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I'm going to change my original answer, since I interpreted the question wrongly. (I hope that's alright.)

Suppose $p: E\to B$ is a Hurewicz fibration, where $F = p^{-1}(\ast)$ is the fiber over the basepoint and $B$ is connected. Then one can cook up a map $\Omega B \times F \to F$ which might be called a "holonomy" in the algbraic topology sense.

The idea is this: Let $$ \Lambda_p = E\times_B B^I $$ be the space of path lifting problems for $p$ (this is the space of pairs $(e,\lambda)$ where $e\in E$ and $\lambda$ is a path starting at $p(e)$. There is a map $$ q: E^I \to \Lambda_p $$ by sending path $\lambda$ in $E$ to $(\lambda(0), p\circ \lambda)$. Then the condition that $p$ be a Hurewicz fibration is tantamount to saying that $q$ has a section. A choice of section might be regarded as {\it parallel transport} along a path in the algebraic topological sense. Choose such a section. This gives a way of associating to each path in $B$, starting at $x$ and ending at $y$, a map $E_x \to E_y$, where $E_x$ is the fiber at $x$. This map is a homotopy equivalence. (When $p$ is a fiber bundle, one can choose the section in such a way that each parallel transport is a homeomorphism).

Evaluating the section when $x=y$ is the basepoint gives the holonomy operation $\Omega B \times F \to F$, or adjointly as $\Omega B \to \text{homeo}(F)$. If $p$ is a fiber bundle with structure group $G$, then the transport operation described above can be factored as $$ \Omega B \to G\to \text{homeo}(F) . $$ If we choose a basepoint in $F$, then the value of the operation on the basepoint gives a map $$ \Omega B \to F \, . $$ This map is well-known: it's the map sitting in the homotopy fiber sequence $$ \Omega B \to F \to E . $$ (this should be in any reasonable text on the subject).

So, in the particular case when $p\: EG \to BG$ and $F = G$, then map $\Omega BG \to G$ will be a homotopy equivalence, using the above homotopy fiber sequence, since $E = EG$ is contractible. We have also seen this map as decribed by the orbit of a point in $G$ under the holonomy operation $\Omega BG \times G \to G$ as given above.

I'm going to change my original answer, since I interpreted the question wrongly. (I hope that's alright.)

Suppose $p: E\to B$ is a Hurewicz fibration, where $F = p^{-1}(\ast)$ is the fiber over the basepoint and $B$ is connected. Then one can cook up a map $\Omega B \times F \to F$ which might be called a "holonomy" in the algbraic topology sense.

The idea is this: Let $$ \Lambda_p = E\times_B B^I $$ be the space of path lifting problems for $p$ (this is the space of pairs $(e,\lambda)$ where $e\in E$ and $\lambda$ is a path starting at $p(e)$. There is a map $$ q: E^I \to \Lambda_p $$ by sending path $\lambda$ in $E$ to $(\lambda(0), p\circ \lambda)$. Then the condition that $p$ be a Hurewicz fibration is tantamount to saying that $q$ has a section. A choice of section might be regarded as parallel transport along a path in the algebraic topological sense. Choose such a section. This gives a way of associating to each path in $B$, starting at $x$ and ending at $y$, a map $E_x \to E_y$, where $E_x$ is the fiber at $x$. This map is a homotopy equivalence. (When $p$ is a fiber bundle, one can choose the section in such a way that each parallel transport is a homeomorphism of fibers.)

Evaluating the section when $x=y$ is the basepoint gives the holonomy operation $\Omega B \times F \to F$, or adjointly as $\Omega B \to G(F)$, where $G(F)$ is the topological monoid of self homotopy equivalences of $F$. If $p$ is a fiber bundle with structure group $G$, then the transport operation described above can be factored as $$ \Omega B \to G\to G(F) . $$ If we choose a basepoint in $F$, then the value of the operation on the basepoint gives a map $$ \Omega B \to F . $$ This map is well-known: it's the map sitting in the homotopy fiber sequence $$ \Omega B \to F \to E . $$ (this should be in any reasonable text on the subject).

So, in the particular case when $p: EG \to BG$ and $F = G$, then map $\Omega BG \to G$ will be a homotopy equivalence, using the above homotopy fiber sequence, since $E = EG$ is contractible. We have also seen this map as decribed by the orbit of a point in $G$ under the holonomy operation $\Omega BG \times G \to G$ as given above.

I'm going to change my original answer, since I interpreted the question wrongly. (I hope that's alright.)

Suppose $p: E\to B$ is a Hurewicz fibration, where $F = p^{-1}(\ast)$ is the fiber over the basepoint and $B$ is connected. Then one can cook up a map $\Omega B \times F \to F$ which might be called a "holonomy" in the algbraic topology sense.

The idea is this: Let $$ \Lambda_p = E\times_B B^I $$ be the space of path lifting problems for $p$ (this is the space of pairs $(e,\lambda)$ where $e\in E$ and $\lambda$ is a path starting at $p(e)$. There is a map $$ q: E^I \to \Lambda_p $$ by sending path $\lambda$ in $E$ to $(\lambda(0), p\circ \lambda)$. Then the condition that $p$ be a Hurewicz fibration is tantamount to saying that $q$ has a section. A choice of section might be regarded as {\it parallel transport} along a path in the algebraic topological sense. Choose such a section. This gives a way of associating to each path in $B$, starting at $x$ and ending at $y$, a map $E_x \to E_y$, where $E_x$ is the fiber at $x$. This map is a homotopy equivalence. (When $p$ is a fiber bundle, one can choose the section in such a way that each parallel transport is a homeomorphism).

