Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

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Tensor structure on category of symmetric monoidal categories, $\mathsf{SymMonCat}$.

Let $\mathsf{SymMonCat}$ denote the category with objects symmetric monoidal categories and morphisms lax symmetric monoidal functors. Can $\mathsf{SymMonCat}$ be endowed with a "tensor" ...
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Further interesting examples? Obtaining (co)monoids from dual objects

1. Context Obtaining (co)monoids from dual objects Let $(C, \otimes, I, a, l,r)$ be a monoidal category. To simplify notation (and work with string diagrams) we assume that $C$ is strict. Let $V \in ...
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Question for a full and faithful functor. [duplicate]

I wonder that if functor $T: \mathbf{C} \to \mathbf{B}$ is full and faithful, then the image of $T$ is a full subcategory of $\mathbf{B}$.
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Dual image map restricts to open sets?

A book I'm reading on category theory says that if $A$ and $B$ are topological spaces and $f:A\to B$ is continuous, then the "dual image" map $$f_*(U)=\{\,b\in B\mid f^{-1}(b)\subseteq U\,\}$...
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Confused about coproduct in $\mathbf{Ab}$: what are the inclusion maps, and what is the unique map from coproduct to arbitrary object?

Note that I have looked at similar questions/answers on this topic, but have not found the clarity I need yet as they have been a bit out of my reach of understanding. I have heard that the coproduct ...
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What are some examples of self-adjoint functors? Is this an example?

I've been trying to figure out what some examples of "self-adjoint" functors are, or when this even happens, since I've never seen this before. What I mean is if $F: \mathcal{C} \to \...
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Is Tropicalization a Functor?

Sorry if this is a basic or incoherent question, I can't seem to find any literature (written at my level of understanding of category theory) that addresses the "tropicalization" (unsure if ...
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Coproducts exist in the category of groups (need help understanding proof)

First let me put down the proof (by Serge Lang) I think this proof is beautiful but there are several points I can't get over. Here are my questions: What does "$S_i$ be denumerable if $G_i$ is ...
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natural isomorphism by right exactness

R is a local ring with maximal ideal $\mathfrak{m}$ and residue field k. M is a finitely generated R-module with a projective cover: $0 \to$ N $\to$ F $\to$ M $\to 0$. Tensor $0\to$ $\mathfrak{m}$ $\...
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Four questions about linear algebra, posets, and related topics

This is a compilation of some questions that appear when dealing with very simple notions from linear algebra. It occurred to me while trying to answer this question with an example of a poset having ...
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When is an isomorphism between representable functors natural?

Let $\mathcal{A},\mathcal{B}$ be two small categories and $\mathcal{C},\mathcal{D}$ two arbitrary categories. Let $F:\mathcal{A}\rightarrow\mathcal{B}$, $G:\mathcal{A}\rightarrow\mathcal{C}$ and $L:\...
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In a pretriangulated category, a morphism is an isomorphism if and only if its homotopy kernel and homotopy cokernel are zero

Let $\mathcal{T}$ be a pretriangulated category with suspension $\Sigma$ (assumed to be an automorphism) and a class of distinguished triangles. $v\colon Y\to Z$ is a homotopy cokernel of $u\colon X\...
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Notation for replacing a substructure by another substructure?

In set theory, suppose $A\subset B$. Then suppose we want to "replace $A$ by $C$". We have a notation for this: $B'=(B\setminus A) \cup C$. However, suppose that $A,B,C$ are groups, or ...
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Kan extenstion and left adjoint

This is a continuation of the question asked here: Kan extension "commutes" with a certain left adjoint. Let $\mathcal{A},\mathcal{B}$ be small categories and $\mathcal{C},\mathcal{D}$ an ...
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Categories with zero morphisms

I am looking to find a bunch of examples for the categories with zero morphisms which they are not additive categories. I guess one of them is the category of pointed topological spaces but I do not ...
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unique up to unique up to unique up to…

For a scheme $S$ and $S$-prestack $\mathscr{X}$, there exists a unique (up to unique (up to unique $2$-isomorphism) $1$-isomorphism.) stackification. In more higher category theory, is there a similar ...
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Kan extension “commutes” with a certain left adjoint

