{\ell+1}_{s^{\frown} k }: \mathcal{V}_{\ell+1}\to \mathcal{U}_{s^{\frown} k } \end{align*}$$

by setting $p^{\ell +1}_{s^{\frown } k }=\mathrm {id}$ , if $k=\ell +1$ ; if $k<\ell +1$ , then notice that $s^{\frown } k\leq \boldsymbol {\bar {\ell }}^{\frown }\ell $ , and we may define

$$\begin{align*}p^{\ell+1}_{s^{\frown} k}:= r^{\boldsymbol{\bar{\ell}}^{\frown}\ell}_{s^{\frown} k}\circ p^{\ell+1}_{\boldsymbol{\bar{\ell}}^{\frown}\ell},\end{align*}$$

where the map $r^{\boldsymbol {\bar {\ell }}^{\frown }\ell }_{s^{\frown } k}$ is given by inductive assumption. Finally, for all $s^{\frown } k \leq t^{\frown } (\ell +1)$ , we set

$$\begin{align*}r^{ t^{\frown} (\ell+1)}_{s^{\frown} k}:=\big(r^t_s\cup p^{\ell+1}_{s^{\frown} k }\big).\end{align*}$$

The fact that our new system of maps which is indexed by pairs $s^{\frown }k, t^{\frown } m\in (\mathbb {N}^{n+1})^{*}$ with $s^{\frown }k\leq t^{\frown } m$ and $m\leq \ell +1$ coheres in the sense of point (2) above, follows from our inductive assumptions and the fact that all new maps which are not equal to $\mathrm {id}$ on $\mathcal {V}_{\ell +1}$ factor through $\mathcal {U}_{\boldsymbol {\bar {\ell }}^{\frown }\ell }$ .

This ends the induction on $\ell $ and consequently the induction on the length of $s,t$ . Let now

$$ \begin{align*}\boldsymbol{\mathcal{U}}=\big{(}(X_n),(\mathcal{U}_{\alpha}),(r^{\beta}_{\alpha})\big{)}:=\mathcal{A}(\boldsymbol{\mathcal{U}}_{\mathrm{fin}}),\end{align*} $$

where $\boldsymbol {\mathcal {U}}_{\mathrm {fin}}=((\mathcal {U}_s),(r^{t}_s))$ . Using points (1) and (2) above, it is easy to see that $\boldsymbol {\mathcal {U}}$ is a covering system for X. For example, (E1) follows from the shrinking diameters in point (1) above and Lebesgue’s covering lemma; the fact that each $\mathcal {U}_{\alpha }$ is locally finite follows from the assumption $X_n\subseteq \mathrm {int}(X_{n+1})$ and the fact that each $\mathcal {U}_s$ is finite.

2.4. Definable cohomology with discrete coefficients

Let G be a countable discrete abelian group, and let X be a locally compact Polish space. We will fix a covering system $\boldsymbol {\mathcal {U}}=\big {(}(X_n),(\mathcal {U}_{\alpha }),(r^{\beta }_{\alpha })\big {)}$ for X and use it to define a Polish cochain complex $C^{\bullet }(\boldsymbol {\mathcal {U}};G)$ . The associated cohomology groups , viewed as groups with Polish cover, form the definable cohomology of X with coefficients in G. If we forget the definable content from , we recover the classical Čech cohomology groups for X. In Section 5, we will show that, up to definable isomorphism, does not depend on the choice of $\boldsymbol {\mathcal {U}}$ . One may extend these ideas to develop definable cohomology groups with coefficients in an arbitrary Polish abelian group G. That project, however, falls beyond the scope of this paper; see Remark 2.13.

For every open cover $\mathcal {U}$ of X, the nerve of $\mathcal {U}$ is the simplicial complex $\mathrm {Nv}(\mathcal {U})$ with $\sigma \in \mathrm {Nv}(\mathcal {U})$ if and only if $\sigma $ is a finite subset of $\mathcal {U}$ with $\cap _{U\in \sigma }U\neq \emptyset .$ Notice that $\mathrm {dom}(\mathrm {Nv}(\mathcal {U}))=\mathcal {U}\setminus \{\emptyset \}$ . Set

$$\begin{align*}N_{\alpha}:= \mathrm{Nv}(\mathcal{U_{\alpha}})\end{align*}$$

for every $\alpha \in \mathcal {N}^{*}$ , and notice that every refinement map $r^{\beta }_{\alpha }$ induces a simplicial map

$$\begin{align*}r^{\beta}_{\alpha}\colon N_{\beta}\to N_{\alpha}.\end{align*}$$

In this way, $\boldsymbol {\mathcal {U}}$ determines an inverse system $((N_{\alpha }),(r^{\beta }_{\alpha }))$ of simplicial maps. We similarly define the finite complexes $N_s:= \mathrm {Nv}(\mathcal {U}_{s})$ and consider the simplicial maps $r^{t}_s\colon N_{t}\to N_s$ , for every $s\leq t$ in $(\mathbb {N}^{<\mathbb {N}})^{*}$ .

For every $n\in \mathbb {N}$ , we will define the Polish group $C^{n}(\boldsymbol {\mathcal {U}};G)$ of n-dimensional cocycles of $\boldsymbol {\mathcal {U}}$ as the quotient of a Polish semigroup $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ by a certain closed semigroup congruence. Let

$$\begin{align*}C_{\mathrm{sem}}^{n}(\boldsymbol{\mathcal{U}};G):=\bigcup_{\alpha\in\mathcal{N}^{*}}C^{n}(N_{\alpha};G)=\bigcup_{\alpha\in\mathcal{N}^{*}}C(N^{(n)}_{\alpha};G)\end{align*}$$

be the collection of all functions from the set of singular n-faces of some $N_{\alpha }$ to G. We endow $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ with a semigroup structure by setting for all $\zeta \colon N^{(n)}_{\alpha }\to G$ and $\eta \colon N^{(n)}_{\beta }\to G$

(4) $$ \begin{align} (\zeta+\eta)\colon N^{(n)}_{\alpha\vee \beta}\to G, \text{ with } (\zeta+\eta)(\bar{U})=\zeta(r^{\alpha\vee \beta}_{\alpha}(\bar{U}))+\eta(r^{\alpha\vee \beta}_{\beta}(\bar{U})). \end{align} $$

We also endow $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ with the topology generated by basic open sets of the form $V_{s,p}$ , with $s\in \mathbb {N}^{<\mathbb {N}}$ and $p\colon N^{(n)}_s\to G$ , which is given by all $\zeta \colon N^{(n)}_{\alpha }\to G$ with $\alpha \in \mathcal {N}_s$ and $\zeta \upharpoonright N^{(n)}_s= p$ .

Proof. Since $\alpha \vee \beta =\beta \vee \alpha $ and G is abelian, so is $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ . It is also a topological semigroup since by (L2), the operation $(\zeta ,\eta )\mapsto (\zeta +\eta )$ is continuous. The topology is clearly second countable since it has, by definition, a basis consisting of countably many open sets. Finally, it is easy to check that $\rho _{\mathrm {sem}}$ below is a complete metric on $C_{\mathrm {sem}}^{(n)}(\boldsymbol {\mathcal {U}};G)$ that is compatible with the topology: for every $\zeta \in C^{n}(N_{\alpha };G)$ and $\eta \in C^{n}(N_{\beta };G)$ , set $\delta _k:=0$ if $\zeta \upharpoonright N^{(n)}_{\alpha |k}=\eta \upharpoonright N^{(n)}_{\beta |k}$ , and set $\delta _k:=1$ otherwise. Let

(5) $$ \begin{align} \rho_{\mathrm{sem}}(\zeta,\eta):= \sum_{k\in\mathbb{N}}\frac{\delta_k}{2^{k}}.\\[-34pt]\notag \end{align} $$

We define a congruence $\sim $ on $C_{\mathrm {sem}}^{(n)}(\boldsymbol {\mathcal {U}};G)$ by setting for all $\zeta \colon N^{(n)}_{\alpha }\to G$ and $\eta \colon N^{(n)}_{\beta }\to G$

$$\begin{align*}\zeta \sim \eta \iff \zeta\circ r^{\alpha\vee \beta}_{\alpha}=\eta\circ r^{\alpha\vee \beta}_{\beta}. \end{align*}$$

Proof. First, we prove that $\sim $ is an equivalence relation on $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ . Symmetry follows from $\alpha \vee \beta =\beta \vee \alpha $ and reflexivity from $\alpha \vee \alpha =\alpha $ . For transitivity, let $\zeta \colon N^{(n)}_{\alpha }\to G$ , $\eta \colon N^{(n)}_{\beta }\to G$ , and $\theta \colon N^{(n)}_{\gamma }\to G$ , with $\zeta \sim \eta $ and $\eta \sim \gamma $ . In particular, we have that

$$\begin{align*}\zeta\circ r^{\alpha\vee \beta}_{\alpha}\circ r^{\alpha\vee \beta\vee \gamma}_{\alpha\vee \beta}=\eta\circ r^{\alpha\vee \beta}_{\beta}\circ r^{\alpha\vee \beta\vee \gamma}_{\alpha\vee \beta}\; \text{ and } \; \eta\circ r^{\beta \vee \gamma}_{\beta}\circ r^{\alpha\vee \beta\vee \gamma}_{\beta \vee \gamma}=\theta\circ r^{\beta \vee \gamma}_{\gamma}\circ r^{\alpha\vee \beta\vee \gamma}_{\beta \vee \gamma}.\end{align*}$$

