Definitions for On the existence of bibundles

QAlgorithms.Papers.Independent.Intakethird100.P93.Bibundle

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$; require also continuous right inverses for $G\to G/\ker t$ and $H\to H/t(G)$ on open neighborhoods of every target point. For topological spaces $M,P$, a bibundle structure consists of a jointly continuous right action $p\cdot g$ of $G$ on $P$, satisfying $p\cdot1=p$ and $(p\cdot g)\cdot k=p\cdot(gk)$, a continuous map $\pi:P\to M$ with $\pi(p\cdot g)=\pi(p)$, and local trivializations as follows: for every $m\in M$, some open neighborhood $U$ of $m$ admits a homeomorphism $e_U:\pi^{-1}(U)\to U\times G$ whose first coordinate has underlying point $\pi(p)$ and for which $e_U(p\cdot g)=((e_U(p))_1,(e_U(p))_2g)$ whenever $p\in\pi^{-1}(U)$, $g\in G$, and $p\cdot g$ is regarded as an element of $\pi^{-1}(U)$. The structure also contains a continuous map $\psi:P\to H$ satisfying $\psi(p\cdot g)=\psi(p)t(g)$ for every $p\in P$ and $g\in G$. Empty base and total spaces $M=P=\varnothing$ are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P93.Bispace

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$; require also that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has, around every point of its target, an open neighborhood $U$ and a continuous section $s$ satisfying $q(s(z))=z$ for all $z\in U$. For a topological space $X$ in that universe, this structure consists of a jointly continuous operation $x\cdot g$ satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$, a requirement that $X$ is nonempty and that for every $x\in X$ there is a homeomorphism $e:G\to X$ with $e(g)=x\cdot g$ for all $g$, and a continuous map $\psi:X\to H$ satisfying

$$\psi(x\cdot g)=\psi(x)\,t(g) \quad(x\in X,\ g\in G).$$

QAlgorithms.Papers.Independent.Intakethird100.P93.BispaceIso

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$; require also that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has a continuous section on an open neighborhood of every target point, with the quotient map composed with that section equal to the identity. Let $X$ and $Y$ be topological spaces in that universe. Suppose each space $Z\in\{X,Y\}$ is equipped with a jointly continuous operation $z\cdot_Z g$ satisfying $z\cdot_Z1_G=z$ and $(z\cdot_Z g)\cdot_Z k=z\cdot_Z(gk)$, is nonempty, has for every $z\in Z$ a homeomorphism $e:G\to Z$ satisfying $e(g)=z\cdot_Z g$, and has a continuous map $\psi_Z:Z\to H$ satisfying $\psi_Z(z\cdot_Z g)=\psi_Z(z)t(g)$. This structure consists of a homeomorphism $F:X\to Y$ together with the requirements

$$F(x\cdot_X g)=F(x)\cdot_Y g \quad(x\in X,\ g\in G),\qquad \psi_Y(F(x))=\psi_X(x) \quad(x\in X).$$

QAlgorithms.Papers.Independent.Intakethird100.P93.CrossedModule

For topological groups $G$ and $H$ whose underlying types lie in the same universe, this structure consists of a continuous group homomorphism $t:G\to H$ and a group homomorphism $\alpha:H\to\operatorname{Aut}(G)$ into the group of multiplicative automorphisms of $G$, such that the map $(h,g)\mapsto\alpha_h(g)$ from $H\times G$ to $G$ is continuous and

$$t(\alpha_h(g))=h\,t(g)\,h^{-1} \quad(h\in H,\ g\in G),\qquad \alpha_{t(g)}(k)=gkg^{-1} \quad(g,k\in G).$$

It also includes the following requirement for both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$, where $\ker t=\{g:t(g)=1_H\}$ and $t(G)=\{t(g):g\in G\}$: every point $b$ of the respective quotient has an open neighborhood $U$ and a continuous map $s$ from $U$ to the source group such that the quotient map sends $s(z)$ to $z$ for every $z\in U$. Trivial groups are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P93.DualTensor

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=h\,t(g)\,h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$. Assume that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has a continuous right inverse on an open neighborhood of every quotient point. Let $X$ be a topological space and $B$ consist of a continuous right $G$-action $x\cdot g$, with $X\ne\varnothing$ and every orbit map $g\mapsto x\cdot g$ a homeomorphism $G\to X$, together with a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. This definition gives the quotient of $X\times X$ by the equivalence relation generated by

$$(x_1,x_2)\sim(y_1,y_2)\quad\text{whenever some }k\in G \text{ satisfies }y_1=x_1\cdot k\text{ and }y_2=x_2\cdot k.$$

