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Feb
4
awarded  Enlightened
Feb
4
comment A nice subcategory of the category of measurable spaces
In situations like this we should a class-locally presentable pretopos. I'm not so sure whether we get a cartesian closed category, or even whether we get a subobject classifier. Restricting to a small "subsite" (not standard terminology) would give a Grothendieck topos but it seems like an arbitrary thing to do.
Feb
4
comment A nice subcategory of the category of measurable spaces
I suppose coaccessibility is what we want, though. Small presheaves on an coaccessible categories are the same as (covariant) accessible functors, if I recall correctly. On the other hand, restricting to small sheaves is not going to give you a Grothendieck topos...
Feb
3
comment A nice subcategory of the category of measurable spaces
Err, the opposite of a locally presentable category is accessible if and only if it is a preorder...
Feb
3
comment A nice subcategory of the category of measurable spaces
Is the category of measurable locales accessible? Or else what do you mean?
Feb
3
comment Ref request: modelling regular theories as an injectivity condition
I've used the converse of this – that injectivity conditions are regular sequents – in my paper on internal homotopy theory.
Feb
3
comment The source-side-opposite of the arrow category
Isn't this the twisted arrow category?
Feb
3
awarded  Nice Answer
Feb
2
answered Why the reflection rule trivializes higher paths in Martin-Löf Extensional Type theory?
Feb
1
comment Definition of ind-schemes
@MatthieuRomagny How do you ensure that the pullback axiom is satisfied? With affine schemes or qcqs schemes there is no problem there.
Feb
1
comment Definition of ind-schemes
Surely it's not true in general. For example, imagine if $Y$ is a disjoint union of infinitely many affine schemes.
Feb
1
answered Definition of ind-schemes
Feb
1
comment Definition of ind-schemes
As you observe, the problem with (1) is that it depends on the choice of site. Perhaps what is intended there is for the site to only include affine schemes. In that case the forgetful functor from sheaves to presheaves preserves filtered colimits.
Jan
31
comment How do you rigidify a Bousfield localization?
As you guessed, an idempotent monad only has trivial automorphisms. (This is forced by compatibility with the unit.)
Jan
27
revised How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?
deleted 1 character in body
Jan
21
comment What structure of a monoidal simplicial model category is preserved by taking the opposite category?
Well, instead of $(-) \otimes Y$ having a right adjoint, it now has a left adjoint...
Jan
21
comment What structure of a monoidal simplicial model category is preserved by taking the opposite category?
The dual of a monoidal closed category is not necessarily monoidal closed, so there's the first problem...
Jan
21
comment Do mathematical objects disappear?
There seems to be some terminological confusion here. A prescheme in the sense of Mumford (or EGA) is exactly what we call a scheme nowadays.
Jan
15
revised Gluing affine schemes
added 16 characters in body
Jan
14
comment What is modern algebraic topology(homotopy theory) about?
@crystalline Quillen's original definition of model category allows for non-trivial (essentially) small examples, such as the category of bounded chain complexes of finitely generated abelian groups. And if one is so inclined, there are ways to modify the definition of model category so that we can get every small $(\infty, 1)$-category – of course, one then starts to wonder what the point is...