But since $\eta \circ r^{\alpha \vee \beta }_{\beta }\circ r^{\alpha \vee \beta \vee \gamma }_{\alpha \vee \beta }=\eta \circ r^{\alpha \vee \beta \vee \gamma }_{\beta }=\eta \circ r^{\beta \vee \gamma }_{\beta }\circ r^{\alpha \vee \beta \vee \gamma }_{\beta \vee \gamma }$ , we have that $\zeta \circ r_{\alpha }^{\alpha \vee \beta \vee \gamma }=\theta \circ r_{\gamma }^{\alpha \vee \beta \vee \gamma }$ , and therefore that $\zeta \circ r_{\alpha }^{\alpha \vee \gamma }\circ r_{\alpha \vee \gamma }^{\alpha \vee \beta \vee \gamma }=\theta \circ r_{\gamma }^{\alpha \vee \gamma }\circ r_{\alpha \vee \gamma }^{\alpha \vee \beta \vee \gamma }$ . Since $r_{\alpha \vee \gamma }^{\alpha \vee \beta \vee \gamma }$ is surjective, we have that $\zeta \circ r_{\alpha }^{\alpha \vee \gamma }=\theta \circ r_{\gamma }^{\alpha \vee \gamma }$ ; hence, $\zeta \sim \theta $ .

The relation $\sim $ is closed as a subset of $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)\times C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ : indeed, if $\zeta \circ r^{\alpha \vee \beta }_{\alpha }=\eta \circ r^{\alpha \vee \beta }_{\beta }$ fails, then there is an $s=(\alpha \vee \beta )|k$ , for some $k\in \mathbb {N}$ , so that the values of some $\overline {U}\in N^{(n)}_{s}$ under $\zeta \circ r^{\alpha \vee \beta }_{\alpha }$ and $\eta \circ r^{\alpha \vee \beta }_{\beta }$ differ. This failure is witnessed by some open condition.

Finally, to see that $\sim $ is a semigroup congruence, let $\zeta \colon N^{(n)}_{\alpha }\to G$ and $\eta \colon N^{(n)}_{\beta }\to G$ and $\zeta '\colon N^{(n)}_{\alpha '}\to G$ and $\eta '\colon N_{\beta '}\to G$ , with $\zeta \sim \eta $ and $\zeta '\sim \eta '$ . We will show that $\zeta +\zeta '\sim \eta +\eta '$ . Indeed, since $\zeta \circ r^{\alpha \vee \beta }_{\alpha }=\eta \circ r^{\alpha \vee \beta }_{\beta }$ and $\zeta '\circ r^{\alpha '\vee \beta '}_{\alpha '}=\eta '\circ r^{\alpha '\vee \beta '}_{\beta '}$ , we have

$$ \begin{align*} \zeta\circ r^{\alpha\vee \beta}_{\alpha} \circ r_{\alpha\vee \beta}^{\alpha\vee \alpha'\vee\beta\vee\beta'} + \zeta'\circ r^{\alpha'\vee \beta'}_{\alpha'}\circ r_{\alpha'\vee \beta'}^{\alpha\vee \alpha'\vee\beta\vee\beta'} &=\eta\circ r^{\alpha\vee \beta}_{\beta} \circ r_{\alpha\vee \beta}^{\alpha\vee \alpha'\vee\beta\vee\beta'} +\eta'\circ r^{\alpha'\vee \beta'}_{\beta'}\circ r_{\alpha'\vee \beta'}^{\alpha\vee \alpha'\vee\beta\vee\beta'}\\ \zeta\circ r_{\alpha}^{\alpha\vee \alpha'\vee\beta\vee\beta'} + \zeta'\circ r_{\alpha'}^{\alpha\vee \alpha'\vee\beta\vee\beta'} &=\eta\circ r_{\beta}^{\alpha\vee \alpha'\vee\beta\vee\beta'} +\eta'\circ r_{\beta'}^{\alpha\vee \alpha'\vee\beta\vee\beta'} \\ (\zeta+\zeta')\circ r_{\alpha\vee\alpha'}^{\alpha\vee \alpha'\vee\beta\vee\beta'} &= (\eta+ \eta' )\circ r_{\beta\vee\beta'}^{\alpha\vee \alpha'\vee\beta\vee\beta'} \\ (\zeta+\zeta') &\sim (\eta+\eta').\\[-36pt] \end{align*} $$

We now define $C^{n}(\boldsymbol {\mathcal {U}};G):=C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)/\sim $ to be the collection of all congruence classes $[\zeta ]$ of elements $\zeta $ of $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ . A priori, $C^{n}(\boldsymbol {\mathcal {U}};G)$ is merely an abelian semigroup which is additionally endowed with the quotient topology: $V\subseteq C^{n}(\boldsymbol {\mathcal {U}};G)$ is open in $C^{n}(\boldsymbol {\mathcal {U}};G)$ if and only if its union $\bigcup V=\{\zeta \colon [\zeta ]\in V\}$ is open in $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ . It turns out that $C^{n}(\boldsymbol {\mathcal {U}};G)$ is a much richer structure.

Proof. First, notice that any two maps $0_{\alpha }\colon N^{(n)}_{\alpha }\to G$ and $0_{\beta }\colon N^{(n)}_{\beta }\to G$ which are constantly equal to $0_{G}$ are congruent to each other. Moreover, it is clear that the associated congruence class $0_{\boldsymbol {\mathcal {U}}}:=[0_{\alpha }]=[0_{\beta }]$ is the identity element of $C^{n}(\boldsymbol {\mathcal {U}};G)$ and that for every $\zeta \colon N^{(n)}_{\alpha }\to G$ , we have that $[\zeta ]+[-\zeta ]=[0_{\alpha }]$ . It follows that $C^{n}(\boldsymbol {\mathcal {U}};G)$ is an abelian group. Moreover, notice that if $\zeta '\sim \zeta $ for some $\zeta \colon N^{(n)}_{\alpha }\to G$ and $\zeta '\colon N^{(n)}_{\beta }\to G$ with $\zeta \upharpoonright N^{(n)}_{\alpha |k}=0_{\alpha }\upharpoonright N^{(n)}_{\alpha |k}$ , then $\eta \upharpoonright N^{(n)}_{\beta |k}=0_{\beta }\upharpoonright N^{(n)}_{\beta |k}$ . Hence, the identity $0_{\boldsymbol {\mathcal {U}}}$ admits a neighborhood basis in $C^{n}(\boldsymbol {\mathcal {U}};G)$ consisting of the following open subgroups:

$$\begin{align*}V_k:=\{[\zeta]\in C^{n}(\boldsymbol{\mathcal{U}};G) \colon \exists \alpha \in \mathcal{N}^{*} \text{ so that } \zeta\colon N^{(n)}\to G \text{ with } \zeta \upharpoonright N^{(n)}_{\alpha|k}=0_{\alpha}\upharpoonright N^{(n)}_{\alpha|k}\}.\end{align*}$$

The topology of a quotient of a separable space is separable. We now define a complete metric on $C^{n}(\boldsymbol {\mathcal {U}};G)$ that is compatible with its topology. Let $\rho _{\mathrm {sem}}$ be the complete metric on $C_{\mathrm {sem}}^{n}(\boldsymbol {\mathcal {U}};G)$ given by (5) above. For every $[\zeta ],[\eta ]\in C^{n}(\boldsymbol {\mathcal {U}};G)$ , we let

(6) $$ \begin{align} \rho([\zeta],[\eta]):= \inf \{\rho_{\mathrm{sem}}(\zeta',\eta') : \zeta'\in[\zeta], \eta'\in [\eta] \}. \end{align} $$

It is easy to see that one can directly compute the value of $\rho ([\zeta ],[\eta ])$ by picking any representatives $\zeta '\colon N^{(n)}_{\alpha }\to G$ and $\eta '\colon N^{(n)}_{\beta }\to G$ of $[\zeta ]$ and $[\eta ]$ and alternatively setting

(7) $$ \begin{align} \rho([\zeta],[\eta])= \rho_{\mathrm{sem}}(\zeta'\circ r^{\alpha\vee \beta}_{\alpha},\eta'\circ r^{\alpha\vee \beta}_{\beta}). \end{align} $$

The fact that the latter quantity does not depend on the choice of $\eta '$ and $\zeta '$ follows by the same argument we used to prove transitivity of $\sim $ , using local surjectivity (L3) in place of surjectivity. By (6), it is clear that $\rho $ is a metric, given that by (7), we have $\rho ([\zeta ],[\eta ])>0$ if $[\zeta ]\neq [\eta ]$ . To see that $\rho $ is compatible with the topology of $C^{n}(\boldsymbol {\mathcal {U}};G)$ , notice that since the topology is first countable, it suffices to show that a subset F of $C^{n}(\boldsymbol {\mathcal {U}};G)$ is closed if and only if for every sequence $([\zeta _n])_{n}$ in F with $[\zeta _n]\to _{\rho } [\zeta ]$ , we have that $[\zeta ]\in F$ .

