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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Forgetful functor applied to a module

I try to find a left adjoint to the forgetful functor $U: R-Mod \longrightarrow Ab$. I considered a functor $F:Ab \longrightarrow R-Mod$ defined by $F(G)=Hom(U(R),G)$. I'm not so sure that in this ...
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equivalence of definitions for connected objects?

For an extensive category, the following conditions are equivalent for an object $C$. The representable copresheaf of $C$ commutes with coproducts. The $C=X\amalg Y\implies X\text{ or }Y$ is $0$ and ...
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The benefit of writing Banach space theory in categorical language!

I was wondering if there exists a special benefit of writing Banach space theory in categorical language? I mean does there arise a hint of the existence of a connection with other mathematical field ...
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Slick Definition of the Category of Cartesian Closed Categories

I can produce elementary definitions by just inspecting the definition on nlab, but is there a readily available abstract definition? I vaguely remember seeing that they could be defined as algebras ...
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Categorical Representation of the Set of All Strings

Let $A$ be a small preordered category. How would we define a preordered category $\mathcal A$ for all strings over $A$ (e.g., Kleene Closure) ordered by the subword order (Def'n $3.1$) $\leqslant$? ...
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Let $\mathcal{A}$ be an abelian category with enough projectives (injectives). I tried to prove that if every element $M$ of $\mathcal{D}^{b}(\mathcal{A})$ satisfies $$M \cong \bigoplus_{i} H^{i}(M)[-... 1answer 42 views Extension of field homomorphisms and pullback square Let E/k and F/k be two subextension of a field extension K/k. The following square induced by restriction functions is always pullback square (in category of sets and functions)?$$\begin{...
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A monad $T$ on $\cal C$ is a lax functor $\mathbb T : \mathbf{1}\to\bf Cat$. The lax colimit and limit of $\mathbb T$ are the Klesili and EM categories ${\cal C}^{\mathbb T}$, ${\cal C}_{\mathbb T}$ ...
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How should automorphisms of monoid actions in $\mathbf{Rel}$ be defined?

To explain the question, I'd like to start by considering monoid actions in $\mathbf{Sets}$. In this case, a monoid action is simply a functor $S$ from a monoid $M$ to $\mathbf{Sets}$. Then, one can ...
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Is there an adjoint functor to the contravariant hom functor in the category of A-modules.

I should start by saying that I don't know any category theory. However, I am reading Atiyah-MacDonald and have just learned that in the category of A-modules (where here A is a commutative unital ...
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Morphism in category theory

Morphisms in category theory map from one object to another object. E.g. for group, object is set, right? Homomorphism is morphism. For topplogy space, object is set. Homeomorphism is morphism. My ...
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Generally in most introductory university courses, finding the inverses of functions, is done in what seems to be to be a very haphazard way. Given any scalar function $f : \mathbb{R^n} \to \mathbb{R}... 0answers 67 views Elegant formalism for gluing spaces over open subsets (Vakil 17.2.B.) This question is about exercise 17.2.B. from Vakil's algebraic geometry notes. Let$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$. ... 2answers 55 views Coproduct in the category of metric spaces While discussing categories without coproducts, we stumbled with the category$\mathbf{Met}$that takes metric spaces as its objects and short maps as its morphisms. It is claimed that$\mathbf{Met}$... 1answer 37 views On essential surjectivity of the restriction functor Consider the following setting:$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$). ... 1answer 36 views Does every surjective morphism from an uncountable into a countable monoid admit a homomorphic right inverse function Let$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$with addition or multiplication). Further ... 1answer 32 views Closed maps in terms of lifting properties (analogousy to formally étale morphisms)? In continuation to this MSE question, where closed maps are characterized by "fiber thickenings", I trying to formulate this fiber thickening condition as some lifting property of$f$against some ... 2answers 56 views Coproducts in Top preserved under pullbacks? The statement which I'd like to prove is as follows. Let$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$a pullback in ... 1answer 79 views Why does this exact sequence of sheaves imply the maps are$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$-equivariant maps as ... 0answers 32 views What notation is usually used to denote the category of$R,S$-bimodules I came across the notation:$\mathbf{Mod}_{(R,S)}$. But generally, when handwritten, long superscripts/subscripts are becoming clumsy. So I'm wondering whether there's an adopted alternative. Is ... 1answer 62 views Functor of points definition of a space modeled on a site I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples. Let$(C,J)$be a grothendieck site and$...
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I'm looking for some example of categories which requires some effort to prove that it is a category (For example it is straightforward to prove that $\mathbf{Set}$ is a category, I don't want that ...
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Is there another way to describe the category $\mathbf{Top}$ of topological spaces?

The category $\mathbf{Top}_\ast$ of pointed topological spaces can be viewed as the comma category $(\bullet\downarrow\mathbf{Top})$. The objects of the category $\mathbf{Top}$ are topological spaces ...
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Localization of triangulated categories

I have been reading from the Stacks project, and Lemma 13.5.4. says: Let $F : \mathcal{D} \to \mathcal{D}'$ be an exact functor of pre-triangulated categories. Let  S = \{f \in \text{...
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Duality of diagrams for fibration and cofibration

According to May's A Concise Course in Algebraic Topology, the diagrams in the following represent cofibration and fibration, respectively if there exists an arrow diagonally (to the upper right ...
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Does this property really characterize monomorphisms?

In the post Does this property characterize monomorphisms?, I do not see how the third condition is equivalent to the others. Specifically, I require that $k_0$ and $k_1$ be isomorphisms in order that ...
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How to write the real projective plane as a pushout of a disk and the mobius strip?

I heard in topology class that the real projective plane is obtained by gluing a disk along the boundary of the mobius strip. I was wondering - how can I write this as a pushout? Also, how can I ...
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Are the components of the unit of an adjunction monomorphisms?

Looking at the list of adjunctions (in CWM) I strongly get the impression that the components $\eta_x$ of the unit $\eta$ involved are all monomorphisms. But uptil now I have missed/overlooked any ...
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Equivalence of definition of product in a category

I was reading Mitchel book on categories and the following observation without proof is given: A family of morphisms given by $\lbrace p_{i}:A \to A_{i} \rbrace$ is the product of $A_{i}$ if and only ...
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Intuition for the definition of a dense functor?

A functor $i:S\to C$ is dense if every object $c$ of $C$ is the vertex of the colimit of the following diagram $\varinjlim((i/c)\stackrel{\mathrm{pr}_S}{\longrightarrow} S \stackrel{i}{\to} C)$. I ...
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Definition 0. By a topological germ, I mean a pointed topological space. Whenever $X$ and $Y$ denote topological germs, by a morphism of topological germs $X \rightarrow Y$, I mean a neighbourhood $U$ ...
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What do you call the category of groups with surjections (injections)?

Take the category $Grp$ and throw out all morphisms that are not surjections. I think what's left is still a category. The composition of two surjections is surjective, and identities exist ...
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How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
Let $H,K$ be Hilbert spaces. Let $S$ be a Hilbert basis for $H$. (Which means that $S$ is orthogonal and the span of $S$ is dense in $H$.) For each arrow $S\rightarrow K$, there exists a unique ...