QAlgorithms.Papers.Independent.Intakethird100.P93.ExtensionSpace

Let $G$ be a group equipped with a topology, let $X$ be a topological space, and let $x\cdot g$ be a continuous right action, satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$. For any subgroup $K\le G$ and any subset $S\subseteq X$, this definition gives the quotient of $S\times G$ by the equivalence relation generated by

$$(x,g)\sim(y,h)\quad\text{whenever there exists }k\in K \text{ with }y=x\cdot k\text{ and }h=k^{-1}g.$$

Here $x,y$ are required to lie in $S$, but no assumption that $S$ is preserved by the $K$-action is imposed. The subset $S$ may be empty, in which case the quotient is empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.HasLocalSections

For types $A$ and $B$ in the same universe, each equipped with a topology, and any function $f:A\to B$, this property means that every $b\in B$ has an open neighborhood $U\subseteq B$ and a continuous map $s:U\to A$, where $U$ has the subspace topology, such that $f(s(z))=z$ for every $z\in U$. There is no continuity hypothesis on $f$. If $B$ is empty, the property holds vacuously; if $B$ is nonempty, the property requires $A$ to be nonempty.

QAlgorithms.Papers.Independent.Intakethird100.P93.HasUnitType

Let $G,H$ be topological groups and $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$, with continuous local right inverses for both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ around every target point. Let $M,P$ be topological spaces and $B$ consist of a jointly continuous right $G$-action $p\cdot g$ on $P$, satisfying $p\cdot1=p$ and $(p\cdot g)\cdot k=p\cdot(gk)$, a continuous invariant projection $\pi:P\to M$, and a continuous map $\psi:P\to H$ with $\psi(p\cdot g)=\psi(p)t(g)$. Assume that every $m\in M$ has an open neighborhood $U$ with a homeomorphism $\pi^{-1}(U)\to U\times G$ preserving the base coordinate and taking the action to right multiplication on the $G$-coordinate. Then $B$ has unit type, by definition, precisely when

$$\forall p\in P,\qquad \psi(p)t(G)=t(G)\quad\text{in }H/t(G).$$

This condition is vacuous if $P$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.IotaIsomorphic

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=ht(g)h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$ for all $h,g,k$. Assume that both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ admit continuous sections near every target point, on open neighbourhoods. Put $K=\ker t$. Let $M,Q,R$ be topological spaces, and let $B,D$ be bundles on $Q,R$, respectively, each consisting of a continuous right $K$-action and a continuous invariant projection to $M$, denoted $\pi_B,\pi_D$. Each projection is required, near every base point on an open set $U$, to admit a homeomorphism of its inverse image with $U\times K$ preserving the base coordinate and carrying the action to right multiplication on $K$. Define $I_B$ and $I_D$ to be the quotients of $Q\times G$ and $R\times G$ generated respectively by $(q,g)\sim(q\cdot_B k,k^{-1}g)$ and $(r,g)\sim(r\cdot_D k,k^{-1}g)$, for $k\in K$, and write their classes as $[q,g]_B$ and $[r,g]_D$. The defined proposition says that there exists a homeomorphism $e:I_B\to I_D$ such that, for every $q\in Q$, $r\in R$, and $g,h\in G$, the equality $e([q,g]_B)=[r,h]_D$ implies both $\pi_B(q)=\pi_D(r)$ and $t(g)=t(h)$, and such that for every additional $a\in G$ the same equality implies $e([q,ga]_B)=[r,ha]_D$. Empty bases and the resulting empty total spaces are included.

QAlgorithms.Papers.Independent.Intakethird100.P93.IotaSpace

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ such that evaluation is continuous, $t(\alpha(h)(g))=ht(g)h^{-1}$, and $\alpha(t(g))(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$. Assume that each of $G\to G/\ker t$ and $H\to H/t(G)$ admits a continuous section over an open neighbourhood of every target point. Put $K=\ker t$, with its subgroup topology. Let $M,Q$ be topological spaces, and let $R$ consist of a continuous right $K$-action on $Q$ and a continuous map $\pi_R:Q\to M$ invariant under that action, such that every $m\in M$ has an open neighbourhood $U$ and a homeomorphism $\pi_R^{-1}(U)\to U\times K$ preserving the base coordinate and identifying the action with right multiplication on the second coordinate. The defined space is the quotient of $Q\times G$ by the equivalence relation generated by the identifications $(q,g)\sim(q\cdot k,k^{-1}g)$ for every $q\in Q$, $g\in G$, and $k\in K$, where $k^{-1}g$ uses the underlying element of $G$. Empty $M$ is allowed and then forces $Q$, and this quotient, to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.IotaToBibundleIsomorphic