If $[\zeta _n]\to _{\rho } [\zeta ]$ and F is closed, then using (7), we may choose the representatives $\zeta _n$ and $\zeta $ so that $\zeta _n\to \zeta $ in $C^{n}_{\mathrm {sem}}(\boldsymbol {\mathcal {U}};G)$ . Since the quotient map $\pi \colon C^{n}_{\mathrm {sem}}(\boldsymbol {\mathcal {U}};G)\to C^{n}(\boldsymbol {\mathcal {U}};G)$ is continuous, we have that $\pi ^{-1}(F)$ is closed. Hence, $\zeta \in \pi ^{-1}(F)$ (i.e., $[\zeta ]\in F$ ). Conversely, assume that whenever $[\zeta _n]\to _{\rho } [\zeta ]$ , we have $[\zeta ]\in F$ . We will show that $\pi ^{-1}(F)$ is closed. Let $(\zeta _n),\zeta $ be elements of $C^{n}_{\mathrm {sem}}(\boldsymbol {\mathcal {U}};G)$ such that $\zeta _n\to \zeta $ . Then by (7), we have that $[\zeta _n]\to _{\rho } [\zeta ]$ . Hence, $[\zeta ]\in F$ , and therefore, $\zeta \in \pi ^{-1}(F)$ . Finally, the fact that $\rho $ is complete follows from completeness of $\rho _{\mathrm {sem}}$ and (7).

Finally, notice that for every $\zeta \colon N^{(n)}_{\alpha }\to G$ and $\zeta '\colon N^{(n)}_{\alpha '}\to G$ with $\zeta \sim \zeta '$ , we have $\eta \sim \eta '$ , where

$$\begin{align*}\eta \big(v_{0},\ldots ,v_{n}\big)=\sum_{i=0}^{n}\left( -1\right) ^{i}\zeta (v_{0},\ldots ,\hat{v}_{i},\ldots ,v_{n}) \text{ and } \eta' \big(v_{0},\ldots ,v_{n}\big)=\sum_{i=0}^{n}\left( -1\right) ^{i}\zeta' (v_{0},\ldots ,\hat{v}_{i},\ldots ,v_{n}).\end{align*}$$

As a consequence, for every $n>0$ , we have a continuous coboundary map

(8) $$ \begin{align} \begin{aligned} \delta^{n}:C^{n-1}(\boldsymbol{\mathcal{U}};G)&\rightarrow C^{n}(\boldsymbol{\mathcal{U}};G), \text{ where } \\ \delta^{n}\left( [\zeta] \right)= [\eta], \text{ with } \eta (v_{0},\ldots ,v_{n})&=\sum_{i=0}^{n}\left( -1\right) ^{i}\zeta (v_{0},\ldots ,\hat{v}_{i},\ldots ,v_{n}). \end{aligned} \end{align} $$

This defines the Polish cochain complex $C^{\bullet }(\boldsymbol {\mathcal {U}};G)$ of G-valued cochains of $\boldsymbol {\mathcal {U}}$ .

Definition 2.11. Let X be a locally compact Polish space, and let G be a countable abelian Polish group. Fix a covering system $\boldsymbol {\mathcal {U}}$ for X, and for every $n\in \mathbb {N}$ , define the n-dimensional definable cohomology group of X with coefficients in G to be the n-dimensional cohomology group of the Polish cochain complex $C^{\bullet }(\boldsymbol {\mathcal {U}};G)$ , viewed as the group with a Polish cover

$$\begin{align*}0\longrightarrow \mathrm{B}^{n}(\boldsymbol{\mathcal{U}};G)\longrightarrow \mathrm{Z}^{n}(\boldsymbol{\mathcal{U}};G) \longrightarrow \mathrm{Z}^{n}(\boldsymbol{\mathcal{U}};G)/ \mathrm{B}^{n}(\boldsymbol{\mathcal{U}};G)\longrightarrow 0\end{align*}$$

where $\mathrm {Z}^{n}(\boldsymbol {\mathcal {U}};G)= \mathrm {ker}(\delta ^{n})$ is the Polish group of n-dimensional G-valued cocycles of $\boldsymbol {\mathcal {U}}$ and $\mathrm {B}^{n}(\boldsymbol {\mathcal {U}};G)=\mathrm {im}(\delta ^{n-1})$ is the Polishable group of n-dimensional G-valued coboundaries of $\boldsymbol {\mathcal {U}}$ .

Notice that while the definition of $\mathrm {H}^{n}(X;G)$ involves a choice of covering system $\boldsymbol {\mathcal {U}}$ for X, the latter is not explicit in the notation . This is because, as we will show in Corollary 5.8, a different choice of $\boldsymbol {\mathcal {U}}$ will induce the exact same definable cohomology groups up to definable isomorphism.

Since the Polish groups in $C^{\bullet }(\boldsymbol {\mathcal {U}};G)$ are non-archimedean, the rigidity results of [Reference Bergfalk and Lupini9] provide a powerful resource for analyzing the definable cohomology groups of any locally compact Polish space. Note that as a sequence of abstract groups, $\mathrm {H}^{\bullet }(X;G)$ coincides with the classical Čech cohomology groups of X with coefficients in G.

Proof. Throughout this proof, every group is viewed as an abstract group. The proof of Theorem 2.12 follows by combining the two usual arguments which show (1) without loss of generality, the colimit in the definition of can be taken over any $\preceq $ –cofinal collection of open covers and (2) the group is isomorphic to the sheaf of germs of constant G-valued functions. We sketch the argument here, as the explicit abstract group isomorphism generated by this argument will be needed in the proof of Theorem 5.7.

Recall that in order to define the n-dimensional Čech cohomology group of X, one first considers the poset $\mathrm {Cov}(X)$ of all open covers $\mathcal {U}$ of X ordered by the relation of refinement $\mathcal {U}\preceq \mathcal {V}$ and observes that any two refinement maps $r^{\mathcal {V}}_{\mathcal {U}},s^{\mathcal {V}}_{\mathcal {U}}\colon \mathcal {V}\to \mathcal {U}$ induce contiguous (see Section 4.2) simplicial maps $r^{\mathcal {V}}_{\mathcal {U}},s^{\mathcal {V}}_{\mathcal {U}}\colon \mathrm {Nv}(\mathcal {V})\to \mathrm {Nv}(\mathcal {U})$ . Hence, while the associated chain maps $(r^{\mathcal {V}}_{\mathcal {U}})^{\bullet },(s^{\mathcal {V}}_{\mathcal {U}})^{\bullet }\colon C^{\bullet }(\mathrm {Nv}(\mathcal {U}),G)\to C^{\bullet }(\mathrm {Nv}(\mathcal {V}),G)$ with $(r^{\mathcal {V}}_{\mathcal {U}})^{\bullet }(\zeta ):=\zeta \circ r^{\mathcal {V}}_{\mathcal {U}}$ and $(s^{\mathcal {V}}_{\mathcal {U}})^{\bullet }(\zeta ):=\zeta \circ s^{\mathcal {V}}_{\mathcal {U}}$ may differ, they induce the same homomorphism $(r^{\mathcal {V}}_{\mathcal {U}})^{*}=(s^{\mathcal {V}}_{\mathcal {U}})^{*}$ on the level of simplicial cohomology $\mathrm {H}^{n}(\mathrm {Nv}(\mathcal {U}),G)\to \mathrm {H}^{n}(\mathrm {Nv}(\mathcal {V}),G)$ for every $n\in \mathbb {N}$ ; see [Reference Eilenberg and Steenrod29, Corollary IX.2.14], for example.

As a consequence, we have a well-defined direct system of group homomorphisms $\mathrm {H}^{n}(\mathrm {Nv}(\mathcal {U}),G)\to \mathrm {H}^{n}(\mathrm {Nv}(\mathcal {V}),G)$ indexed by the poset $(\mathrm {Cov}(X),\preceq )$ . By definition, see [Reference Eilenberg and Steenrod29, Definition IX.3.1], we have that

(9)

where $(r^{\mathcal {V}}_{\mathcal {U}}\colon \mathcal {V}\to \mathcal {U})_{\mathcal {U}\preceq \mathcal {V}}$ is any fixed pre-chosen collection of refinement maps.

