You're are talking about action of a groupoid on a space via homeomorphisms. Let $G$ be a groupoid. Then, in general, one may replace the category $\mathcal{Top}$ by a bicategory $\mathcal{A}$ and talk about an action of $G$ on $\mathcal{A}$. A good reference for this purpose is the following paper of Meyer, Buss and Zhu: https://arxiv.org/pdf/0908.0455v1.pdf wherein they define and study actions of group(oid)s on bicategories.

An interesting case of these actions is when $\mathcal{A}$ is the bicategory of $C^*$-algebras, say $\mathcal{C}^*$. In $\mathcal{C}^*$, the 'objects' are $C^*$-algebras, a 'functor' or 1-arrow from a $C^*$-algebra $A$ to $B$ is a $C^*$-correspondence from $A$ to $B$, and a 2-arrow is an isomorphism of $C^*$-correspondences. Meyer, Buss and Zhu show that, in discrete cases, these actions are same as equivalent to saturated Fell bundles on groupoids. This result is true when $G$ is a topological group. Thus your question has a significant meaning in some other category.

Moving back to your example, Meyer-Buss-Zhu's work shows that $\mathcal{Top}$ is kind of smaller category to work with. You may form the (bi)category $\mathcal{Q}$ of topological quivers introduced by Muhly and Tomford : http://arxiv.org/abs/math/0312109. Then and action of $G$ on on $\mathcal{Q}$ gives, what we may call, a `topological' version of Fell bundles, $B\to G$. Let $B_{\eta}$ denote the fibre on $\eta\in G$. If $G$ acts on the subcategory $\mathcal{Top}\subseteq \mathcal{Q}$, then the equivalence between $B_{s(\gamma)}$ and $B_{r(\gamma)}$ is just a homeomorphism.

You're are talking about action of a groupoid on a space via homeomorphisms. Let $G$ be a groupoid. Then, in general, one may replace the category $\mathcal{Top}$ by a bicategory $\mathcal{A}$ and talk about an action of $G$ on $\mathcal{A}$. A good reference for this purpose is

• Buss, A., Meyer, R., & Zhu, C., A higher category approach to twisted actions on $C^*$-algebras, Proceedings of the Edinburgh Mathematical Society, 56(2) (2013) pp 387-426, doi:10.1017/S0013091512000259, arXiv:0908.0455

wherein they define and study actions of group(oid)s on bicategories.

An interesting case of these actions is when $\mathcal{A}$ is the bicategory of $C^*$-algebras, say $\mathcal{C}^*$. In $\mathcal{C}^*$, the 'objects' are $C^*$-algebras, a 'functor' or 1-arrow from a $C^*$-algebra $A$ to $B$ is a $C^*$-correspondence from $A$ to $B$, and a 2-arrow is an isomorphism of $C^*$-correspondences. Meyer, Buss and Zhu show that, in discrete cases, these actions are same as equivalent to saturated Fell bundles on groupoids. This result is true when $G$ is a topological group. Thus your question has a significant meaning in some other category.

Moving back to your example, Meyer–Buss–Zhu's work shows that $\mathcal{Top}$ is kind of smaller category to work with. You may form the (bi)category $\mathcal{Q}$ of topological quivers introduced in:

• Paul S. Muhly, Mark Tomforde, Topological Quivers, International Journal of Mathematics 16 No. 7, (2005) pp. 693-755, doi:10.1142/S0129167X05003077, arXiv:math/0312109.

Then an action of $G$ on on $\mathcal{Q}$ gives, what we may call, a `topological' version of Fell bundles, $B\to G$. Let $B_{\eta}$ denote the fibre on $\eta\in G$. If $G$ acts on the subcategory $\mathcal{Top}\subseteq \mathcal{Q}$, then the equivalence between $B_{s(\gamma)}$ and $B_{r(\gamma)}$ is just a homeomorphism.

Your are talking about action of a groupoid on a space via homeomorphisms. Let $G$ be a groupoid. Then, in general, one may replace the category $\mathcal{Top}$ by a bicategory $\mathcal{A}$ and talk about an action of $G$ on $\mathcal{A}$. A good reference for this purpose is the following paper of Meyer, Buss and Zhu: https://arxiv.org/pdf/0908.0455v1.pdf wherein they define and study actions of group(oid)s on bicategories.

An interesting case of these actions is when $\mathcal{A}$ is the bicategory of $C^*$-algebras, say $\mathcal{C}^*$. In $\mathcal{C}^*$, the 'objects' are $C^*$-algebras, a 'functor' or 1-arrow from a $C^*$-algebra $A$ to $B$ is a $C^*$-correspondence from $A$ to $B$, and a 2-arrow is an isomorphism of $C^*$-correspondences. Meyer, Buss and Zhu show that, in discrete cases, these actions are same as equivalent to saturated Fell bundles on groupoids. This result is true when $G$ is a topological group. Thus your question has a significant meaning in some other category.

Moving back to your example, Meyer-Buss-Zhu's work shows that $\mathcal{Top}$ is kind of smaller category to work with. You may form the (bi)category $\mathcal{Q}$ of topological quivers introduced by Muhly and Tomford : http://arxiv.org/abs/math/0312109. Then and action of $G$ on on $\mathcal{Q}$ gives, what we may call, a `topological' version of Fell bundles, $B\to G$. Let $B_{\eta}$ denote the fibre on $\eta\in G$. If $G$ acts on the subcategory $\mathcal{Top}\subseteq \mathcal{Q}$, then the equivalence between $B_{s(\gamma)}$ and $B_{r(\gamma)}$ is just a homeomorphism.

You're are talking about action of a groupoid on a space via homeomorphisms. Let $G$ be a groupoid. Then, in general, one may replace the category $\mathcal{Top}$ by a bicategory $\mathcal{A}$ and talk about an action of $G$ on $\mathcal{A}$. A good reference for this purpose is the following paper of Meyer, Buss and Zhu: https://arxiv.org/pdf/0908.0455v1.pdf wherein they define and study actions of group(oid)s on bicategories.

An interesting case of these actions is when $\mathcal{A}$ is the bicategory of $C^*$-algebras, say $\mathcal{C}^*$. In $\mathcal{C}^*$, the 'objects' are $C^*$-algebras, a 'functor' or 1-arrow from a $C^*$-algebra $A$ to $B$ is a $C^*$-correspondence from $A$ to $B$, and a 2-arrow is an isomorphism of $C^*$-correspondences. Meyer, Buss and Zhu show that, in discrete cases, these actions are same as equivalent to saturated Fell bundles on groupoids. This result is true when $G$ is a topological group. Thus your question has a significant meaning in some other category.

Moving back to your example, Meyer-Buss-Zhu's work shows that $\mathcal{Top}$ is kind of smaller category to work with. You may form the (bi)category $\mathcal{Q}$ of topological quivers introduced by Muhly and Tomford : http://arxiv.org/abs/math/0312109. Then and action of $G$ on on $\mathcal{Q}$ gives, what we may call, a `topological' version of Fell bundles, $B\to G$. Let $B_{\eta}$ denote the fibre on $\eta\in G$. If $G$ acts on the subcategory $\mathcal{Top}\subseteq \mathcal{Q}$, then the equivalence between $B_{s(\gamma)}$ and $B_{r(\gamma)}$ is just a homeomorphism.