Linked Questions
10 questions linked to/from When size matters in category theory for the working mathematician
110
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Reflection principle vs universes
In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...
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17
votes
2
answers
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Why do we care about small sets?
I have been a user of category theory for a long time. I recently started studying a rigorous treatment of categories within ZFC+U. Then I become suspecting the effect of the smallness of sets.
We ...
- 445
16
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2
answers
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Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?
Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with.
So, my understanding is that category theory and related fields of higher mathematics ...
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13
votes
4
answers
2k
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Set-Theoretic Issues/Categories
It is a major bummer that one cannot strictly speaking talk about the category of all categories without saying "it is not really a category, since the morphisms between objects may form a class" and "...
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21
votes
1
answer
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Surmounting set-theoretical difficulties in algebraic geometry
The category $\text{AffSch}_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(\...
- 3,189
9
votes
3
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Which categories are injective with respect to fully faithful functors?
Recall that a poset $K$ is a complete lattice if and only if $K$ is injective with respect to poset embeddings in that sense that for any poset $B$, any embedded subposet $A \subseteq B$, and any ...
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12
votes
2
answers
594
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Is the $\infty$-category $N_{dg}(\mathrm{Ch}(\mathcal{A}))$ presentable?
(See Jacob Lurie's "Higher Algebra", section 1.3.5 for context.)
Let $\mathcal{A}$ be a Grothendieck abelian category. Then the stable $\infty$-category $\mathcal{D}(\mathcal{A})$ is a ...
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7
votes
2
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Set theoretical foundations for derived categories
A modern approach to derived functors, that has been shown to be useful in a number of different circumstances, is that of a derived category (see the book by Yakutieli, for example, here).
However, ...
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6
votes
1
answer
507
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Are infinitary monads monadic?
As discussed here, Are monads monadic?, in "On the monadicity of finitary monads" by Steve Lack, the following is shown, the forgetful functor from $Mnd_f(C) \rightarrow Endo_f(C)$ is ...
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12
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1
answer
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Can passing to a larger Grothendieck universe ever lead to category-theoretic complications?
Suppose we use Tarski-Grothendieck set theory as a foundation for category theory (ZFC together with the assumption that every set is contained in a Grothendieck universe, or equivalently that there ...
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