# Sheaves and presheaves

Sheaves and presheaves are important structures at the intersection of logic and geometry. Presheaves fit into categorical logic as a relatively inexpressive logic, less expressive than algebraic theories, yet they still encompass many important examples. Presheaves are equivalent to discrete fibrations.

## Theory

**General**

- Tennison, 1975:
*Sheaf theory*(doi)- Commentary on my blog

- Reyes, Reyes, Zolfaghari, 2004:
*Generic figures and their glueings: A constructive approach to functor categories*(online , pdf) - Borceux, 1994:
*Handbook of categorical algebra*, Vol 3: Categories of sheaves - Mac Lane & Moerdijk, 1992:
*Sheaves in geometry and logic: A first introduction to topos theory*(doi, nLab )

**Relational presheaves**, the relational counterpart of presheaves

A lax functor \(B \to \mathbf{Rel}\) is variously known as a *relational
presheaf*, *relational variable set*, *specification structure*, or *dynamic
set*. Likewise, a lax functor \(B \to \mathbf{Par}\) is a *partial presheaf* or
*partial variable set*.

- Rosenthal, 1996:
*The theory of quantaloids*, Sec. 3.3: Categories enriched in a free quantaloid- Based on Rosenthal, 1991: Free quantaloids (doi, pdf)
- The paper uses the term
*nondeterministic functor* - For related references, see quantaloids

- Abramsky, Gay, Nagarajan, 1996: Specification structures and propositions-as-types for concurrency (doi)
- Ghilardi and Meloni, 1996: Relational and partial variable sets and basic predicate logic (doi, jstor )
- Niefield, 2004: Change of base for relational variable sets (pdf)
- Theorem 3.1: Equivalence between relational variable sets and full subcategory \(\mathbf{Cat}_f/B\) of \(\mathbf{Cat}/B\) consisting of faithful functors over \(B\)

- Niefield, 2010: Lax presheaves and exponentiability (pdf)
- Studies lax presheaves valued in \(\mathbf{Rel}\), \(\mathbf{Par}\), and \(\mathbf{Span}\)
- Sec. 3: Further equivalences between lax functors and subcategories of \(\mathbf{Cat}/B\)

- Sobociński, 2015: Relational presheaves, change of base and weak simulation (doi, pdf)

**Displayed categories**

Closely related to a relational presheaf, a displayed category is a lax functor \(B \to \mathbf{Span}\). Displayed categories over \(B\) are equivalent to slice categories over \(B\):

\[ \mathrm{Lax}(B,\mathbf{Span}) \simeq \mathbf{Cat}/B. \]

## Examples

**Presheaf toposes**

Every category of presheaves is an elementary topos, known as a *presheaf
topos*. Much of the literature on topos theory is related sheaves and
presheaves.

**Graphs**

The category of graphs is an important example of a “combinatorial” presheaf topos.

- Lawvere, 1989: Qualitative distinctions between some toposes of generalized
graphs (pdf)
- Lawvere, 1991: More on graphic toposes (online , pdf)
- Context and further references on nLab pages for graphic category and, bizarrely, Hegelian tacos

- Vigna, 2003: A guided tour of the topos of graphs (arxiv, pdf)
- Sebastiano Vigna’s personal page on graph fibrations
- Boldi & Vigna, 2002: Fibrations of graphs (doi, pdf)
- Boldi, Lonati, Santini, Vigna, 2006: Graph fibrations, graph isomorphism, and PageRank (doi, pdf)

- Ronnie Brown’s papers on graphs of graph morphisms, which explicitly invoke the topos of graphs

Not only do graphs form a category of presheaves; there are also sheaves *on*
graphs.

- Hansen, 2019: A gentle introduction to sheaves on graphs (pdf)
- A very nice introduction, both readable and rigorous
- Covers sheaves of vector spaces on graphs, their morphisms and Laplacians, and hints at possible applications

- Hansen & Ghrist, 2018: Towards a spectral theory of cellular sheaves (arxiv, pdf)
- Friedman, 2015: Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture (doi, arxiv)

**Simplicial and cubical sets**

Simplicial sets and semi-simplicial sets form presheaf toposes, generalizing (reflexive) graphs to dimension greater than one. Likewise for cubical sets. For references, see simplicial stuff and cubical stuff.

## Applications

**Presheaves as models of networks**

- Spivak, 2009: Higher-dimensional models of networks (arxiv)
- Network models as categories of presheaves, with functorial changes of model
- Main examples: graphs, hypergraphs, semi-simplicial sets, simplicial sets

**Sheaves in topological data analysis**

- Robinson, 2014:
*Topological signal processing*(doi)- DARPA tutorial on sheaves in data analytics , including slides and video
- Robinson, 2018: Assignments to sheaves of pseudometric spaces (arxiv)

- Curry, 2014, PhD thesis:
*Sheaves, cosheaves and applications*(arxiv)- Thesis by student of Robert Ghrist developing “cellular sheaves”
- Sec 4.2, relating classical and cellular sheaves, presents an alternate proof of: Ladkani, 2008: On derived equivalences of categories of sheaves over finite posets (doi, arxiv)

- Curry, 2018: Dualities between cellular sheaves and cosheaves (doi, arxiv)
- Partly extracted from Curry’s thesis
- Correction issued in: Curry, 2019: Functors on posets left Kan extend to cosheaves: An erratum (arxiv)

- Vepstas, 2017: Sheaves: A topological approach to big data (doi, arxiv,
GitHub )
- A gentle, informal introduction to graphs and cellular sheaves, motivated by data analysis
- Plenty of good intuition but certain analogies make little sense to me (e.g., comparing graphs and lambda calculus as “similar concepts”)
- Also lots of mistakes, not merely typographical (assuming an adjacency list representation when claiming that graphs have bad computational properties compared to sections; claiming that contractions are what make tensor algebras into a monoidal category, rather than a compact closed category; claiming that “simply typed” means there is only one type; and so on)

**Sheaves in systems theory**

- Goguen, 1992: Sheaf semantics for concurrent interacting objects (doi)
- Brief exposition in Spivak, 2014:
*Category Theory for the Sciences*, Sec. 7.2.3: Sheaves, especially Sec 7.2.3.13: Time

- Brief exposition in Spivak, 2014:
- Robinson, 2017: Sheaves are the canonical datastructure for sensor integration (doi, arxiv)
- Schultz & Spivak, 2019:
*Temporal type theory: A topos-theoretic approach to systems and behavior*(doi, arxiv)