Category Theory For Beginners: Topos Theory Essentials
In this video we introduce topos theory in a systematic way, before going for a faster less rigorous tour of some of the deeper ideas in the subject. We start by introducing the idea of a subobject classifier, and how it lets us link ideas about subobjects and logic. We define an elementary topos. We define logical AND, IMPLIES, and FOR ALL. We prove many results interrelating these ideas. Next we discuss other logical notions like FALSE, NOT, and OR. We also discuss epimorphism-monomorphism factorization. Finally, we quickly tour deeper ideas in topos theory, like the idea of forming a Heyting algebra by ordering an object's subobjects by containment. We also discuss notions like the fundamental theorem of topos theory, about how every slice category formed from a topos is also a topos.
Here are some more videos exploring more aspects of topos logic:
Understanding False and Not
• Understanding False and Not
More results about exponential and power objects
• More results about exponential and po...
"If then" statements in a topos
• "If then" statements in a topos
Implications of implication 1
• Implications of implication 1
Regarding the latter, the result expressing when an arrow w is in [q double-arrow r] , in terms of when [w after [the pullback of q along w]] is in r, can be used together with the Yoneda lemma to determine the D-type elements of [q double-arrow r], in the category of functors from C to Set (for an object D of C).
Correction at time 3:32:12
In fact, I do not think (w times 1_A) is always monic, so I should write
(w times 1_A) in r, rather than (w times 1_A) contained in r, within Theorem 13.9.
A proof of Theorem 13.9 can be found here
"Universal Quantification Proof"
• Universal Quantification Proof
Proofs involved in the first part of the video (up until and including the discussion of power objects), can be found in the following four videos:
• Mclarty Topos Basics 1
• Mclarty Topos Basics 2
• Mclarty Topos Basics 3
• Maclarty Topos Basics 4
Here is a link in the description
to a video explaining how to
construct subobject classifiers
and exponential objects for a category of functors from C into Set (for some category C)
• Category Theory For Beginners: Preshe...
Proofs of the results about when arrows are in arrows into power objects can be found here:
• top es inA proof
An (rough) introduction to the Mitchell-Bénabou language can be found in the following videos (I will probably release a higher quality video on this topic later):
• Mitchell-Benabou Language Rules 1
• Mitchell-Benabou Language Rules 2
• Mitchell-Benabou Language Rules 3
• Mitchell-Benabou Language Rules 4
• Mitchell-Benabou Language Rules 5
• Mitchell-Benabou Language Rules 6
• Mitchell-Benabou Language Rules 7
• Mitchell-Benabou Language Rules 8
An attempt to explain how the a partial ordering can be associated with a power object, to form an internal Heyting Algebra can be found here:
• Power objects as internal Heyting Alg...
A very rich set of examples of applications can be found in this series of videos
on categories of structured sets:
• fun 1
• fun 2
• fun 3
• fun 4
• fun 5
• fun 6
• fun 7
• fun 8
• fun 9
• Monic proofs 1
• Monic proofs 2
• Monic proofs 3
• fun 10 and pullback proof
• Subobject classifier proof
• fun 12
• fun 13q
• fun 14a
• fun 15
• fun 16
• fun 17
• fun 18
• fun 19
• fun 20
• Containment, And, Implies proofs
• fun 21
• fun 22
• fun 23
• Coproduct proof
• fun 24
• fun 25
• fun 26
• fun 27
• fun 28
• fun 29
• fun 30
• fun 31
• fun 32
• Functor categories 1
• Functor categories 2
• fun new intro
• fun fun 1 NNO
• fun fun 2 forall exists
• fun fun 3 flip
• fun fun 2 pt 1 forall
• fun fun 2 pt 2 exists
• fun fun 2 pt 3 epic
• fun fun 2 pt 2 exists more examples
• Power objects 1
• power objects 2
• power objects 3
• power objects 4 examples
• power objects 3 pt 5
coequalizer documents and videos
https://drive.google.com/open?id=17OO...
• Coequalizers for structured sets
• Coequalizers for structured sets 2
• Coequalizers for structured sets 3
• Coequalizers for structured sets 4
Geometric Morphisms
• Geometric morphisms 1
• Geometric morphisms 2
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