Reputation
Next tag badge:
392/400 score
103/80 answers
Badges
2 51 126
Newest
 Nice Answer
Impact
~355k people reached

Apr
9
comment Function on a Power Set
Well, you can continue your construction transfinitely...
Apr
9
answered What's up with this endofunctor $\mathbf{Aff}_k \rightarrow \mathbf{Aff}_k$?
Apr
9
comment The definition of the $false$ truth value
I have added a paragraph.
Apr
9
revised The definition of the $false$ truth value
added 473 characters in body
Apr
9
comment What's up with this endofunctor $\mathbf{Aff}_k \rightarrow \mathbf{Aff}_k$?
The definition of morphism in (1) is not complete – you should allow the vector space to vary between the domain and the codomain.
Apr
9
answered The definition of the $false$ truth value
Apr
9
comment functors with a morphism lifting property
I don't know any name for that. Grothendieck (pre)fibrations have an additional condition on the lift.
Apr
8
comment functors with a morphism lifting property
Isn't this a discrete fibration?
Apr
8
comment Is there a characterization of coverings in subcanonical pretopologies?
I think some of this is explained in Shulman's paper on exact completions.
Apr
8
comment Is category theory constructive?
No, that's even more inconvenient. Then all the time you have to check that things are cofinally small. It's far from obvious whether, say, the sheaf associated with a cofinally small presheaf is again a cofinally small presheaf. Even the fact that cofinally small presheaves are closed under limits requires a hard theorem. You would know all this if you actually tried to work with these things.
Apr
8
comment How to intrinsically think about simplicial objects.
Yes, things work much better for presheaf toposes.
Apr
8
comment Do hom-sets really live in the category Set?
No, that doesn't work. To define colimit you need to quantify over all cocones, which cannot be done in first order logic.
Apr
8
comment Is there a categorical characterization of differentiation?
There was an MO question about Kähler differentials of smooth functions. The conclusion was that it doesn't work for non-polynomials.
Apr
8
comment Is there a categorical characterization of differentiation?
How do we know that $d$ acts on non-polynomials correctly?
Apr
8
comment Is category theory constructive?
I don't agree at all that ZFC alone suffices. You of all people know how convenient it is to think about very large categories like $[\mathbf{CRing}, \mathbf{Set}]$.
Apr
8
comment Do hom-sets really live in the category Set?
I don't really agree that first-order category theory suffices to describe colimits. Finite colimits, maybe. There is a reason why CWM uses a universe axiom. Anyway, you should look at indexed/fibred category theory à la Bénabou.
Apr
8
comment Motivation for the definition of an infinitesimal object
It is an example of an infinitesimal object in a presheaf category. But then all representables are infinitesimal.
Apr
8
comment How to intrinsically think about simplicial objects.
That procedure only works well for models of cartesian theories. As is well known, a sheaf of local rings (= internal local ring in the topos of sheaves) is not literally a presheaf of local rings that happens to be a sheaf.
Apr
7
answered Is there a characterization of coverings in subcanonical pretopologies?
Apr
7
comment Is there a characterization of coverings in subcanonical pretopologies?
Another phrase that is used is "universally effective epimorphic sieves" – you'll find that in the Elephant, for example. If you have kernel pairs, then strictly universal epimorphisms are the same as universally effective epimorphisms. Anyway, universality is an important condition – otherwise you will not get a Grothendieck topology.