Let $\mathcal{A},\mathcal{B}$ be small categories, $\mathcal{C}$ a cocomplete category and $\mathcal{D}$ an arbitrary category. Consider functors $F:\mathcal{A}\rightarrow\mathcal{B}$, $G:\mathcal{A}\...
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Right exactness of quotienting out the maximal divisible subgroup

For every abelian groups $G$ let $\mathrm{d}G$ be its maximal divisible subgroup. Then $G \mapsto G/\mathrm{d}G$ is a right exact functor $\mathbf{Ab} \to \mathbf{Ab}$. Let $$ 0 \to G \...
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Non-associativeness of composition in deductive systems?

WARNING: The first three and last two paragraphs of this question concern historical/philosophical matters related to a secondary aim of the question. If you are more interested in the properly ...
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Degenerate simplex is the degeneracy of a unique non-degenerate simplex

Preface: Note that although this is a proof verification question, I'd also be happy with a link to a reference outlining a proof of the following statement. As far as I can tell this fact has not ...
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“Being an algebraic space” is local property on the target

Let $S$ be a scheme and let $\require{AMScd}$ \begin{CD} X' @>{p}>> X\\ @V{f'}VV @V{f}VV\\ Y' @>{q}>> Y \end{CD} be a Cartesian diagram in the category of sheaves on $\mathscr{S}ch/S$...
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Representability by algebraic spaces of a morphism of sheaves

Let $S$ be a scheme, $ f : X \to Y, g : Y \to Z$ a morphism of sheaves on the big etale site on $S$. Assume that $Y \times _Z Y \to Y \times _S Y$ and $g \circ f$ are representable. Then $f$ is ...
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Bounded chain complexes and the bounded derived category

Let $\mathcal{A}$ be an abelian category and consider the following categories: $\mathbf{Ch} (\mathcal{A})$, the category of cochain complexes in $\mathcal{A}$. The full subcategories $\mathbf{Ch}^\...
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Why isn't the Disjoint Union in Set a *product* in addition to being a coproduct?

So I'm understanding that in $Set$ that the cartesian product is a categorical product, and further get why the disjoint union is a categorical coproduct, but why is it also not a product? I want to ...
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Proving Segal's Category $\Gamma \simeq \mathbf{FinSet}_*^{op}$

I'm trying to show that Segal's category $\Gamma$ is equivalent to $\mathbf{FinSet}_*^{op}$, the opposite of the category of finite pointed sets with basepoint preserving morphisms. I'm intuitively ...
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What vector bundles are tangent bundles of smooth manifolds?

Given a smooth manifold $M$, we can naturally associate to it a vector bundle ${\rm T}M\rightarrow M$ called the tangent bundle of $M$. This operation induces a functor $\rm T:\rm Diff\rightarrow\rm ...
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Corestriction of a full and faithful functor

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a full and faithful functor. Consider the corestriction $F:\mathcal{C}\rightarrow F(\mathcal{C})$ of $F$ to its image. Note that for $D\in F(\mathcal{C})$, ...
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Pullback in the category of graphs

Consider the category of (undirected) multigraphs (possibly with loops) and multigraph homomorphisms. What are pullbacks in such a category? Is there an informal, colloquial and intuitive way to ...
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Adjunction isomorphisms imply full and faithful

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$ be two functors such that $\alpha:1_{\mathcal{B}}\cong F\circ G$ and $\beta:G\circ F\cong 1_{\mathcal{C}}$. I want ...
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Prove that $\Bbb Q$ is an injective object in $\Bbb Z\text{-mod}$ using the following definition.

How can I show that $\Bbb Q$ is an injective object in $\Bbb Z\text{-mod}$ using the following definition: $A$ is an injective object in $\Bbb Z\text{-mod}$ iff the morphism Hom$_{\Bbb Z}(\Bbb Z,A) \...
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Dual of a statement involving adjoint functors

Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a functor. The following conditions are equivalent: $F$ is full and faithful and has a full and faithful left adjoint $G$. $F$ has a left adjoint $G$ and ...
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Is a continuous function with this property injective?