Let now $(U_{\alpha }\colon \alpha \in \mathcal {N}^{*})$ and $(r^{\beta }_{\alpha }\colon \alpha \leq \beta )$ be the open covers and refinement maps appearing in the covering system $\boldsymbol {\mathcal {U}}$ used in the definition of , and set $N_{\alpha }:=\mathrm {Nv}(\mathcal {U}_{\alpha })$ . Since the colimit in (9) is isomorphic to the colimit taken over any cofinal collection of $\mathcal {U}\in \mathrm {Cov}(X)$ , by Lemma 2.6(3), we have that

(10)

However, the system $(r^{\beta }_{\alpha }\colon \alpha \leq \beta )$ coheres (i.e., for all $\alpha \leq \beta \leq \gamma $ , we have $r^{\gamma }_{\alpha }=r^{\beta }_{\alpha }\circ r^{\gamma }_{\beta }$ ); see Definition 2.3. Moreover, if $\delta ^{\bullet }_{\alpha }$ and $\delta ^{\bullet }_{\beta }$ are the coboundary maps of the cochain complexes $\mathrm {C}^{\bullet }(N_{\alpha };G)$ and $\mathrm {C}^{\bullet }(N_{\beta };G)$ with $\alpha \leq \beta $ , then for all $n\in \mathbb {N}$ , we have that $(r^{\beta }_{\alpha })^{n+1}\circ \delta ^{n}_{\alpha }=\delta ^{n}_{\beta }\circ (r^{\beta }_{\alpha })^{n}$ . As a consequence, we have a direct system of chain complexes $\big (\mathrm {C}^{\bullet }(N_{\alpha };G) ,(r^{\beta }_{\alpha })^{\bullet }\big )_{\alpha \in \mathcal {N}^{*}}$ . But then, by the definition of a colimit of a directed system of abelian groups (see, for example, [Reference Eilenberg and Steenrod29, Definition IX.4.1]), for every $n\in \mathbb {N}$ , we have that

(11) $$ \begin{align} \mathrm{colim}_{\;\alpha \in\mathcal{N}^{*}} \;\big( \mathrm{C}^{n}(N_{\alpha};G) ,(r^{\beta}_{\alpha})^{n} \big)\quad :=\quad \bigg(\bigsqcup_{\alpha \in\mathcal{N}^{*}} C^{n}(N_{\alpha};G) \; \bigg)\bigg/\sim \quad = \quad C^n_{\mathrm{sem}}(\boldsymbol{\mathcal{U}};G)/\sim \quad = \quad C^{n}(\boldsymbol{\mathcal{U}};G), \end{align} $$

where $\sim $ is the congruence we defined earlier in this section, and similarly have that $\delta ^n= \mathrm {colim}_{\alpha }\delta ^n_{\alpha }$ . Consequently,

(12) $$ \begin{align} \mathrm{colim}_{\;\alpha \in\mathcal{N}^{*}} \;\big( \mathrm{Z}^{n}(N_{\alpha};G) ,(r^{\beta}_{\alpha})^{n} \big) = \mathrm{Z}^{n}(\boldsymbol{\mathcal{U}};G) \quad \text{ and } \quad \mathrm{colim}_{\;\alpha \in\mathcal{N}^{*}} \;\big( \mathrm{B}^{n}(N_{\alpha};G) ,(r^{\beta}_{\alpha})^{n} \big) = \mathrm{B}^{n}(\boldsymbol{\mathcal{U}};G). \end{align} $$

But for every direct system of pairs of abelian groups $(H_i\leq G_{i},i\in I)$ , colimits commute with quotients (i.e., $\mathrm {colim}_i(G_i)/\mathrm {colim}_i( H_i) \cong \mathrm {colim}_i(G_i/H_i)$ ); see [Reference Eilenberg and Steenrod29, Theorem IX.6.3]. Hence, by combining (10) and (12), we have

(13)

An explicit formula for the composition of these isomorphisms, , can be given as follows. First, let be any element of , where $\zeta \in \mathrm {Z}^{n}(\mathrm {Nv}(\mathcal {U});G)$ for some $\mathcal {U}\in \mathrm {Cov}(X)$ , and is the image of $a\in \mathrm {H}^{n}(\mathrm {Nv}(\mathcal {U});G)$ under the inclusion . Then,

(14)

where $\alpha $ can be taken to be any element of $\mathcal {N}^{*}$ with $\mathcal {U}\preceq \mathcal {U}_{\alpha }.$ Such $\alpha $ exists by Lemma 2.6(3).

2.5. Definable cohomology of pairs of spaces

The process of enriching the Čech cohomology groups of locally compact spaces with definable content readily extends to the Čech cohomology groups of locally compact pairs (i.e, pairs $(X,A)$ in which X is a locally compact Polish space and A is closed in X); see Section 4.1. In this section, we sketch the details. We begin by generalizing the theory from Section 2.1 to pairs $(K,L)$ of simplicial complexes.

Let K be a simplicial complex. A subcomplex L of K is any simplicial complex with $\mathrm {dom}(L)\subseteq \mathrm {dom}(K)$ for which $\sigma \in L$ implies $\sigma \in K$ . By a simplicial pair we mean a pair $(K,L)$ of simplicial complexes in which L is a subcomplex of K. A simplicial map (of pairs) $f\colon (K,L)\to (K',L')$ is any simplicial map $f\colon K\to K'$ such that $\{f(v)\colon v\in \sigma \}\in L'$ for all $\sigma \in L$ . For any abelian Polish group G and $n\in \mathbb {N}$ , let

$$\begin{align*}C^{n}(K,L;G):=\{\zeta \in C^{n}(K;G) \colon \zeta(\bar{v})=0_G \text{ for all } \bar{v}\in L^{(n)} \}.\end{align*}$$

$C^{n}(K,L;G)$ is a closed subgroup of $C^{n}(K;G)$ with respect to the product topology of the $K^{(n)}$ -fold product of copies of G; hence, it is a Polish group. The coboundary map $ \delta ^{n}:C^{n}(K;G) \rightarrow C^{n+1}(K;G)$ clearly restricts to a map $C^{n}(K,L;G) \rightarrow C^{n+1}(K,L;G)$ which we will also denote by $\delta ^{n}$ . This induces a Polish cochain complex

$$\begin{align*}C^{\bullet}(K,L;G):=( \quad \quad \quad \quad \quad \quad\cdots\longrightarrow \,C^{n-1}(K,L;G)\overset{\delta^{n-1}}{\longrightarrow} C^n(K,L;G)\overset{\delta^{n}}{\longrightarrow} C^{n+1}(K,L;G) {\longrightarrow} \cdots \quad \quad \quad \quad \quad ). \quad \quad\end{align*}$$

The definable cohomology groups of $(K,L)$ with coefficients in G are the groups with a Polish cover

$$\begin{align*}\mathrm{H}_{\mathrm{def}}^n(K,L;G):= \mathrm{Z}^{n}(K,L;G)/ \mathrm{B}^{n}(K,L;G) := \mathrm{ker}(\delta^{n})/\mathrm{im}(\delta^{n-1}). \end{align*}$$

Note that the natural homomorphisms determining the short exact sequences

(15) $$ \begin{align} 0\longrightarrow C^n(K,L;G) \overset{i^n}{\longrightarrow} C^n(K;G) \overset{r^n}{\longrightarrow} C^n(L;G) \longrightarrow 0 \end{align} $$

are continuous and aggregate into a short exact sequence of Polish cochain complexes

(16) $$ \begin{align} 0\longrightarrow C^{\bullet}(K,L;G) \longrightarrow C^{\bullet}(K;G) \longrightarrow C^{\bullet}(L;G) \longrightarrow 0. \end{align} $$

For the rest of this section, let G be a countable abelian group and $(X,A)$ be a locally compact pair. Fix any covering system $\boldsymbol {\mathcal {U}}=\big {(}(X_n),(\mathcal {U}_{\alpha }),(r^{\beta }_{\alpha })\big {)}$ for X, and notice that the triple $\boldsymbol {\mathcal {V}}=\big {(}(A_n),(\mathcal {V}_{\alpha }),(s^{\beta }_{\alpha })\big {)}$ , where $A_n:=A\cap X_n$ , $\mathcal {V}_{\alpha }:=\mathcal {U}_{\alpha }\upharpoonright A\subseteq \mathcal {U}_{\alpha }$ and $s^{\beta }_{\alpha }$ denotes the restriction of $r^{\beta }_{\alpha }$ to $\mathcal {V}_{\beta }$ , is a covering system for A. Moreover, the inclusion $\mathcal {V}_{\alpha }\subseteq \mathcal {U}_{\alpha }$ induces a simplicial pair $(N_{\alpha },M_{\alpha }):=\big {(}\mathrm {Nv}(\mathcal {U}_{\alpha }),\mathrm {Nv}(\mathcal {V}_{\alpha })\big {)}$ . For every $n\in \mathbb {N}$ , we set

$$\begin{align*}C^{n}( \boldsymbol{\mathcal{U}}, \boldsymbol{\mathcal{V}};G):=\{[\zeta] \in C^{n}( \boldsymbol{\mathcal{U}},;G) \colon \; \;\forall \alpha\in \mathcal{N}^{*} \; \; \forall \bar{v}\in M_{\alpha}^{(n)} \; \;\forall \zeta' \in [\zeta] \cap C^n(N_{\alpha};G), \text{ we have that } \zeta' (\bar{v})=0_G \}.\end{align*}$$