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=ht(g)h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$ for all $h,g,k$. Assume that $G\to G/\ker t$ and $H\to H/t(G)$ each admit a continuous section over an open neighbourhood of every target point. Put $K=\ker t$. Let $M,Q,P$ be topological spaces. Let $R$ consist of a continuous right $K$-action on $Q$ and a continuous invariant projection $\pi_R:Q\to M$; let $B$ consist of a continuous right $G$-action on $P$, a continuous invariant projection $\pi_B:P\to M$, and a continuous map $\psi:P\to H$ satisfying $\psi(p\cdot_B g)=\psi(p)t(g)$. For each projection and every base point, assume an open neighbourhood $U$ on which its inverse image is homeomorphic to $U\times K$, respectively $U\times G$, preserving the base coordinate and carrying the action to right multiplication on the group coordinate. Let $I_R$ be the quotient of $Q\times G$ generated by $(q,g)\sim(q\cdot_R k,k^{-1}g)$ for $k\in K$, and write $[q,g]$ for its classes. The defined proposition says that there exists a homeomorphism $e:I_R\to P$ such that $\pi_B(e([q,g]))=\pi_R(q)$, $e([q,gk])=e([q,g])\cdot_B k$, and $\psi(e([q,g]))=t(g)$ for every $q\in Q$ and $g,k\in G$. Empty bases, and hence empty total spaces, are allowed.

QAlgorithms.Papers.Independent.Intakethird100.P93.IsCrossedSubmodule

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ whose evaluation is continuous, satisfying $t(\alpha(h)(g))=h\,t(g)\,h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$. Assume also that the quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ each admit a continuous right inverse on an open neighborhood of every quotient point. For arbitrary subgroups $K\le G$ and $L\le H$, the defined property is precisely the conjunction that $t(k)\in L$ for every $k\in K$, and that $\alpha(h)(k)\in K$ for every $h\in L$ and every $k\in K$. No closedness assumption is imposed on either subgroup.

QAlgorithms.Papers.Independent.Intakethird100.P93.IsInducedGBispace

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for every $h\in H$ and $g,k\in G$; require also that both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ admit continuous right-inverse sections on open neighborhoods of every point of their respective targets. Let $X$ be a topological space in that universe, equipped with a jointly continuous right operation $x\cdot g$ satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$, with $X$ nonempty and every map $g\mapsto x\cdot g$ a homeomorphism $G\to X$, and with a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. Put $L_g(x)=x\cdot\alpha_{\psi(x)}^{-1}(g)$. This property is the conjunction of the following requirements: $X$ is nonempty and, for every $x\in X$, there exists a homeomorphism $e:G\to X$ with $e(g)=x\cdot g$ for all $g$; the map $(g,x)\mapsto L_g(x)$ is continuous; $L_{1_G}(x)=x$ for every $x$; $L_{gk}(x)=L_g(L_k(x))$ for all $g,k,x$; for every $x$, there exists a homeomorphism $e:G\to X$ with $e(g)=L_g(x)$ for every $g$; and

$$L_g(x\cdot k)=L_g(x)\cdot k \quad(g,k\in G,\ x\in X),\qquad x\cdot g=L_{\alpha_{\psi(x)}(g)}(x) \quad(x\in X,\ g\in G).$$

QAlgorithms.Papers.Independent.Intakethird100.P93.IsReduction

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=h\,t(g)\,h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$. Assume that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has a continuous right inverse on some open neighborhood of each quotient point. Let $X$ be a topological space and $B$ consist of a continuous right $G$-action $x\cdot g$, with $X\ne\varnothing$ and every orbit map $g\mapsto x\cdot g$ a homeomorphism $G\to X$, and a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. For subgroups $K\le G$, $L\le H$, and an arbitrary subset $S\subseteq X$, the defined property means that there exists $x\in X$ such that $S=\{x\cdot k:k\in K\}$, and that $\psi(y)\in L$ for every $y\in S$. In particular, this property requires $S$ to be nonempty, since $1_G\in K$.

QAlgorithms.Papers.Independent.Intakethird100.P93.IsRightTorsor

Let $G$ be a group equipped with a topology, let $X$ be a topological space in the same universe, and let $R$ be a jointly continuous operation $x\cdot g$ satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$ for all $x\in X$ and $g,k\in G$. This property means that $X$ is nonempty and that, for every $x\in X$, there exists a homeomorphism $e:G\to X$ satisfying $e(g)=x\cdot g$ for every $g\in G$. Thus the nonemptiness condition is explicitly required even though the condition quantified over $x$ would by itself be vacuous for an empty space.

