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### Fiber product of nilmanifolds

Let $M_1$ and $M_2$ be nilmanifolds. We can see them as total spaces of torus bundles $\pi_i:M_i \to B_i\ \ i=1,2$. Suppose that $B_1=B_2$ and that the fibers are torus of the same dimension and ...
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In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states: Indeed, the UMP of pullbacks essentially states that composition along any function α is left adjoint to pullback ...
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I previously asked this on Math.SE but didn't receive a satisfactory answer. Let $p:E\rightarrow B$ be a fibration (i.e. have the homotopy lifting property with respect to all spaces), and $f: B'\... 1answer 167 views ### finite generation of a certain type of subring Let$k$be a field, and let$R$be a finitely generated$k$-algebra. (If it helps, you may assume$R$is an integral domain.) Let$I$be an ideal of finite colength. Note that$A:=k+I$is a subring ... 0answers 184 views ### Is continuity of a functor stable under pullback? Let$p:C\rightarrow D$,$i:F\rightarrow D$be functors of 2-categories, and we form the lax pullback of$p$along$i$$$\bar{p}:C\times_D^{lax} F\rightarrow F$$ Q1: Is it true that if$p$... 2answers 409 views ### Pullback of a constant sheaf Let$\varphi:X\to Y$be a surjective morphism of schemes which are connected and of finite type. Let$A$be an abelian group,$\mathscr{F}$be the constant sheaf on$X$with fibers$A$and$\mathscr{...
I am reading Awodey (Category Theory, 1st edition), p 175, and I have difficulties to understand the paragraph about the subobject classifier of $\mathbf{Sets}^{\mathbf{C^{op}}}$. First let me quote ...