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Summer Fall 2020
Thursday 3:00-4:00 pm
@ Google.Hangouts
We are continuing with working through Steven Awodey's book on Category Theory(*) that we started on Spring-2020.
(*) Note that we reference a freely available draft copy of the book, but we actually used the final version published by OxUP. There are differences, especially in the number of exercises.
In the Fall, we kept going through the book, slowly...
Present: Artem, Cameron, Donald, Julia, Olek.
Summary: Awodey's book, Chap. 2.6–7 (products and HOM-sets).
Present: Artem, Cameron, Julia, Olek.
Summary: Awodey's book, Chap. 2.4–5 (products and product examples).
We left wondering about "Curry-Howard non-isomorphism" (the book claims there is
a functor from the category of proofs to the category of types,
but not an isomorphism unless extra restrictions are imposed).
It appears that the category of types has a lot of equivalences between terms,
i.e. beta-eta equivalent terms correspond to the same arrow.
This, in particular, is important for the uniqueness of identity.
But in the category of proofs, different proofs of the same proposition
represent different arrows. Can we not build multiple identities then? Hmm.
Present: Artem, Cameron, Julia.
Summary: Awodey's book, Chap. 2.3.
Present: Artem, Cameron, Julia, Olek.
Summary: Awodey's book, Chap. 2, started section 2.3 (completed the ultrafilters example).
- discussed the last statement in 2.1 about free monoid being projective -- an observation that, following ncatlab, employs adjunction between the free monoid and forgetful functors -- something we have not covered yet,
- went through 2.2 (initial/terminal objects),
- started 2.3, got through the first couple examples, and finished with the example of ultrafilters in boolean algebras corresponding to arrows into the terminal object there.
We will go on with 2.3 next week.
Present: Artem, Donald, Julia, Olek.
Summary: Awodey's book, Chap. 2, finished section 2.1.
Note. We got stuck on the statement that "free objects in many categories of algebras are also projective".
Present: Artem, Cameron, Julia.
Summary: Awodey's book, Chap. 2, up to (including) Example 2.5.
Present: Artem, Cameron, Julia, Olek.
Summary: Awodey's book, Chap. 1, exercises 9–10.
Present: Artem, Cameron, Julia, Olek.
Summary: Awodey's book, Chap. 1, exercises 4–8.