QAlgorithms.Papers.Independent.Intakethird100.P93.IsTrivialBibundle

Let $G,H$ be topological groups and $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for every $h\in H$ and $g,k\in G$; assume the quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ have continuous right inverses on open neighborhoods of every target point. Let $M,P$ be topological spaces, and let $B$ provide a jointly continuous right action $p\cdot g$ satisfying the identity and action laws, a continuous projection $\pi:P\to M$ with $\pi(p\cdot g)=\pi(p)$, local homeomorphisms $\pi^{-1}(U)\to U\times G$ over open neighborhoods $U$ of every base point that preserve the base coordinate and take the action to right multiplication on $G$, and a continuous map $\psi:P\to H$ satisfying $\psi(p\cdot g)=\psi(p)t(g)$. The predicate that $B$ is trivial means that there exists a homeomorphism $e:P\to M\times G$ such that, for every $p\in P$ and $g\in G$,

$$(e(p))_1=\pi(p),\qquad e(p\cdot g)=\bigl((e(p))_1,(e(p))_2g\bigr),\qquad t((e(p))_2)=\psi(p).$$

The definition includes the case $M=P=\varnothing$.

QAlgorithms.Papers.Independent.Intakethird100.P93.IsTrivialIota

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation $(h,g)\mapsto\alpha_h(g)$, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for every $h\in H$ and $g,k\in G$. Assume also that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has, around every point of its target, an open neighborhood $U$ and a continuous map $s:U\to G$, respectively $s:U\to H$, whose quotient is the inclusion of $U$. Put $K=\ker t=\{g\in G:t(g)=1\}$. Let $M,Q$ be arbitrary topological spaces and $R$ a principal $K$-bundle: it has a continuous right action $q\cdot k$, satisfying $q\cdot1=q$ and $(q\cdot k)\cdot l=q\cdot(kl)$, and a continuous projection $\pi:Q\to M$ satisfying $\pi(q\cdot k)=\pi(q)$; every $m\in M$ has an open neighborhood $U$ with a homeomorphism $\pi^{-1}(U)\cong U\times K$ preserving the first coordinate $\pi$ and carrying the action to $(u,a)\cdot k=(u,ak)$. Form the quotient space $E$ of $Q\times G$ by the identifications $(q,g)\sim(q\cdot k,k^{-1}g)$ for $k\in K$, and denote its classes by $[q,g]$. The property $\operatorname{IsTrivialIota}(C,R)$ means that there exists a homeomorphism $e:E\to M\times G$ such that, for every $q\in Q$ and $g,a\in G$,

$$\operatorname{pr}_M e([q,g])=\pi(q),\qquad e([q,ga])=\bigl(\operatorname{pr}_M e([q,g]),\operatorname{pr}_G e([q,g])\,a\bigr),\qquad t\bigl(\operatorname{pr}_G e([q,g])\bigr)=t(g).$$

No nonemptiness assumption is imposed on $M$ or $Q$; if $M$ is empty, the projection forces $Q$, and hence $E$, to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.IsTrivialPrincipal

Let $K$ be a group equipped with a topology, and let $M,P$ be topological spaces; continuity of the group operations on $K$ is not assumed. Let $R$ consist of a continuous right $K$-action on $P$ and a continuous invariant projection $\pi:P\to M$, such that every $m\in M$ has an open neighbourhood $U$ and a homeomorphism $\pi^{-1}(U)\to U\times K$ preserving the base coordinate and carrying the action to right multiplication on $K$. The defined proposition says that there exists a homeomorphism $e:P\to M\times K$ such that its first coordinate satisfies $e_1(p)=\pi(p)$ for every $p\in P$, and $e(p\cdot k)=(e_1(p),e_2(p)k)$ for every $p\in P$ and $k\in K$. Empty $M$ is included; in that case $P$ must be empty and the proposition holds.

QAlgorithms.Papers.Independent.Intakethird100.P93.IsTrivialUnitLevel

Let $G,H$ be topological groups and $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$, with continuous right inverses for $G\to G/\ker t$ and $H\to H/t(G)$ on open neighborhoods of every target point. Let $M,P$ be topological spaces, and let $B$ consist of a jointly continuous right $G$-action $p\cdot g$ satisfying the identity and action laws, a continuous invariant projection $\pi:P\to M$ locally homeomorphic to $U\times G$ over an open neighborhood $U$ of every base point, with those homeomorphisms preserving the base and taking the action to right multiplication on $G$, and a continuous map $\psi:P\to H$ satisfying $\psi(p\cdot g)=\psi(p)t(g)$. Put $P_1=\{p\in P:\psi(p)=1\}$ and $K=\ker t$, both with their subspace topologies. The predicate that the unit level is trivial means that there exists a homeomorphism $e:P_1\to M\times K$ such that $(e(p))_1=\pi(p)$ for every $p\in P_1$, and such that for every $p\in P_1$, every $k\in K$, and every witness of $\psi(p\cdot k)=1$,

$$e(p\cdot k)=\bigl((e(p))_1,(e(p))_2k\bigr),$$

where $p\cdot k$ is regarded as a point of $P_1$ using that witness and the product in the second coordinate is multiplication in $K$. The definition allows $M=P=\varnothing$.

