David Corfield fibration

Idea

Points regarding the use of fibrations in applications.

Definition

Graph fibrations as generating fibrations on free categories. Local input isomorphisms.

Properties

Given a functor $F:C\rightarrow B$, we can construct a new category called the “category of elements” of $F$, denoted as $(*/F)$. Objects of $(*/F)$ are pairs $(c,x)$, where $c$ is an object in $C$ and $x$ is a morphism in $B$ with codomain $F(c)$. A morphism $(c_{1},x_{1})\rightarrow (c_{2},x_{2})$ in $(*/F)$ is a morphism $f:c_{1}\rightarrow c_{2}$ in $C$ such that $F(f)\circ x_{1}=x_{2}$.

The projection functor $\pi :(*/F)\rightarrow B$ defined by $\pi (c,x)=\text{cod}(x)$ (where $\text{cod}(x)$ is the codomain of $x$) is a Grothendieck fibration. This means that for any object $b$ in $B$ and any morphism $g:b^{\prime }\rightarrow b$ in $B$, there exists a morphism $f$ in $(*/F)$ such that $\pi (f)=g$ and $f$ is a cartesian lifting of $g$. 

The original functor $F:C\rightarrow B$ can be factored as $F=\pi \circ \iota$, where $\iota :C\rightarrow (*/F)$ is a functor defined by $\iota (c)=(c,1_{F(c)})$ (where $1_{F(c)}$ is the identity morphism at $F(c)$). This factorization shows that any functor can be factored through a Grothendieck fibration.  

Fibrations in Language

The FibLang work of @Fabrizio Romano Genovese,  @fosco and @Caterina Puca (here and here). These look to understand natural language and its acquisition via fibrations. They rely on the result that any functor factors through a fibration. There’s also work on locating obstructions to a functor being a fibration by Fabrizio and others.

Fibrations in biology

On the other hand, those using graph fibrations in biology are also interested in slight departures, which get called “13.5.1 Pseudosymmetries 246, 13.5.2 Quasifibrations 249, 13.5.3 Pseudo-balanced colorings”.

Perhaps some useful cross-over. Grammatical language would resemble the latter’s reduced gene networks, the logic of cognition and of the cell.

Fibrations in the mind

References

• Fabrizio Genovese, Fosco Loregian, Caterina Puca, Fibrational linguistics: First concepts arXiv:2201.01136

• Fabrizio Genovese, Fosco Loregian, Caterina Puca, Fibrational Linguistics (FibLang): Language Acquisition arXiv:2207.06765

• Benjamin Merlin Bumpus, James Fairbanks, Caterina Puca, Fabrizio Genovese, Daniel Rosiak, How Nice is this Functor? Two Squares and Some Homology go a Long Way

• Sebastiano Vigna The Graph–Fibrations Home Page

• Interacting Conceptual Spaces I : Grammatical Composition of Concepts Joe Bolt, Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Marsden, Robin Piedeleu, https://arxiv.org/abs/1703.08314