Then $C^{n}( \boldsymbol {\mathcal {U}}, \boldsymbol {\mathcal {V}};G)$ is a Polish group, as it is a closed subgroup of $C^{n}( \boldsymbol {\mathcal {U}};G)$ . Moreover, the coboundary map $ \delta ^{n}:C^{n}( \boldsymbol {\mathcal {U}};G) \rightarrow C^{n+1}( \boldsymbol {\mathcal {U}};G)$ restricts to $ \delta ^{n}:C^{n}( \boldsymbol {\mathcal {U}},\boldsymbol {\mathcal {V}};G) \rightarrow C^{n+1}( \boldsymbol {\mathcal {U}},\boldsymbol {\mathcal {V}};G)$ , inducing a Polish cochain complex

$$\begin{align*}C^{\bullet}( \boldsymbol{\mathcal{U}}, \boldsymbol{\mathcal{V}};G):=( \quad \quad \quad \quad \quad \quad\cdots\longrightarrow \,C^{n-1}( \boldsymbol{\mathcal{U}}, \boldsymbol{\mathcal{V}};G)\overset{\delta^{n-1}}{\longrightarrow} C^n( \boldsymbol{\mathcal{U}}, \boldsymbol{\mathcal{V}};G)\overset{\delta^{n}}{\longrightarrow} C^{n+1}( \boldsymbol{\mathcal{U}}, \boldsymbol{\mathcal{V}};G) {\longrightarrow} \cdots \quad \quad \quad \quad \quad ). \quad \quad\end{align*}$$

The definable cohomology groups of $(X,A)$ with coefficients in G are the groups with a Polish cover

Arguing exactly as for Theorem 2.12 shows that, as abstract groups, these are precisely the Čech cohomology groups of the pair $(X,A)$ ; see [Reference Eilenberg and Steenrod29, Chapter IX] for a definition of the latter. Moreover, these groups array into a definable long exact sequence of the usual form; put differently, by the following theorem, they satisfy the Exactness Axiom as maps to the category of groups with a Polish cover.

Theorem 2.14. For any countable discrete group G and locally compact pair $(X,A)$ , there exists a long exact sequence of groups with a Polish cover and definable group homomorphisms

Theorem 2.14 is an instance of Theorem 2.16 below which, in turn, is a direct consequence of the following definable version of the Snake Lemma:

Lemma 2.15. Consider the following diagram of continuous homomorphisms of abelian Polish groups:

If the rows are exact, then there exists a Borel function $\varepsilon \colon \mathrm {ker}(c)\rightarrow A^{\prime }$ so that the map $x\mapsto \big (\varepsilon (x)+\mathrm {im}(A)\big )$ links the middle terms in the following exact sequence of groups with a Polish cover and definable group homomorphisms:

$$ \begin{align*} \mathrm{ker}(a)\overset{e}{\longrightarrow }\mathrm{ker}(b)\overset{f}{ \longrightarrow }\mathrm{ker}(c)\longrightarrow \mathrm{coker}(a)\overset{ g}{\longrightarrow }\mathrm{coker}(b)\overset{h}{ \longrightarrow }\mathrm{coker}(c). \end{align*} $$

Observe that we have notationally conflated all other maps with those which they induce on subgroups and quotients.

Theorem 2.16. Let $0\longrightarrow A^{\bullet } \overset {i^{\bullet }}{\longrightarrow } B^{\bullet } \overset {r^{\bullet }}{\longrightarrow } C^{\bullet } \longrightarrow 0$ be a short exact sequence of Polish cochain complexes where the group homomorphisms $i^n,r^n$ are continuous for all $n\in \mathbb {Z}$ . Then there is a long exact sequence of definable homomorphisms between the associated groups with a Polish cover

(17) $$ \begin{align} \dots\longrightarrow \mathrm{H}^{n-1}(C^{\bullet})\overset{\partial^{n-1}}{\longrightarrow} \mathrm{H}^{n}(A^{\bullet}) \overset{\widehat{\iota}^{n}}{\longrightarrow} \mathrm{H}^{n}(B^{\bullet}) \overset{\widehat{\rho}^{n}}{\longrightarrow} \mathrm{H}^{n}(C^{\bullet})\overset{\partial^{n}}{\longrightarrow} \mathrm{H}^{n+1}(A^{\bullet})\longrightarrow\cdots \end{align} $$

Proof. On the level of abstract cochain complexes and abstract abelian groups, the long exact sequence (17) can be formed by taking $\widehat {\iota }^{n}$ and $\widehat {\rho }^{n}$ to be the maps induced by the homomorphisms $\iota _n\upharpoonright \mathrm {Z}^{n}(A^{\bullet })$ and $\rho _n\upharpoonright \mathrm {Z}^{n}(B^{\bullet })$ , and setting $\partial ^{n}$ to be the map induced by an application of the classical Snake Lemma on the diagram

But then, $\widehat {\iota }^{n}$ and $\widehat {\rho }^{n}$ are both definable since they admit $\iota _n\upharpoonright \mathrm {Z}^{n}(A^{\bullet })$ and $\rho _n\upharpoonright \mathrm {Z}^{n}(B^{\bullet })$ as continuous lifts. The homomorphism $\partial ^n$ is also definable as a consequence of Lemma 2.15.

Theorem 2.14 now follows by an application of Theorem 2.16 to the short exact sequence (16).

3.1. Idealistic equivalence relations

We say that an equivalence relation E on a Polish space X is open, closed or Borel, respectively, if it is an open, closed or Borel subset of $X\times X$ endowed with the product topology. We will denote by $[x]_E$ , or simply by $[x]$ if E is understood, the E-equivalence class of x in X.

The following is a slight generalization of the classical notion of an idealistic equivalence relation; it coincides with [Reference Gao31, Definition 5.4.9] if we require that the map $\zeta :X\rightarrow X$ in Definition 3.1 is the identity map. And while our definition’s allowance for a potentially wider range of functions $\zeta $ appears indispensible to the proof of Lemma 3.6 below, for example, it remains unclear whether [Reference Gao31, Definition 5.4.9] is, in fact, equivalent to Definition 3.1. Recall that a $\sigma $ -filter of subsets of a set C is a nonempty collection $\mathcal {F} $ of nonempty subsets of C which is closed under supersets and countable intersections.

A common convention facilitates statements like the second bulleted point above: if $ \mathcal {F}$ is a filter of subsets of C and $P(x) $ is a property which elements of C may or may not have, write ‘ $\mathcal {F}x\:\:P( x) $ ’ for the assertion ‘ $\{x\in C:P( x) $ holds $\}\in \mathcal {F}$ ’. We may then rephrase the second item of Definition 3.1 as the requirement that for every Borel $A\subseteq X\times X$ , the set $\{x\in X:\mathcal {F}_{\left [ x\right ] }x^{\prime }\:\:( \zeta ( x) ,x^{\prime }) \in A\}$ is Borel.

The collection of idealistic equivalence relations enjoys several desirable closure properties. It is clear, for example, that if E is an idealistic equivalence relation on X and $ X_{0}\subseteq X$ is an E-invariant Borel subset, then $E|_{X_0}$ is idealistic.

Proof. Let the assignments $[x]\mapsto \mathcal {F}_{[x]}^{X}$ and $\zeta ^{X}:X\rightarrow X$ witness that E is idealistic, and let $[y]\mapsto \mathcal {F }_{[y]}^{Y}$ and $\zeta ^{Y}:Y\rightarrow Y$ witness that F is idealistic. Observe that an $( E\times F) $ -class $[(x,y)]$ is of the form $[x]\times [y]$ where $[x]$ is an E-class and $[y]$ is an F-class. Define $\mathcal {F}_{[(x,y)]}$ by setting $S\in \mathcal {F}_{[(x,y)]}$ if and only if $[\mathcal {F}_{[x]}^{X}x'\:\:\mathcal {F}_{[y]}^{Y}y'\:\:( x',y') \in S]$ , and observe that $\mathcal {F}_{[(x,y)]}$ is indeed a $\sigma $ -filter. Define also the function $\zeta :X\times Y\rightarrow X\times Y$ , $( x,y) \mapsto ( \zeta ^{X}( x) ,\zeta ^{Y}( y) ) $ .

