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Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
That's not a boolean algebra, however.
Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
I don't believe that. What's the canonical algebra structure on the $\mathbb{F}_2$-vector space of countable dimension?
Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
It isn't true. Boolean algebras are $\mathbb{F}_2$-algebras but not every $\mathbb{F}_2$-algebra is a boolean algebra. (There's an extra equation.)
Jul
17
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
No, it is not. Boolean algebras are $\mathbb{F}_2$-algebras.
Jul
16
comment Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?
Isn't every finitely generated boolean algebra free?
Jul
16
comment Definition of (left) resolution
I think you are missing intuition. Did you try thinking concretely in terms of abelian groups?
Jul
16
comment Definition of (left) resolution
The proof works both ways, no?
Jul
15
comment Definition of (left) resolution
Do you know the so-called first isomorphism theorem?
Jul
15
comment Definition of (left) resolution
Well, did you use the hypothesis of exactness?
Jul
15
comment Definition of (left) resolution
OK, and did you show that $H_n = 0$ for $n \ne 0$?
Jul
15
comment Algebra structure on dual to coalgebra
Use the unit–counit equation to get rid of the S-shaped bend, and then push past the braiding.
Jul
15
comment A proof that right adjoints preserve limits?
There is something to be extracted here. First of all, the limit cone is encoded into the counit of the adjunction $\Delta \dashv \varprojlim$. Secondly, the proof that adjoints compose also tells you something about the counits. So you just need to unfold these proofs a bit.
Jul
15
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
Let me be more explicit: you should change your notation. That might alleviate some confusion.
Jul
14
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
Then maybe you should think about it more carefully.
Jul
14
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
There is a problem with your notation – $s$ and $p$ both depend on both $C$ and $D$.
Jul
14
comment Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$
Keep unwinding the definitions!
Jul
13
comment Is the category of (pre)sheaves over a singleton isomorphic to the category of sets?
Yes, sheaves of sets on a point are equivalent to sets. For presheaves, well, that depends on what you mean...
Jul
13
comment In what kinds of categories is a monic epi an isomorphism?
A monomorphism that is an extremal epimorphism is an isomorphism, more or less by definition. Extremal epimorphisms seem to be the weakest commonly used notion of epimorphism with this property, and quite conveniently, it seems that every reasonably well-behaved epimorphism is extremal.
Jul
13
comment In what kinds of categories is a monic epi an isomorphism?
This is precisely the definition of a balanced category. Incidentally, your characterisation of epimorphisms in the category of topological spaces is incorrect. (They are exactly the surjective continuous maps.)
Jul
13
comment Simple question in “Sheaves in geometry and logic”
It is a general fact about commutative squares involving binary products of that form. You can deduce it from the pullback pasting lemma, if you like.