Item 5 (0143) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

Interaction With Groupoids. If $\mathcal{C}$ and $\mathcal{D}$ are groupoids, then the following conditions are equivalent:
1. The functor $F$ is an equivalence of groupoids.
2. The following conditions are satisfied:
  1. The functor $F$ induces a bijection
    \[ \pi _{0}\webleft (F\webright )\colon \pi _{0}\webleft (\mathcal{C}\webright )\to \pi _{0}\webleft (\mathcal{D}\webright ) \]
    
    of sets.
  2. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the induced map
    \[ F_{x,x}\colon \mathrm{Aut}_{\mathcal{C}}\webleft (A\webright )\to \mathrm{Aut}_{\mathcal{D}}\webleft (F_{A}\webright ) \]
    
    is an isomorphism of groups.

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