For any $( x,y) \in X\times Y$ and Borel $A\subseteq ( X\times Y) \times ( X\times Y) $ , we have that $[\mathcal {F}_{ \left [ ( x,y) \right ] }( x^{\prime },y^{\prime }) \:\: ( \zeta ( x,y) ,( x^{\prime },y^{\prime }) ) \in A]$ if and only if $[\mathcal {F}_{\left [ x\right ] }^{X}x^{\prime }\:\:\mathcal {F}_{\left [ y\right ] }^{Y}y^{\prime }\:\:( ( \zeta ^{X}( x^{\prime }) ,\zeta ^{Y}( y^{\prime }) ) ,( x^{\prime },y^{\prime }) ) \in A]$ . The set

$$ \begin{align*} A_{\mathcal{F}}=\{( x,y) \in X\times Y:\mathcal{F}_{\left[ x \right] }^{X}x^{\prime }\:\:\mathcal{F}_{\left[ y\right] }^{Y}y^{\prime }\:\:( ( \zeta^{X}( x) ,\zeta^{Y}( y) ) ,( x^{\prime },y^{\prime }) ) \in A\} \end{align*} $$

is Borel by the assumption that the assignments $[x]\mapsto \mathcal {F}_{[x]}^{X}$ and $\zeta ^{X}$ and $ [y]\mapsto \mathcal {F}_{[y]}^{Y}$ and $\zeta ^{Y}$ , respectively, witness that E and $ F$ are idealistic. Hence, $[(x,y)]\mapsto \mathcal {F}_{[(x,y)]}$ and $ \zeta :X\times Y\rightarrow X\times Y$ witness that $E\times F$ is idealistic.

Proof. Suppose that the assignments $[x]\mapsto \mathcal {F}_{[x]}$ and $\zeta :Y\rightarrow Y$ witness that F is idealistic. Fix a Polish topology $\tau _Z$ on Z that is compatible with the Borel structure inherited from Y. Let C be an $F_Z$ -equivalence class. Set $\mathcal {F}_{C}^{\prime }:= \mathcal {F}_{C}$ if $C=[x]$ with $x\not \in Z$ , and let $\mathcal {F}_Z^{\prime }=\{S\subseteq Z:S$ contains a $\tau _Z$ -comeager Borel subset of $Z\}$ . Notice that for a Borel subset A of $Y\times Y$ and $x\in Y$ , we have $[\mathcal {F}_{\left [ x\right ] }^{\prime }x^{\prime }\:\:( \zeta ( x) ,x^{\prime }) \in A]$ if and only if either $x\in Z$ and $A_{\zeta (x)}=\left \{ x^{\prime }\in Y:( \zeta ( x) ,x^{\prime }) \in A\right \} $ is comeager in Z or $x\notin Z$ and $[\mathcal {F}_{\left [ x\right ] }x^{\prime }\:\:( \zeta ( x) ,x^{\prime }) \in A]$ . It then follows from the reasoning, for example, of [Reference Kechris44, Theorem 16.1] that

$$ \begin{align*} A_{\mathcal{F}^{\prime }}=\{x\in Y:\mathcal{F}_{\left[ x\right] }^{\prime }x^{\prime }\:\:( \zeta ( x) ,x^{\prime }) \in A\} \end{align*} $$

is Borel. Thus, the assignments $C\mapsto \mathcal {F}_{C}^{\prime }$ and $ \zeta $ witness that $F_Z$ is idealistic.

We now recall several notions of reduction between equivalence relations; see [Reference Gao31, Definition 5.1.1] and [Reference Ros66, Definitions 2.1 and 2.2]. Suppose that E and F are equivalence relations on Polish spaces X and Y, respectively, and let $ X/ E$ and $Y/ F$ be the set of E and F-equivalence classes on X and Y, respectively. If $f:X/ E \rightarrow Y/ F$ is a function, then a lift of f is a function $\hat {f}:X\rightarrow Y$ such that $f( [x]_{E}) =[\hat {f} ( x) ]_{F}$ for every $x\in X$ ; if some such $\hat {f}$ is, furthermore, Borel, then we say that f lifts to a Borel function $ X\rightarrow Y$ .

By the following lemma, Borel and idealistic equivalence relations are downwards closed with respect to Borel reducibility and classwise Borel embedding relations, respectively.

Proof. Suppose that F is idealistic and E classwise Borel embeds into F. Without loss of generality, we may assume E and F to be classwise Borel isomorphic. Thus, there exists a bijection $f:X/ E \rightarrow Y/ F$ such that f lifts to a Borel function $\hat {f}:X\rightarrow Y$ and $f^{-1}$ lifts to a Borel function $\hat {f}^{\ast }:X\rightarrow Y$ .

Suppose that the assignment $[y]\mapsto \mathcal {F}_{[y]}$ and the map $\tau :Y\rightarrow Y$ witness that F is idealistic. We define an assignment $[x]\mapsto \mathcal {E}_{[x]}$ from E-classes to $\sigma $ -filters as follows: for each E-class $[x]$ and subset S of $[x]$ , let S be in $\mathcal {E}_{[x]}$ if and only if $\hat {f}^{\ast -1}( S) \in \mathcal {F}_{f([x])}$ (i.e., $ \mathcal {F}_{[f(x)]}\:\:\hat {f}^{\ast }( y) \in S$ ). Notice that for a Borel subset A of $X\times X$ and $x\in X$ , we have that $[\mathcal {E} _{\left [ x\right ] }x^{\prime }\:\:( x,x^{\prime }) \in X]$ if and only if $[\mathcal {F}_{f( \left [ x\right ] ) }y^{\prime }\:\:(x, \hat {f}^{\ast }( y^{\prime }) )\in A]$ . We also define the Borel map $\zeta =\hat {f}^{\ast }\circ \tau \circ \hat {f}:X\rightarrow X$ . Notice that $\zeta (x)\,E\,x$ for every $x\in X$ .

Let A be a Borel subset of $X\times X$ . Consider the Borel subset B of $Y\times Y$ defined by $( y,y^{\prime }) \in B$ if and only if $(\hat {f}^{\ast }( y) ,\hat {f}^{\ast }( y^{\prime }) )\in A$ . Then, by assumption, we have that $\{y\in Y: \mathcal {F}_{\left [ y\right ] }y^{\prime }\:\:( \tau ( y) ,y^{\prime }) \in B\}$ is Borel. Hence, $\hat {f}^{-1}( \{y\in Y:\mathcal {F}_{\left [ y\right ] }y^{\prime }\:\:( \tau ( y) ,y^{\prime }) \in B\}) \subseteq X$ is Borel. We then have

$$ \begin{align*} x\in \hat{f}^{-1}( \{y\in Y:\mathcal{F}_{\left[ y\right] }y^{\prime }\:\:( \tau ( y) ,y^{\prime }) \in B\}) \end{align*} $$

if and only if

$$ \begin{align*} \mathcal{F}_{f( \left[ x\right] ) }y^{\prime }\:\:((\tau \circ \hat{f })( x) ,y^{\prime })\in B \end{align*} $$

if and only if

$$ \begin{align*} \mathcal{F}_{f( \left[ x\right] ) }y^{\prime }\:\:((\hat{f} ^{\ast }\circ \tau \circ \hat{f})( x) ,\hat{f}^{\ast }( y^{\prime }) )\in A \end{align*} $$

if and only if

$$ \begin{align*} \mathcal{E}_{\left[ x\right] }x^{\prime }\:\:(\zeta ( x) ,x^{\prime })\in A. \end{align*} $$

Hence, $\mathcal {E}$ and $\zeta $ witness that E is idealistic.

We now recall some results from [Reference Gao31] and [Reference Kechris and Macdonald45]. We will always assume E and F to be analytic equivalence relations on Polish spaces $X,Y$ , respectively. The following result is essentially [Reference Kechris and Macdonald45, Lemmas 3.7, 3.8], although a slightly more restrictive notion of idealistic equivalence relation is considered there; its proof readily adapts to our more generous notion.

Proof. Suppose that $[x]\mapsto \mathcal {F}_{[x]}$ and $\zeta :X\rightarrow X$ witness that E is idealistic. Suppose that $f:X/E \rightarrow Y/F $ is a Borel reduction from E to F, and let $\hat {f} :X\rightarrow Y$ be a Borel lift of f. We will show that the F-saturation

$$ \begin{align*} \lbrack \hat{f}( X) ]_{F} &=\{y\in Y:\exists x\in X\:\,y\,F\hat{f} ( x) \} \\ &=\{y\in Y:\exists x\in X\:\,y\,F\,(\hat{f}\circ \zeta )( x) \} \end{align*} $$

of $\hat {f}( X) $ is a Borel subset Z of Y and that there exists a Borel lift $\hat {f}^{*}$ of $f^{-1}:Z/F\to X/E$ .

