Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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### If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?

I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...

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### Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra:
The space of ...

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108 views

### How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...

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162 views

### What is the definition of a $\mathcal{U}$-category?

Let $\mathcal{U}$ be a universe. The adaptation of the concept of a locally small category to universes is a $\mathcal{U}$-category.
There are two definitions of $\mathcal{U}$ category I've met.
$...

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**1**answer

340 views

### Endomorphisms in the derived category

(Apologies if this question is trivial, but I'm way outside my area here.)
Let $R$ be a commutative ring, $C^{\bullet}(R)$ the category of complexes of $R$-modules, and $D^{\bullet}(R)$ its derived ...

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### DG functors along which contractions can be lifted

For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...

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230 views

### Understanding the functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point.
Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...

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107 views

### Quantum groups at $q=-1$

For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...

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113 views

### In a fibration, how do properties of arrows downstairs affect the base-change functors?

Fix a fibration of categories. Suppose $f:A\to B$ is an arrow in the base.
What are the relations between the following pairs?
$$f\text{ epi}\qquad f^\ast \text{ faithful}$$
$$f\text{ mono}\qquad f^...

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142 views

### Testing for equivalences of $\infty$-categories on strictifications?

It is in general not too hard to show that maps between finite $CW$-complexes/finite simplicial sets are homotopy equivalences.
Question : Can we do something similar for:
quasi-categorical ...

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200 views

### Constructive proof of exponential law in a category

I'm trying to write a constructive proof of the isomorphism ${Z}^{X \times Y} \equiv (Z^Y)^X$ in a category with exponential objects.
I can construct a map $(Z^Y)^X \rightarrow {Z}^{X \times Y}$, but ...

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144 views

### Tannaka duality for $DG$ indschemes

In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism
$$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$
where $X$ and ...

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407 views

### Turning a category into a semigroup

Consider a small (that is objects and morphisms belonging to a Grothendieck universe) category $\mathcal{C}$, whose objects are all small sets, and an identity-on-objects functor ${\uparrow}: \mathbf{...

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62 views

### sequences of iterated orthogonals (lifting property) in a category

I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property.
For example, several iterated orthogonals of $ \emptyset\...

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**1**answer

78 views

### Equivalent conditions to be a lax idempotent contravariant monads

$\require{AMScd}$As already said, a contravariant monad is "like a monad, but a contravariant functor":
the multiplication and unit are dinatural transformations
an algebra is a map $a : TA\to A$ ...