Category theory reading

Reading references saved in October 2024. The researcher titles and affiliations below reproduce that historical list, not a current directory.

Sheaves and presheaves

Facets of descent I by George Janelidze & Walter Tholen.

Sheaves are presheaves with an equalizer diagram

The Yoneda embedding $Y:\mathrm{Open}\left(X\right)\to {\mathrm{Set}}^{{\mathrm{Open}\left(X\right)}^{\text{op}}}$ sends each open set to its representable presheaf in the ambient presheaf category.

There is an adjunction $L⊣\Gamma :{\mathrm{Set}}^{{\mathrm{Open}\left(X\right)}^{\text{op}}}\rightleftarrows (\mathrm{Top}\downarrow X)$, where $L$ forms the étalé space (sheafification) of a presheaf and $\Gamma$ assigns to a space over $X$ its presheaf of sections.

Restricting this adjunction to sheaves and local homeomorphisms yields the classical equivalence $\mathrm{Shv}\left(X\right)\simeq \mathrm{Etale}\left(X\right)$.

Given an open cover ${\left\{U_i\right\}}_{i\in I}$ of $U$, the restriction morphism $\delta :F\left(U\right)\to \prod _{i\in I}F\left(U_i\right)$ records the sections on each element of the cover.

The matching condition is encoded by the parallel morphisms $\alpha ,\beta :\prod _{i\in I}F\left(U_i\right)\to \prod _{(i,j)\in I\times I}F\left(U_i\cap U_j\right)$ defined by $\alpha ((s_i))=\left(s_i\right|U_i\cap U_j\left)\right)$ and $\beta ((s_i))=\left(s_j\right|U_i\cap U_j\left)\right)$.

The sheaf condition asserts that $F$ makes the above diagram an equalizer: compatible families glue uniquely to a section on $U$.

Researchers and papers

Prof. George Janelidze. (University of Cape Town)

• Effective descent morphisms of filtered preorders.
• Strict monadic topology II: Descent for closure spaces.
• Strict monadic topology I: First separation axioms and reflections.
• Split extensions and semidirect products of unitary magmas.
• Picard-Vessiot and categorically normal extensions in differential-difference Galois theory.
• The monads of classical algebra are seldom weakly cartesian.

Prof. Marino Gran. (Université catholique de Louvain)

Prof. Zurab Janelidze. (Stellenbosch University)

Dr. Michael Hoefnagel. (Stellenbosch University)

Dr. Pierre-Alain Jacqmin. (Université catholique de Louvain)

• Embedding theorems in non-abelian categorical algebra.

Thomas Mbewu. (University of Cape Town)

• Coverings and descent theory of finite spaces.

Jacob Lund. (University of Cape Town)

• Finite covering spaces. (unpublished).

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