QAlgorithms.Papers.Independent.Intakethird100.P93.LeftUnitTensor

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=h\,t(g)\,h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$. Assume that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has a continuous right inverse on an open neighborhood of every quotient point. Let $X$ be a topological space and $B$ consist of a continuous right $G$-action $x\cdot g$, with $X\ne\varnothing$ and every orbit map $g\mapsto x\cdot g$ a homeomorphism $G\to X$, together with a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. This definition gives the quotient of $G\times X$ by the equivalence relation generated by

$$(g,x)\sim(h,y)\quad\text{whenever some }k\in G \text{ satisfies }h=gk \text{ and }y=x\cdot\alpha(\psi(x))^{-1}(k^{-1}),$$

where $\alpha(\psi(x))^{-1}$ denotes the inverse automorphism of $G$.

QAlgorithms.Papers.Independent.Intakethird100.P93.LocalCocycleData

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h,g,k$. Require each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ to admit, around every target point, a continuous section on an open neighborhood whose quotient is the inclusion. Let $M,P$ be arbitrary topological spaces, and let $B$ comprise a continuous right $G$-action on $P$ satisfying the identity and multiplication laws, a continuous invariant projection $\pi:P\to M$ locally equipped around every base point with projection-preserving, equivariant homeomorphisms $\pi^{-1}(U)\cong U\times G$ for right multiplication on the second factor, and a continuous map $\psi:P\to H$ with $\psi(p\cdot g)=\psi(p)t(g)$. A value of $\operatorname{LocalCocycleData}(C,B)$ consists of an index type $I$ in the same universe, open subsets $U_i\subseteq M$ covering $M$, continuous maps $s_i:U_i\to P$, continuous maps $g_{ij}:U_i\cap U_j\to G$ for every ordered pair $i,j$, and continuous maps $h_i:U_i\to H$. They must satisfy $\pi(s_i(m))=m$ and $h_i(m)=\psi(s_i(m))$ for every $i$ and $m\in U_i$, together with

$$s_j(m)=s_i(m)\cdot g_{ij}(m),\qquad h_j(m)=h_i(m)\,t(g_{ij}(m)) \quad(m\in U_i\cap U_j), \qquad g_{ij}(m)g_{jk}(m)=g_{ik}(m) \quad(m\in U_i\cap U_j\cap U_k).$$

These requirements apply to all indices, including repeated ones. Individual open sets and intersections may be empty, in which case the corresponding pointwise requirements are vacuous; the index type itself may be empty only if $M$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.OrbitSpace

Let $G$ be a group equipped with a topology, let $X$ be a topological space, and let $R$ specify a continuous map $X\times G\to X$, written $x\cdot g$, satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$ for every $x\in X$ and $g,k\in G$. For any subgroup $K\le G$, the definition gives the quotient of $X$ by the equivalence relation generated by the pairs $x,y$ for which there exists $k\in K$ with $y=x\cdot k$. Thus it identifies points belonging to the same $K$-orbit. No nonemptiness or torsor assumption is required of $X$; if $X$ is empty, this quotient is empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.PrincipalBundle

For a group $G$ equipped with a topology and topological spaces $M,P$, all in the same universe, a principal-bundle structure consists of a jointly continuous map $P\times G\to P$, written $p\cdot g$, satisfying $p\cdot1=p$ and $(p\cdot g)\cdot k=p\cdot(gk)$ for every $p\in P$ and $g,k\in G$; a continuous map $\pi:P\to M$ satisfying $\pi(p\cdot g)=\pi(p)$; and the following local condition. For every $m\in M$, there exist an open subset $U\subseteq M$ containing $m$ and a homeomorphism $e:\pi^{-1}(U)\to U\times G$ such that the underlying point of the first coordinate of $e(p)$ is $\pi(p)$ for every $p\in\pi^{-1}(U)$, and, for every such $p$, every $g\in G$, and every witness that $\pi(p\cdot g)\in U$,

$$e(p\cdot g)=\bigl((e(p))_1,(e(p))_2g\bigr),$$

where $p\cdot g$ on the left is regarded as a point of $\pi^{-1}(U)$ using that witness. The definition allows $M=P=\varnothing$, when the local condition is vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P93.PrincipalHom