Evaluating the section when $x=y$ is the basepoint gives the holonomy operation $\Omega B \times F \to F$, or adjointly as $\Omega B \to \text{homeo}(F)$. If $p$ is a fiber bundle with structure group $G$, then the transport operation described above can be factored as $$ \Omega B \to G\to \text{homeo}(F) . $$ If we choose a basepoint in $F$, then the value of the operation on the basepoint gives a map $$ \Omega B \to F \, . $$ This map is well-known: it's the map sitting in the homotopy fiber sequence $$ \Omega B \to F \to E . $$ (this should be in any reasonable text on the subject).

So, in the particular case when $E$ is contractible, the map $\Omega B \to F$ will be a homotopy equivalence, and this map is decribed by orbit the holonomy operation as given above.

I'm going to change my original answer, since I interpreted the question wrongly. (I hope that's alright.)

Suppose $p: E\to B$ is a Hurewicz fibration, where $F = p^{-1}(\ast)$ is the fiber over the basepoint and $B$ is connected. Then one can cook up a map $\Omega B \times F \to F$ which might be called a "holonomy" in the algbraic topology sense.

The idea is this: Let $$ \Lambda_p = E\times_B B^I $$ be the space of path lifting problems for $p$ (this is the space of pairs $(e,\lambda)$ where $e\in E$ and $\lambda$ is a path starting at $p(e)$. There is a map $$ q: E^I \to \Lambda_p $$ by sending path $\lambda$ in $E$ to $(\lambda(0), p\circ \lambda)$. Then the condition that $p$ be a Hurewicz fibration is tantamount to saying that $q$ has a section. A choice of section might be regarded as {\it parallel transport} along a path in the algebraic topological sense. Choose such a section. This gives a way of associating to each path in $B$, starting at $x$ and ending at $y$, a map $E_x \to E_y$, where $E_x$ is the fiber at $x$. This map is a homotopy equivalence. (When $p$ is a fiber bundle, one can choose the section in such a way that each parallel transport is a homeomorphism).

Evaluating the section when $x=y$ is the basepoint gives the holonomy operation $\Omega B \times F \to F$, or adjointly as $\Omega B \to \text{homeo}(F)$. If $p$ is a fiber bundle with structure group $G$, then the transport operation described above can be factored as $$ \Omega B \to G\to \text{homeo}(F) . $$ If we choose a basepoint in $F$, then the value of the operation on the basepoint gives a map $$ \Omega B \to F \, . $$ This map is well-known: it's the map sitting in the homotopy fiber sequence $$ \Omega B \to F \to E . $$ (this should be in any reasonable text on the subject).

So, in the particular case when $p\: EG \to BG$ and $F = G$, then map $\Omega BG \to G$ will be a homotopy equivalence, using the above homotopy fiber sequence, since $E = EG$ is contractible. We have also seen this map as decribed by the orbit of a point in $G$ under the holonomy operation $\Omega BG \times G \to G$ as given above.

Something like this is worked out in papers of Pawel Gajer:

Gajer, Paweł Higher holonomies, geometric loop groups and smooth Deligne cohomology. Advances in geometry, 195–235, Progr. Math., 172, Birkhäuser Boston, Boston, MA, 1999.

 (I suggest you also see its Math Review)

The rough idea is that there's a bona-fide topological group model for the loop space $\Omega M$ when $M$ is smooth and compact. Holonomy is then a bona-fide topological group homomorphism $\Omega M \to G$.

I'm going to change my original answer, since I interpreted the question wrongly. (I hope that's alright.)

Suppose $p: E\to B$ is a Hurewicz fibration, where $F = p^{-1}(\ast)$ is the fiber over the basepoint and $B$ is connected. Then one can cook up a map $\Omega B \times F \to F$ which might be called a "holonomy" in the algbraic topology sense.

The idea is this: Let $$ \Lambda_p = E\times_B B^I $$ be the space of path lifting problems for $p$ (this is the space of pairs $(e,\lambda)$ where $e\in E$ and $\lambda$ is a path starting at $p(e)$. There is a map $$ q: E^I \to \Lambda_p $$ by sending path $\lambda$ in $E$ to $(\lambda(0), p\circ \lambda)$. Then the condition that $p$ be a Hurewicz fibration is tantamount to saying that $q$ has a section. A choice of section might be regarded as {\it parallel transport} along a path in the algebraic topological sense. Choose such a section. This gives a way of associating to each path in $B$, starting at $x$ and ending at $y$, a map $E_x \to E_y$, where $E_x$ is the fiber at $x$. This map is a homotopy equivalence. (When $p$ is a fiber bundle, one can choose the section in such a way that each parallel transport is a homeomorphism).

Evaluating the section when $x=y$ is the basepoint gives the holonomy operation $\Omega B \times F \to F$, or adjointly as $\Omega B \to \text{homeo}(F)$. If $p$ is a fiber bundle with structure group $G$, then the transport operation described above can be factored as $$ \Omega B \to G\to \text{homeo}(F) . $$ If we choose a basepoint in $F$, then the value of the operation on the basepoint gives a map $$ \Omega B \to F \, . $$ This map is well-known: it's the map sitting in the homotopy fiber sequence $$ \Omega B \to F \to E . $$ (this should be in any reasonable text on the subject).

So, in the particular case when $E$ is contractible, the map $\Omega B \to F$ will be a homotopy equivalence, and this map is decribed by orbit the holonomy operation as given above.