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Apr
7
comment how do contravariant 2-functors preserve adjunctions?
It depends on what you mean by "contravariant", because there are two compositions in a 2-category.
Apr
7
comment Categories of defintion for sites on spaces and sites on schemes
It's a cultural thing. In algebraic geometry one works over a base scheme, which may be something simple like a "point" ($\operatorname{Spec} k$ for some algebraically closed field $k$) or something more complicated like a space of parameters.
Apr
4
comment Pulling back along surjective étale maps vs being “locally in $\mathcal M$” vs being “locally in $\Sigma \mathcal M$”
Q1 is not really answerable for two reasons: first, because the standard notion of internal logic does not handle predicates with object variables – for that you need stack semantics – and second, because stack semantics presupposes that your predicates are local in the sense you are asking about.
Apr
4
comment Sheaf cohomology via resolutions vs. derived categories
I like to believe that the principle of conservation of work applies even in pure mathematics. The derived category approach is elegant but is difficult to understand concretely; the resolution approach is concrete but difficult to understand elegantly.
Apr
4
comment elementary question concerning definition of sifted colimit
Why do you bring up coends? You could just as well use the notation $\varinjlim_\mathcal{C} \varinjlim_\mathcal{D}$ and $\varinjlim_{\mathcal{C} \times \mathcal{D}}$.
Apr
4
comment Categorical interpretation of equality type
The equality type of $A$ is interpreted as the diagonal $A \to A \times A$ (regarded as an object over $A \times A$).
Apr
3
comment Flat Modules are Filtered Colimits of Free Modules
Between 2 and 3 you pass from $\mathbf{Ab}$-valued profunctors to $\mathbf{Set}$-valued profunctors. Why is this allowed?
Apr
1
comment Representable Functors and Upper Sets (Final Segments)
This is a wild guess, but it seems to me that you have some misconceptions about categorification.
Apr
1
comment Representable Functors and Upper Sets (Final Segments)
That's not what the statement is about.
Mar
31
comment What is the most general category in which exist short exact sequences?
While it makes sense to talk about short exact sequences as soon as you have kernels and cokernels, whether or not it is a useful concept is another story...
Mar
31
comment Why are left/right proper model categories called so?
There's that. More generally, colimits and cofibrations have to do with the left (first) variable of the hom functor while limits and fibrations have to do with the right (second) variable of the hom functor.
Mar
31
comment Why are left/right proper model categories called so?
It's the same left/right as left/right adjoint (but the opposite of left/right exact, unfortunately).
Mar
30
comment Equalizers in abelian categories
In fact it is enough. The other axioms force the canonical comparison to be an isomorphism.
Mar
29
comment A question about filtered colimits in a category of representations
You also need the fact that the forgetful functor is conservative.
Mar
29
comment Sufficient conditions for the category of group objects to have coproducts
You don't need $\mathcal{C}$ to be cartesian closed – locally presentable is enough.
Mar
29
comment Sufficient conditions for the category of group objects to have coproducts
It's not automatic – you do have to check it.
Mar
29
comment Sufficient conditions for the category of group objects to have coproducts
If $\mathcal{C}$ is locally presentable then you have a left adjoint.
Mar
29
comment A question about filtered colimits in a category of representations
The category of finite-dimensional representations is not cocomplete, obviously.
Mar
29
comment A question about filtered colimits in a category of representations
If they exist then you would have infinite direct sums, but you don't.
Mar
29
comment A question about filtered colimits in a category of representations
There are not that many filtered colimits in the category of finite-dimensional representations to begin with. Did you mean to ask about the category of all representations?