Let $K$ be a group equipped with a topology, and let $M,Q,R$ be topological spaces; no assumption that the group operations on $K$ are continuous is imposed. Let $B,D$ consist respectively of continuous right $K$-actions on $Q,R$ and continuous action-invariant projections $\pi_B:Q\to M$, $\pi_D:R\to M$. Assume that every base point has an open neighbourhood $U$ on which each projection admits a homeomorphism of its inverse image with $U\times K$, preserving the base coordinate and carrying the action to right multiplication on the second coordinate. The defined structure consists of a continuous map $f:Q\to R$, together with the requirements $\pi_D(f(q))=\pi_B(q)$ for every $q\in Q$, and $f(q\cdot_B k)=f(q)\cdot_D k$ for every $q\in Q$ and $k\in K$. Empty bases are allowed and force both total spaces to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.PrincipalIsomorphic

Let $K$ be a group equipped with a topology, and let $M,Q,R$ be topological spaces; continuity of the group operations on $K$ is not an additional assumption. Let $B$ on $Q$ and $D$ on $R$ each consist of a continuous right $K$-action and a continuous action-invariant projection, respectively $\pi_B:Q\to M$ and $\pi_D:R\to M$. For each of these projections, assume that every $m\in M$ has an open neighbourhood $U$ on which the inverse image is homeomorphic to $U\times K$, with first coordinate the projection and with the action carried to right multiplication on the second coordinate. The defined proposition says that there exists a homeomorphism $e:Q\to R$ such that $\pi_D(e(q))=\pi_B(q)$ for every $q\in Q$, and $e(q\cdot_B k)=e(q)\cdot_D k$ for every $q\in Q$ and $k\in K$. Empty bases are allowed; if $M$ is empty, both total spaces must be empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.PullbackSpace

Let $G,H$ be topological groups and $C=(t,\alpha)$, where $t:G\to H$ is a continuous homomorphism and $\alpha:H\to\operatorname{Aut}(G)$ is a homomorphism with continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h,g,k$. Assume that both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ admit continuous sections on open neighborhoods of every target point, with quotient equal to inclusion. Let $M,N,P$ be arbitrary topological spaces. Let $B$ consist of a continuous right $G$-action on $P$ satisfying the identity and multiplication laws, a continuous invariant projection $\pi:P\to N$ with projection-preserving, equivariant local homeomorphisms $\pi^{-1}(U)\cong U\times G$ around every point of $N$, where $G$ acts by right multiplication on the second factor, and a continuous map $\psi:P\to H$ satisfying $\psi(p\cdot g)=\psi(p)t(g)$. For any continuous map $f:M\to N$, $\operatorname{PullbackSpace}(C,B,f)$ is the subtype

$$P_f=\{(m,p)\in M\times P:f(m)=\pi(p)\},$$

with the topology inherited from $M\times P$. Its elements are precisely pairs satisfying the displayed equality. Empty spaces are permitted; in particular, $P_f$ is empty when $M$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.PullbacksIsomorphic

Let $G,H$ be topological groups, with a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ having continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h,g,k$. Assume that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ admits a continuous section on an open neighborhood of every target point, composing with the quotient to the inclusion. Let $M,N,P$ be arbitrary topological spaces. Suppose $B$ comprises a continuous right $G$-action on $P$ satisfying the identity and multiplication laws, a continuous invariant projection $\pi:P\to N$ locally admitting projection-preserving, equivariant homeomorphisms $\pi^{-1}(U)\cong U\times G$ around every point of $N$, with right multiplication on the second factor, and a continuous map $\psi:P\to H$ satisfying $\psi(p\cdot k)=\psi(p)t(k)$. For continuous maps $f,g:M\to N$, put $P_f=\{(m,p):f(m)=\pi(p)\}$ and $P_g=\{(m,p):g(m)=\pi(p)\}$, each with its subspace topology. The property $\operatorname{PullbacksIsomorphic}(C,B,f,g)$ means that there exists a homeomorphism $e:P_f\to P_g$ preserving the $M$-coordinate and the value of $\psi$, and satisfying the following equivariance requirement: for every $(m,p)\in P_f$, every $k\in G$, and the equality $f(m)=\pi(p\cdot k)$ required to regard $(m,p\cdot k)$ as an element of $P_f$, writing $e(m,p)=(m,p')$, one has

$$\psi(p')=\psi(p),\qquad e(m,p\cdot k)=(m,p'\cdot k).$$

The membership equality holds automatically by invariance of $\pi$. Empty pullback spaces are allowed; if $M$ is empty, both are empty and all the pointwise requirements are vacuous.

