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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
answered Sufficient conditions for the category of group objects to have coproducts
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?
Mar
28
comment Characterization of fully faithful functors as objects in a functor category
If you remember the evaluation functor $D^C \times C \to D$ then yes, but that's somehow tautological.
Mar
28
comment Characterization of fully faithful functors as objects in a functor category
Obviously not. For instance, suppose all of the categories involved are actually sets. Then fully faithful functors are the same as injective maps, so you are asking if bijections of function-sets preserve injectivity.
Mar
24
comment How, intuitively, does commuting with filtered colimits capture “smallness”?
Of course, it is worth pointing out that compact topological spaces are not literally compact objects in $\mathbf{Top}$...
Mar
24
awarded  Popular Question
Mar
24
comment Intuitive meaning for Kan fibration
It is a morphism that has a combinatorial version of the homotopy lifting property.
Mar
24
awarded  Notable Question
Mar
23
comment on the necessity of gluing conditions
Well, it depends on what you expect to get after you glue. Suppose the result is $X$; by abuse of notation, consider each $X_i$ as a subobject of $X$; then one would expect $X_i \cap X_j = U_{i,j} = U_{j,i}$.
Mar
23
awarded  Revival
Mar
22
comment Effective equivalence relations in a topos
I don't think so. The isomorphism should involve both symmetry and transitivity.
Mar
22
comment Effective equivalence relations in a topos
Have you tried to see what happens in $\mathbf{Set}$?