Tags: #MOC

Classical Category Theory

References

Topics

• natural transformation
• Yoneda embedding
• Yoneda lemma
• adjoint (categorical)
• monad
• Limit and Colimit
• Cartesian closed category
• Monoidal category
• Symmetric monoidal category
• Pushout
  • Limit definition
• pullback
  • Limit definition
• equivalence of categories
  • Need to state this precisely!
• equivalence of categories
• Unsorted/adjoint (categorical)
• Limits and universal properties
  • coproduct
  • cokernel
  • colimit
  • monomorphism
• Homological algebra
  • additive functor
  • abelian category
  • additive category
  • monomorphism
  • mapping cone
• Yoneda lemma
• isomorphism of functors
• subfunctor
• exponential object
• monads
• natural transformation
• Yoneda embedding
• Yoneda lemma
• Unsorted/adjoint (categorical)
• monad
• Limit and Colimit
• Cartesian closed category
• monoidal category
• Symmetric monoidal category
• Pushout
  • Limit definition
• pullback
  • Limit definition
• equivalence of categories
  • Need to state this precisely!

Notes

| Category | Set | Grp | CRing | Ring | Field | Ab | \({ \mathsf{Vect} }_k\) | R-Mod | \(R{\hbox{-}}\)cAlg | Sch | Top | \({\mathsf{Top}}_*\) | |

————— | ———————– | ————— | ––––––– | ———– | —– | —————– | —————– | —————– | —————— | ———– | —————– | ———— | | Product | \(\prod_i A_i\) | \(\prod_i A_i\) | | | None | | | \(\prod_i A_i\) | | | \(\prod_i A_i\) | | | Coproduct | \(\coprod_i A_i\) | \(A\ast B\) | | \(A\star B\) | None | \(\bigoplus_i A_i\) | \(\bigoplus_i A_i\) | \(\bigoplus_i A_i\) | \(\bigotimes_i A_i\) | | \(\coprod A_i\) | \(\vee_i A_i\) | | Pullback | \(A\times_C B, A \cap B\) | \(A\times_C B\) | \(A\times_C B\) | | | | | \(A\times_C B\) | | | | | | Pushout | \(A \coprod B/\sim\) | \(A \ast B/\sim\) | \(A\otimes_C B\) | | | | | | | | \(A \coprod_{f} B\) | | | Initial Object | \(\emptyset\) | \(\left\{{1}\right\}\) | | \({\mathbf{Z}}\) | None | | | \(\left\{{1}\right\}\) | | \(\operatorname{Spec}(0)\) | \(\emptyset\) | | | Terminal Object | \(\left\{{a_1}\right\}\) | | | \(\left\{{0}\right\}\) | None | | | | | \(\operatorname{Spec}{\mathbf{Z}}\) | \({\operatorname{pt}}\) | | | Zero Object | | \(\left\{{1}\right\}\) | | \(\left\{{0}\right\}\) | None | | | | | | | |

\begin{align*} A\star B \cong A \oplus B \oplus (A \otimes B) \oplus (B \otimes A) \oplus (A \otimes A \otimes B) \oplus (A \oplus B \oplus A) \oplus (B \oplus A \oplus A) \oplus ... \end{align*}

• One regards a category \(\mathsf{C}\) as an infinity category via its nerve.

  • The nerve lands in simplicial sets, but everything in its image satisfies the Kan extension condition.

• Categories are special cases of a simplicial set

• Initial objects: \(\emptyset\).

• Terminal objects: \(*\).