QAlgorithms.Papers.Independent.Intakethird100.P93.RightAction

For types $G$ and $X$ in the same universe, with $G$ a group equipped with a topology and $X$ a topological space, this structure consists of a function $(x,g)\mapsto x\cdot g$ from $X\times G$ to $X$ that is jointly continuous and satisfies

$$x\cdot 1_G=x,\qquad (x\cdot g)\cdot k=x\cdot(gk) \quad(x\in X,\ g,k\in G).$$

The space $X$ may be empty, in which case these identities hold vacuously.

QAlgorithms.Papers.Independent.Intakethird100.P93.RightUnitTensor

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=h\,t(g)\,h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$. Assume that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has a continuous right inverse on an open neighborhood of every quotient point. Let $X$ be a topological space and $B$ consist of a continuous right $G$-action $x\cdot g$, with $X\ne\varnothing$ and every orbit map $g\mapsto x\cdot g$ a homeomorphism $G\to X$, together with a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. This definition gives the quotient of $X\times G$ by the equivalence relation generated by

$$(x,g)\sim(y,h)\quad\text{whenever some }k\in G \text{ satisfies }y=x\cdot k\text{ and }h=k^{-1}g.$$

QAlgorithms.Papers.Independent.Intakethird100.P93.TypeGroup

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for every $h\in H$ and $g,k\in G$. Assume also that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ has a continuous section on an open neighborhood of every point of its target, meaning $q\circ s$ is the identity on that neighborhood. The abbreviation denotes the quotient

$$H/t(G),\qquad t(G)=\{t(g):g\in G\}.$$

In particular, it is a singleton when $t$ is surjective.

QAlgorithms.Papers.Independent.Intakethird100.P93.TypePullback

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$. Assume both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ have continuous right inverses on open neighborhoods of every target point. For any topological space $M$ and continuous map $\phi:M\to H/t(G)$, the type pullback is the subtype

$$\{(m,h)\in M\times H:\phi(m)=h\,t(G)\}.$$

Thus its elements are pairs consisting of $m\in M$ and $h\in H$ together with a witness that $\phi(m)$ equals the quotient class of $h$; it is empty when $M$ is empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.bispaceType

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$; both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ are required to admit continuous sections on open neighborhoods of every target point, with $q\circ s$ equal to the identity there. Let $X$ be a topological space in that universe, equipped with a jointly continuous operation $x\cdot g$ satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$, with $X$ nonempty and every orbit map $g\mapsto x\cdot g$ a homeomorphism $G\to X$, and with a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. This definition uses classical choice on the supplied nonemptiness of $X$ to select a point $x_0\in X$, and returns

$$[\psi(x_0)]\in H/t(G),\qquad t(G)=\{t(g):g\in G\}.$$

QAlgorithms.Papers.Independent.Intakethird100.P93.extensionClass

Let $G$ be a group with a topology, let $X$ be a topological space, and let $x\cdot g$ be a continuous right action satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$. Given a subgroup $K\le G$, a subset $S\subseteq X$, a point $x\in S$, and an element $g\in G$, this definition returns the class of $(x,g)$ in the quotient of $S\times G$ by the equivalence relation generated by

$$(a,b)\sim(c,d)\quad\text{if some }k\in K \text{ satisfies }c=a\cdot k\text{ and }d=k^{-1}b.$$

There is no requirement that $S$ be preserved by $K$; the supplied point $x\in S$ ensures that $S$ is nonempty for this particular input.

QAlgorithms.Papers.Independent.Intakethird100.P93.inducedLeft

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for every $h\in H$ and $g,k\in G$; require also continuous sections of $G\to G/\ker t$ and $H\to H/t(G)$ on open neighborhoods of every target point, with each section right-inverse to the quotient map on its domain. Let $X$ be a topological space in that universe, equipped with a jointly continuous operation $x\cdot g$ satisfying $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$, with $X$ nonempty and every orbit map $g\mapsto x\cdot g$ a homeomorphism $G\to X$, and with a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. For each $g\in G$ and $x\in X$, this definition returns the point

$$L_g(x)=x\cdot\alpha_{\psi(x)}^{-1}(g),$$

where $\alpha_{\psi(x)}^{-1}$ is the inverse automorphism of $\alpha_{\psi(x)}$.

QAlgorithms.Papers.Independent.Intakethird100.P93.iotaClass

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=ht(g)h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$ for all $h,g,k$. Assume that both $G\to G/\ker t$ and $H\to H/t(G)$ have continuous sections on open neighbourhoods of every target point. Put $K=\ker t$. For topological spaces $M,Q$, let $R$ consist of a continuous right $K$-action on $Q$ and a continuous invariant map $\pi_R:Q\to M$, locally admitting homeomorphisms $\pi_R^{-1}(U)\to U\times K$ over an open neighbourhood $U$ of each base point, preserving the base coordinate and carrying the action to right multiplication on $K$. Given $q\in Q$ and $g\in G$, the defined element is the class $[q,g]$ of $(q,g)$ in the quotient of $Q\times G$ generated by $(q,g)\sim(q\cdot k,k^{-1}g)$ for all $k\in K$, with $k$ interpreted in $G$ in the second coordinate. No nonemptiness is assumed for $M$ or $Q$; when $Q$ is empty there are no inputs $q$ to this definition.