Define the subset P of $Y\times X$ by setting $( y,x) \in P$ if and only if $y\,F\,(\hat {f}\circ \zeta )( x) $ . Then $[\hat {f}( X) ]_{F}$ is the projection of P onto the first coordinate. By the ‘large sections’ uniformization theorem figuring in the proof of [Reference Kechris and Macdonald45, Lemma 3.7], it will suffice to show that there is a map $y\mapsto \mathcal {G}_{y}$ that assigns to each $y\in \lbrack \hat {f}( X) ]_{F}$ a $\sigma $ -filter $\mathcal {G} _{y}$ of subsets of $f^{-1}( [ y]) $ such that for each Borel subset R of $Y\times X$ , there exist a $\boldsymbol {\Sigma } _{1}^{1}$ set $S\subseteq Y$ and a $\boldsymbol {\Pi }_{1}^{1}$ set $ T\subseteq Y$ such that, setting

$$ \begin{align*} R_{y}:=\left\{ x\in X:( y,x) \in R\right\} \text{,} \end{align*} $$

one has

$$ \begin{align*} R_{y}\in \mathcal{G}_{y}\Leftrightarrow y\in S\Leftrightarrow y\in T \end{align*} $$

for every $y\in \lbrack \hat {f}( X) ]_{F}$ . The uniformization given by the aforementioned theorem will then be such an $\hat {f}^{*}$ as we had desired.

To this end, define $\mathcal {G}_{y}=\mathcal {F}_{f^{-1}([y])}$ for $y\in \lbrack \hat {f}( X) ]_{F}$ . To see that the assignment $y\mapsto \mathcal {F}_{f^{-1}([y])}$ satisfies the above requirement, suppose $R\subseteq Y\times X$ is Borel. Then for $y\in \lbrack \hat {f}( X) ]_{F}$ , we have that $R_{y}\in \mathcal {F} _{f^{-1}([y])}$ if and only if $[\mathcal {F}_{f^{-1}([y])}x^{\prime }\:\:( y,x^{\prime }) \in R]$ , if and only if

$$ \begin{align*} \exists x\in X\:\big[(( f\circ \zeta ) ( x)\, F\,y)\wedge ( \mathcal{F}_{f^{-1}([y])}x^{\prime }\:\:( ( f\circ \zeta ) ( x) ,x^{\prime }) \in R)\big], \end{align*} $$

if and only if

$$ \begin{align*} \forall x\in X\:\big[(( f\circ \zeta ) ( x) \,F\,y\Rightarrow \mathcal{F}_{f^{-1}([y])}x^{\prime }\:\:( ( f\circ \zeta ) ( x) ,x^{\prime }) \in R)\big]\text{.} \end{align*} $$

This concludes the proof.

The following related result is [Reference Ros66, Proposition 2.3].

Recall that a Borel selector for a Borel equivalence relation E on a Polish space X is a function $s:X\rightarrow X$ such that $s(x)\,E\,x$ and $ x\,E\,y\Leftrightarrow s( x) =s( y) $ for every $x,y\in X$ . The principal filters that selectors determine on the classes of E witness that E is idealistic.

3.2. Definable sets

A definable set is a pair $\left ( X,E\right ) $ such that X is a Polish space and E is a Borel and idealistic equivalence relation on X. We will indulge below in the common abuse whereby the notation $X/E$ may stand either for this pair or for its associated quotient set, as context will indicate. A definable subset of a definable set $X/E$ is a definable set of the form $Y/E \subseteq X/E$ where Y is an E-invariant Borel subset of $ X $ . We regard a Polish space X, in particular, as a definable set $ X/E$ , where E is the relation of equality.

If $X/E$ and $Y/F$ are definable sets, then a definable function $X/E \rightarrow Y/F$ is a Borel homomorphism from E to F (i.e., a function $X/E \rightarrow Y/F$ that lifts to a Borel function $ X\rightarrow Y$ ). We let $\mathsf {DSet}$ be the category that has definable sets as objects and definable functions as morphisms. The identity morphism on $X/E$ is the identity function, and composition of morphisms is given by composition of functions. We now observe that this category has similar properties to the paradigmatic category $\mathsf {Set}$ of sets.

In the framework of $\mathsf {DSet}$ , for example, Corollary 3.8 takes the following form:

The following also is a consequence of Proposition 3.7.

Proposition 3.9 above, similarly, may be rephrased as a Cantor–Bernstein theorem for definable sets.

The category $\mathsf {DSet}$ has a number of further pleasant and $\mathsf {Set}$ -like properties. Observe, for example, that if the definable sets $X/E$ and $Y/F$ are countable sets, then there is a definable bijection from $ X/E$ to $Y/F$ if and only if there is a bijection from $X/E$ to $Y/F$ .

Recall that a morphism $f:x\rightarrow y$ in a category $\mathcal {C}$ is

• • monic if for any two morphisms $g_{0},g_{1}:z\rightarrow x$ , if $fg_{0}=fg_{1}$ , then $g_{0}=g_{1}$ , and

• • epic if for any two morphisms $g_{0},g_{1}:y\rightarrow z$ , if $g_{0}f=g_{1}f$ , then $g_{0}=g_{1}$ .

It is easy to see that a definable function is monic in $\mathsf {DSet}$ if and only if it is injective, and epic in $\mathsf {DSet}$ if and only if it is surjective. The initial object is $\mathsf {DSet}$ is the empty definable set, and a final object in $\mathsf {DSet}$ is a definable set with only one element.

If $X/E$ and $Y/F $ are definable sets, then the coproduct $X/E\amalg Y/F $ is the definable set obtained as the quotient of the disjoint union $X\amalg Y$ of $ X$ and Y by the equivalence relation on $X\amalg Y$ that restricts to E on X and to F on Y. Coproducts of sequences $\left ( X_{n}/ E_{n}\right ) _{n\in \omega }$ of definable sets are defined similarly.

The product of $X/E$ and $Y/F$ is the definable set obtained as the quotient of the product $X\times Y$ by the equivalence relation $E\times F$ defined as in Lemma 3.3. In our conclusion, we record the question of whether the category $\mathsf {DSet}$ possesses arbitrary countable products. Given two definable functions $f,g:X/ E \rightarrow Y/ F$ , the equalizer of f and g is the definable subset of $X/E$ given by

$$ \begin{align*} \left\{ x\in X/ E :f( x) =g( x) \right\} \text{.} \end{align*} $$

The existence of products and equalizers ensures the existence of pullbacks of definable functions $f:X/E\to Y/F$ and $f^{\prime }:X^{\prime }/E^{\prime } \rightarrow Y/F$ . These are the definable subsets of $X/E \times X^{\prime }/E^{\prime }$ consisting of pairs $(x,y) $ such that $f( x) =f^{\prime }( x^{\prime })$ .

3.3. Pointed definable sets

We will also consider pointed definable sets, which are just definable sets $X/E$ with a distinguished point (the basepoint). A basepoint-preserving definable function between pointed definable sets is simply a function that maps the basepoint to the basepoint. This defines a category $\mathsf {DSet}_{\ast }$ , which may be seen as the under category $\left \{ \ast \right \} \downarrow \mathsf {DSet}$ of $\mathsf {DSet}$ -morphisms from $\left \{ \ast \right \} $ , where $\left \{ \ast \right \} $ is a definable set with just one point. In $\mathsf {DSet}_{\ast }$ , the set $\left \{ \ast \right \} $ is a null object (i.e., it is both initial and terminal). For any two pointed definable sets, there exists a unique basepoint-preserving constant function or zero arrow. Thus, we may define the cokernel of a basepoint-preserving injective definable function $f:X/E \rightarrow Y/F$ . This is the definable set obtained as follows. We have that $f(X/E) =Z/F$ for some definable subset $Z/F$ of $Y/F$ . Define the equivalence relation $F_{Z}$ on Y in terms of Z as in Lemma 3.4. The cokernel of f is then $Y/F_{Z}$ . Intuitively, $ Y/F_{Z}$ is the definable set obtained from $Y/F$ by identifying the definable subset $Z/F$ to a single basepoint.

3.4. Definable monoids and definable groups

One may interpret definable monoids and definable groups as monoids and groups in $\mathsf {DSet}$ , as a particular instance of monoids and groups in a category in the sense of [Reference MacLane53, Section III.6]. Explicitly, a definable monoid is a pointed definable set $X/E$ together with a definable function $\mu :X/E\times X/E \rightarrow X/E$ that is an associative binary operation with the basepoint as identity element. A definable group is a definable monoid $ \left ( X/E,\mu \right ) $ that is a also a group, and such that the function $X/E \rightarrow X/E$ mapping each element to its inverse is a definable function. Evidently, definable monoids and groups form subcategories of the category of pointed definable sets. A definable group homomorphism between definable groups is a definable function that is also a group homomorphism. A definable subgroup of $ X/E$ is a simply a definable subset $Y/E$ that is also a subgroup. Notice that the kernel of a definable group homomorphism is a definable subgroup. A definable exact sequence of definable groups is a sequence

$$ \begin{align*} X^{\prime }/ E^{\prime }\overset{f}{\rightarrow }X/E \overset{g}{\rightarrow }X^{\prime \prime }/ E^{\prime \prime } \end{align*} $$

of definable groups and definable group homomorphisms such that f is injective, g is surjective, and the kernel of g is the image of f. Notice that the category of groups with a Polish cover $G/N$ is a full subcategory of the category of definable groups. We say that a definable group $X/E$ is essentially a group with a Polish cover if it definably isomorphic to a group with a Polish cover.

