Following \cite{grothendieck-kansas}, define a fiber space p:E→B to be a morphism in cat. A map of fiber spaces is a commutative square. Thus, the category of fiber spaces is isomorphic to the diagram category $\textup{\textsf{cat}}^\mathbbe{2}$. We are also interested in the non-full subcategory $\textup{\textsf{cat}}/B \subset \textup{\textsf{cat}}^\mathbbe{2}$ of fiber spaces over B and maps whose codomain component is the identity. Prove the following:
- A map \(\xymatrix{ E' \ar[d]_{p'} \ar[r]^g & E \ar[d]^{p} \\ B' \ar[r]_f & B}\)of fiber spaces induces a canonical map from the fiber over a point b∈B’ to the fiber over its image f(b)∈B.
- The fiber of a product of fiber spaces is the product of the fibers.
A projection B×F→B defines a trivial fiber space over B, a definition that makes sense for any space F.
- Show that the fiber of a trivial fiber space B×F→B is isomorphic to F.
- Characterize the isomorphisms in cat/B between two trivial fiber spaces (with a priori distinct fibers) over B.
- Prove that the assignment of the set of continuous sections of a fiber space over B defines a functor fun:cat/B→cat.
- Consider the non-full subcategory $\textup{\textsf{cat}}^\2{\text{pb}}offiberspacesinwhichthemorphismsarethepullbacksquares.Provethattheassignmentofthesetofcontinuoussectionstoafiberspacedefinesafunctor\textup{fun}\colon (\textup{\textsf{cat}}^\2{\text{pb}})^\mathrm{op} \to \textup{\textsf{cat}}$.
- Describe the compatibility between the actions of the ``sections’’ functors just introduced with respect to the map g of fiber spaces p and q over B and their restrictions along f:B’→B.
\(\xymatrix@=10pt{ E' \ar@{}[dr]\mid(.2){\displaystyle\lrcorner} \ar[rr] \ar[ddd]_{p'} & & E \ar'[d][ddd]_p \ar[dr]^g \\ & F' \ar@{}[dr]\mid(.2){\displaystyle\lrcorner} \ar[ddl]^{q'} \ar[rr] & & F \ar[ddl]^q \\ & & \\ B' \ar[rr]_f & & B}\)