# Tagged Questions

Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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### $F$ is an equivalence of categories implies that $F$ is fully faithful and essentially surjective

I read in wikipedia that: One can show that a functor $F : C → D$ yields an equivalence of categories if and only if $F$ is full, faithful and essnetially surjective. I'm trying to prove this but I ...
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Any extensive category admits a notion of connected object and hence a disconnected object. However, not all disconnected objects are presentable as disjoint unions of connected objects. Among ...
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### Homology as categorification of Euler characteristic

I am trying to understand: "Thus, the homology of a manifold M can be seen, in a sense, as a categorification of its Euler characteristic: a more sophisticated and richly structured bearer of ...
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### What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where ...
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### Finitely generated projective modules form exact category

I am looking for a proof of the fact that the category of f.g. projective modules over some commutative ring is exact. Actually, I would already be happy to know why this category is closed under ...
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### Exact adjoint functors of triangulated categories

Let $T$ and $S$ be triangulated categories. Let $F:T\rightarrow S$ and $G:S\rightarrow T$ be two adjoint functors. Assume that one of them is exact (i.e. sends exact triangles to exact triangles and ...
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### Why does the name “epimorphism” refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are ...
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### Non-injective monomorphisms

I am reading Borceux, vol. 1, I found this example at page 27: we consider the category whose object are the pairs $\langle X,x\rangle$ where $X$ is a topological space and $x$ a point of $X$ (base ...
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### Is there some kind of deep relationship between substitution and recursion?

Define $\mathbb{N}$ as the initial object in the following category: Objects. Sets $X$ equipped with a function $S : X \rightarrow X$ and an element $0:X$. Morphisms. Functions that preserves ...
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### Representable bifunctors

Is there a notion of representability for functors in the form $F:C^{op} \times C \to Set$? Can anyone please give me a reference? Thanks.
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### Regular coimages in a finitely complete and finitely cocomplete category

According to the nLab, in a finitely complete and finitely cocomplete category $\mathcal{A}$, for every morphism $f : A \to B$ a regular coimage exists and is given by the coequalizer of $f$'s kernel ...
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### Free cocompletion of thin categories

Let $C$ be a thin category. Is there a cocomplete thin category $C\to C'$ such that for every functor $C\to D$ into a cocomplete thin category there is a unique, up to isomorphism, cocontinous functor ...
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### Characterisation of adjunction via initial object of a comma category

The following is a definition given in my lecture notes that I'm not sure is standard, so I'll write it out: Given a functor $G: \mathcal{D} \to \mathcal{C}$ and an object $x$ in $\mathcal{C}$, we ...
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### Existence of some categories in the category of categories

I'm studying category of categories. I read that when there are categories $A,B$, it is allowed to define the product $A\times B$. Equalizers and coequalizers also exist. However, there are some ...
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### Elegant formalism for gluing spaces over open subsets (Vakil 17.2.B.)

This question is about exercise 17.2.B. from Vakil's algebraic geometry notes. Let $X$ be a scheme. The following data is given: For each affine open set $U \subset X$ a scheme $\pi_U :Z_U \to U$. ...
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### Coproduct in the category of metric spaces

While discussing categories without coproducts, we stumbled with the category $\mathbf{Met}$ that takes metric spaces as its objects and short maps as its morphisms. It is claimed that $\mathbf{Met}$ ...
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### On essential surjectivity of the restriction functor

Consider the following setting: $P$ and $Q$ are two finite posets, and $i\colon P\to Q$ is a fully faithful embedding of $P$ in $Q$ (that is, $p\leq p'$ in $P$, if and only if $p\leq p'$ in $Q$). ...
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### Does every surjective morphism from an uncountable into a countable monoid admit a homomorphic right inverse function

Let $M$ be a an uncountable monoid (like $\mathbb R$ with addition or multiplication) and $N$ be a countable monoid (like $\mathbb N_0$, or $\mathbb Z$ with addition or multiplication). Further ...
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### Closed maps in terms of lifting properties (analogousy to formally étale morphisms)?

In continuation to this MSE question, where closed maps are characterized by "fiber thickenings", I trying to formulate this fiber thickening condition as some lifting property of $f$ against some ...
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### Coproducts in Top preserved under pullbacks?

The statement which I'd like to prove is as follows. Let $A=\coprod_{i \in I} A_i$ be the coproduct of the sets $\left(A_i\right)_{i \in I}$ and suppose that we have for all $i \in I$ a pullback in ...
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### Why does this exact sequence of sheaves imply the maps are $G$-equivariant?

I'm confused about something in Tamme's Introduction to Étale Cohomology (page 27). Let $G$ be a group and let $G$-$\mathsf{Set}$ denote the category of left $G$-sets with $G$-equivariant maps as ...
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### What notation is usually used to denote the category of $R,S$-bimodules

I came across the notation: $\mathbf{Mod}_{(R,S)}$. But generally, when handwritten, long superscripts/subscripts are becoming clumsy. So I'm wondering whether there's an adopted alternative. Is ...
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