3.5. Semidefinable sets

In the following, we will also consider analytic equivalence relations on Polish spaces that we do not know or have not yet shown to be Borel or idealistic. We will call a quotient $X/E$ of a Polish space by an analytic equivalence relation a semidefinable set. Such pairs form the objects of a category $\mathsf {SemiDSet}$ of semidefinable sets in which morphisms are, as above, definable functions $f:X/E\to Y/F$ (i.e., functions f which lift to Borel functions $X\to Y$ ). Notice that an isomorphism in this category is a definable bijection whose inverse is also definable. By definition, the category of definable sets is a full subcategory of the category of semidefinable sets. The notions of semidefinable subset, semidefinable group and semidefinable subgroup may be defined just as in the definable case. The following is immediate from Lemma 3.6.

As in Section 3.1, the existence of a Borel selector may entail that a semidefinable object is, in fact, in $\mathsf {DSet}$ .

Proof. We let $X\times X\rightarrow X$ , $\left ( x,y\right ) \mapsto x\cdot y$ be a Borel lift of the group operation on $X/E$ . Let s be a Borel selector for F. Notice that the relation E is finer than F; hence, the Borel selector s for F induces a definable function $ \hat {s}:X/E\rightarrow X$ , $\left [ x\right ] \mapsto s( x) $ . Observe that if $T=\left \{ s( x) :x\in X\right \} $ , then $T\subseteq X$ is a Borel transversal for F (i.e., T meets each F-equivalence class in exactly one point). Notice that the map

$$ \begin{align*} Y/ E\times T &\rightarrow X/ E\\ \left( \left[ y\right] _{E},t\right) &\mapsto [y\cdot t]_{E} \end{align*} $$

is a definable bijection with definable inverse

$$ \begin{align*} X/E &\rightarrow Y/E \times T \\ \left[ x\right] _{E} &\mapsto (\left[ x\right] _{E}\cdot \left[ s( x) \right] _{E}^{-1},s(x) )\text{.} \end{align*} $$

This shows that the semidefinable set $X/ E$ is isomorphic in $ \mathsf {SemiDSet}$ to $Y/E \times T$ . If $Y/E$ is Borel or idealistic, respectively, then $Y/E\times T$ is Borel or idealistic, respectively; see Lemma 3.3. The conclusion thus follows from Lemma 3.6.

4.1. Homotopy

Let $\mathsf {LC}$ denote the category of locally compact Polish spaces and continuous functions; we will often term the latter maps, simply. For any two locally compact spaces X and Y, we endow the hom-set $\mathsf {LC}( X,Y)$ with the compact-open topology. This is the topology with subbasis the collection of all sets of the form $\{f\in \mathsf {LC}(X,Y):f(K) \subseteq U\} $ for some compact subset $K\subseteq X$ and open subset $U\subseteq Y$ . With this topology, each such $\mathsf {LC}( X,Y) $ is a Polish space, and what is more, the composition functions $\mathsf {LC}( Y,Z) \times \mathsf {LC}( X,Y) \rightarrow \mathsf {LC}(X,Z)$ are all continuous. In this way, we regard $\mathsf {LC}$ as a Polish category, by which we mean a category enriched over the category of Polish spaces [Reference Kelly46]. We denote by $\mathsf {C}$ the full subcategory of $\mathsf {LC}$ consisting of compact spaces.

Let $\mathcal {C}$ be a category with small (i.e., set-sized) hom-sets. A congruence $\equiv $ on $\mathcal {C}$ [Reference MacLane53, Section I.8] is given by an assignment to each pair of objects $x,y$ of $\mathcal {C}$ of an equivalence relation $\equiv _{x,y}$ on $\mathcal {C}( x,y) $ such that for objects $x,y,z$ and morphisms $f_{1},f_{2}:x\rightarrow y$ and $ g_{1},g_{2}:y\rightarrow z$ in $\mathcal {C}$ , if $f_{1}\equiv _{x,y}f_{2}$ and $g_{1}\equiv _{y,z}g_{2}$ , then $g_{1}f_{1} \equiv _{x,z} g_{2}f_{2} $ . One may regard the pair $\left ( \mathcal {C},\equiv \right ) $ as a (strict) $2$ -category by declaring that for objects $ x,y$ and morphisms $f,g:x\rightarrow y$ in $\mathcal {C}$ , there exists a unique $2$ -cell $f\Rightarrow g$ if and only if $f\equiv _{x,y}g$ . The corresponding quotient category $\mathcal {C}/\!\equiv $ is the category whose objects are those of $\mathcal {C} $ and whose morphisms from x to y are the quotient set $\mathcal {C}(x,y)/\!\equiv _{x,y}$ .

Let X and Y be topological spaces; henceforth, we will let I denote the closed unit interval $[0,1]$ . A homotopy $h:f\Rightarrow g:X\rightarrow Y$ from the map $f:X\to Y$ to the map $g:X\to Y$ is a function $h:X\times I\rightarrow Y$ such that $h\left ( -,0\right ) =f$ and $ h\left ( -,1\right ) =g$ . The homotopy relation $\simeq _{\mathsf {LC}}$ on $\mathsf {LC}(X,Y)$ is defined by setting $ f\simeq _{\mathsf {LC}}g$ if and only if there exists a homotopy $ h:f\Rightarrow g:X\rightarrow Y$ ; evidently, $\simeq _{\mathsf {LC}}$ defines a congruence on $\mathsf {LC}$ . We let $\mathsf {Ho}( \mathsf {LC}) $ denote the corresponding quotient category, which may be regarded as a category enriched over the category $\mathsf {SemiDSet}$ ; $[X,Y] $ will denote the semidefinable set of homotopy classes of maps $X\rightarrow Y$ or, equivalently, of the path-components of $\mathsf {LC}(X,Y)$ . A homotopy equivalence $f\in \mathsf {LC}(X,Y)$ is a map whose image in the quotient category $\mathsf {Ho}(\mathsf {LC})$ is an isomorphism.

More generally, we consider locally compact pairs; these are pairs of topological spaces $(X,A)$ in which X is locally compact and Polish and A is a closed subspace of X. These form the objects of the Polish category $\mathsf {LCP}$ of locally compact pairs. Its morphisms $(X,A)\to (Y,B)$ are those maps $f:X\rightarrow Y$ such that $f(A) \subseteq B$ . The hom-sets $\mathsf {LCP}((X,A),(Y,B)) $ then form $G_{\delta }$ subsets of $\mathsf {LC}(X,Y) $ , and their subspace topologies are consequently Polish. Note also that the identification $X\mapsto (X,\varnothing )$ embeds $\mathsf {LC}$ as a full subcategory of $\mathsf {LCP}$ . We write $f:(X,A) \rightarrow (Y,B) $ to indicate that $f\in \mathsf {LCP}(( X,A),(Y,B))$ . We let $\mathsf {CP}$ denote the full subcategory of $\mathsf {LCP}$ consisting of compact pairs (i.e., of locally compact pairs $(X,A)$ in which X is compact). Given maps $f,g:( X,A) \rightarrow (Y,B) $ a homotopy – sometimes termed pair homotopy – $h:f\Rightarrow g:( X,A) \rightarrow (Y,B)$ from f to g is a homotopy $h:f\Rightarrow g:X\rightarrow Y$ such that $h( -,t) :( X,A) \rightarrow (Y,B) $ for every $t\in I$ . This defines a congruence relation $\simeq _{\mathsf {LCP}}$ on $\mathsf {LCP}$ . We let Ho $(\mathsf {LCP}) $ denote the corresponding quotient category; note that Ho $(\mathsf {LC})$ and Ho $(\mathsf {LCP})$ are the homotopy categories associated to the Hurewicz model category structures on $\mathsf {LC}$ and $\mathsf {LCP}$ , respectively (see [Reference Dwyer and Spaliński24]). We let $\left [(X,A),(Y,B) \right ] $ be the semidefinable set of homotopy classes of maps $(X,A) \rightarrow (Y,B) $ . Just as above, a homotopy equivalence $f\in \mathsf {LCP}((X,A),(Y,B))$ is a map whose image in the quotient category $\mathsf {Ho}(\mathsf {LCP})$ is an isomorphism.

A pointed locally compact Polish space is a locally compact pair $(X,A) $ in which A is a singleton $\{*\}$ ; we call $*$ the basepoint of this space. Such spaces comprise a full subcategory $\mathsf {LC}_{\ast }$ of the category $\mathsf {LCP}$ of locally compact pairs. The morphisms in $\mathsf {LC}_{\ast }$ are maps which are basepoint-preserving (i.e., they are exactly those maps which send basepoints to basepoints). Similarly, one defines the notion of basepoint-preserving homotopy, and the corresponding congruence relation $\simeq _{\mathsf {LC}_{\ast }}$ on $\mathsf {LC}_{\ast }$ (which is simply the restriction of $\simeq _{\mathsf {LCP}}$ to the full subcategory $\mathsf {LC}_{\ast }$ ).