# 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)
• 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)
• Conference version: Sobociński, 2012: Relational presheaves as labelled transition systems (doi, pdf, slides)

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)
• 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)
• A more detailed version of Hansen’s “gentle introduction”
• See also: Hansen & Ghrist, 2019: Learning sheaf Laplacians from smooth signals (doi, 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)
• 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
• Robinson, 2017: Sheaves are the canonical datastructure for sensor integration (doi, arxiv)
• Robinson, 2017: Sheaf and duality methods for analyzing multi-model systems (doi, arxiv)
• Purvine, Joslyn, Robinson, 2016: A category theoretical investigation of the type hierarchy for heterogeneous sensor integration (arxiv)
• Schultz & Spivak, 2019: Temporal type theory: A topos-theoretic approach to systems and behavior (doi, arxiv)
• Spivak, Vasilakopoulou, Schultz, 2016: Dynamical systems and sheaves (arxiv)
• Speranzon, Spivak, Varadarajan, 2018: Abstraction, composition and contracts: A sheaf theoretic approach (arxiv)