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Sunday, July 25, 2021

Kan extensions

Draft of just one, but key, diagram concerning the pointwise right Kan extension at (one object 1lL).


I1lL11!11λ0R31αS30Iright lax fiber12r[λ](lR)1011Q=rR0G(e.g., R)CIrR

Given the initial data, the span (0R:LRC:G0)
we wish to form 
(the right Kan extension 20RanRG:LC
of G over R.
We do this "pointwise".
Given (an object 1lL),
form (its right lax fiber 10(lR)) as shown in the diagram above.
Then form 
(the limit 20((Q=r)G)limC in C), 
an arrow running from the top left in the above diagram southeast to the somewhat lower right,
together with 
(its universal cone 21π:((Q=r)G)limC(Q=r)G), 
a 2-cell just below the just-mentioned arrow.


I1lL11!11λ0R311R1L30Iright lax fiber12r[λ](lR)1011Q=rRR0LIrR

Tuesday, July 20, 2021

Algebraic set theory versus ETCS

Very much a preliminary draft, with errors!

This is a first cut at lining up and comparing various set theories and their models.

referenceAwodeyLeinsterSGL VI.10theoriesBIST: Basic Intuitionistic Set TheoryETCS: Elementary Theory of the Category of SetsRZCmodelscategories of classesWPTNNC

Awodey-Butz-Simpson-Streicher 2014 BIST APAL
https://www.sciencedirect.com/science/article/pii/S0168007213000730

Saturday, July 17, 2021

Spaces, frames, locales

Here are some of the key concepts from Mac Lane and Moerdijk's SGL related to the above.

(Spaces) (arrows go in geometric direction), 
(Frames) (arrows go in algebraic direction), 
(Locales)=(Frames)op (arrows go in geometric direction),

Quoting Mac Lane-Moerdijk, Section IX.1,

For a map f:ST of spaces, the locale-map Loc(f):Loc(S)Loc(T) is given by the frame morphism f1:O(T)O(S)
i.e.
(f:ST)(Spaces)Loc(Locales)(Loc(f):Loc(S)Loc(T))=((f1:O(T)O(S))op)(Frames)op.
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