QAlgorithms.Papers.Independent.Intakethird100.P93.iotaHom

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with continuous evaluation, satisfying $t(\alpha(h)(g))=ht(g)h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$ for all $h,g,k$. Assume that $G\to G/\ker t$ and $H\to H/t(G)$ each admit a continuous section over an open neighbourhood of every target point. Put $K=\ker t$. Let $M,Q,R$ be topological spaces, and let $B,D$ consist of continuous right $K$-actions on $Q,R$ and continuous invariant projections $\pi_B,\pi_D$ to $M$, each locally admitting homeomorphisms to $U\times K$ over an open neighbourhood $U$ of every base point, preserving the base coordinate and carrying the action to right multiplication on $K$. Let $f:Q\to R$ be continuous and satisfy $\pi_D(f(q))=\pi_B(q)$ and $f(q\cdot_B k)=f(q)\cdot_D k$ for every $q\in Q$ and $k\in K$. Form $I_B$ and $I_D$ as the quotients of $Q\times G$ and $R\times G$ generated respectively by $(q,g)\sim(q\cdot_B k,k^{-1}g)$ and $(r,g)\sim(r\cdot_D k,k^{-1}g)$, $k\in K$. The definition gives the function $I_B\to I_D$ specified on every representative by $[q,g]_B\mapsto[f(q),g]_D$. Its well-definedness uses the stated equivariance of $f$; the declared output is a function. Empty bases and the resulting empty quotients are included.

QAlgorithms.Papers.Independent.Intakethird100.P93.orbitClass

Let $G$ be a group with a topology, let $X$ be a topological space, and let $x\cdot g$ be a continuous right action of $G$ on $X$, so that $x\cdot1_G=x$ and $(x\cdot g)\cdot k=x\cdot(gk)$. Given a subgroup $K\le G$ and a point $x\in X$, this definition returns the class of $x$ in the quotient of $X$ by the equivalence relation generated by $y=z\cdot k$ for $k\in K$. Its value is therefore the $K$-orbit class represented by $x$.

QAlgorithms.Papers.Independent.Intakethird100.P93.typeProjection

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for every $h\in H$ and $g,k\in G$. Assume also that both quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ admit a continuous section on an open neighborhood of each target point, with the quotient map composed with that section equal to the identity there. For every $h\in H$, this function returns its quotient class

$$[h]\in H/t(G),\qquad t(G)=\{t(g):g\in G\}.$$

QAlgorithms.Papers.Independent.Intakethird100.P93.unitLevel

Let $G,H$ be topological groups, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ whose evaluation map $H\times G\to G$ is continuous, satisfying $t(\alpha(h)(g))=h\,t(g)\,h^{-1}$ and $\alpha(t(g))(k)=gkg^{-1}$ for all $h\in H$ and $g,k\in G$. Assume also that each of the quotient maps $G\to G/\ker t$ and $H\to H/t(G)$ has, around every point of its codomain, an open neighborhood $U$ and a continuous map $s$ from $U$ into the source with $q(s(z))=z$. Let $X$ be a topological space and $B$ consist of a continuous right $G$-action $x\cdot g$, with $X\ne\varnothing$ and $g\mapsto x\cdot g$ a homeomorphism $G\to X$ for every $x\in X$, together with a continuous map $\psi:X\to H$ satisfying $\psi(x\cdot g)=\psi(x)t(g)$. The definition assigns to $B$ the subset $\{x\in X:\psi(x)=1_H\}$; this subset is allowed to be empty.

QAlgorithms.Papers.Independent.Intakethird100.P93.unitType

Let $G$ and $H$ be topological groups in the same universe, and let $C$ consist of a continuous homomorphism $t:G\to H$ and a homomorphism $\alpha:H\to\operatorname{Aut}(G)$ with jointly continuous evaluation, satisfying $t(\alpha_h(g))=h\,t(g)\,h^{-1}$ and $\alpha_{t(g)}(k)=gkg^{-1}$ for every $h\in H$ and $g,k\in G$. Assume also that each quotient map $G\to G/\ker t$ and $H\to H/t(G)$ admits, around every target point, an open neighborhood with a continuous section whose composite with the quotient map is the identity. This definition returns the class of the identity element of $H$:

$$[1_H]\in H/t(G),\qquad t(G)=\{t(g):g\in G\}.$$