Suppose that I have a continuous $f$ in a topological space with the following property: for every continuous $g$ and $h$ such that $f \circ g = f \circ h$ $\Rightarrow$ $g=h$ It is true that a ...
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Pairs of notation: Which one is the co-thing?

Many notaions come in pairs, to mention the most classical example, sine/cosine are related by the the fact that $\sin(\alpha)=\cos(\beta)$ if $\alpha+\beta=\pi/2$, i.e., the angles are complementary. ...
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morphism is epic?

If we have three morphisms f, g and h between objects of a category. Suppose gf = h. If g and h are epic, can we conclude that so is f ? Any help would be appreciated!
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Identities with respect to composition

In my abstract algebra textbook, when introducing category it says that morphisms should satisfy several properties and two of them are: For every object $A$ of $C$, there exists (at least) one ...
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Solution set condition and the forgetful functor $U:\textbf{Gr}\rightarrow\textbf{Set}$

Let $U:\textbf{Gr}\rightarrow\textbf{Set}$ be the forgetful functor from the category of groups to the category of sets. Let $X\in\textbf{Set}$. I want to construct the solution set $S_X\subset\text{...
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Joy of Cats Corollary 10.50

I have been studying category theory from Joy of Cats. I am stuck at proving Corollary 10.50 from Proposition 10.49. Which says that, Embeddings of concretely reflective subcategories preserve ...
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Pullback of maps out of a connected & locally path connected space

I'm reading Peter May's "Concise Course in Algebraic Topology", and I'm having trouble interpreting this yellow-highlighted line. When I've seen the expression "pullback of $f$ along $g$...
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Flabby representable sheaves

Let $S$ be a scheme. Consider some representable moduli functor $\mathcal{M}:(Sch/S)^{op}\rightarrow Set$ represented by some scheme $M$. Then for each $V\in (Sch/S)^{op}$, let define $$\mathcal{M}^{...
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arrow category and functor category

Let A be an abelian category and D the category having two objects and only one nonidentity morphism between them. The functor category A$^D$ is also abelian and it is called an arrow category with ...
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equality of subobjects

Suppose A $\xrightarrow{f}$ B $\xrightarrow{g}$ C is exact in an abelian category. f and g has factorization f = me and g = m'e' where m and m' are monic and e and e' are epic. Then e = m' as ...
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How do I understand this direct limit?

As in Hartshorne page 72, we defined the morphism between locally ringed spaces, say $(f,f^{\sharp})$ is a morphism between $X$ and $Y$. Then we have, for all $P\in X$, an induced homomorphism between ...
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What conditions on the coefficient category allow one to check isomorphisms for sheaves on stalks?

Let $\mathscr{C}$ be a category which admits small limits and small filtered colimits. For a topological space $X$, one can define $\operatorname{Sh}(X;\mathscr{C})$ to be the subcategory of $\...
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What does it mean for F(Lim X) to be 'naturally isomorphic' to Lim(F(X))

Let F be a left adjoint functor. Given any diagram X, what does it mean for F(Lim X) to be naturally isomorphic to Lim(F(X)). What are the functors here which are naturally isomorphic?
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the definition of invertible sheaf on a functorial scheme in category theory

We define a functorial scheme as in "Two functorial definitions of schemes".A invertible sheaf is important and I'm interested in category theory, so I hope to define invertible sheaf in ...
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What is the algebraic interpretation of a contracted product?

Suppose you have a linear reductive group $G$ acting on an algebraic variety $X$. Let $P\leq G$ be an algebraic subgroup and let $Y \subseteq X$ be a closed subvariety, invariant under $P$. Then a ...
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Right adjoint to the forgetful functor $\text{Ob}$

Let $\text{Ob}:\textbf{Cat}\rightarrow\textbf{Set}$ be the forgetful functor mapping a small category to its set of objects. Consider the functor $R:\mathbf{Set}\rightarrow\textbf{Cat}$ mapping a set $...
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Functor between small functor categories

Let $\textbf{Cat}$ denote the category of small categories and functors between them. Fix $\mathcal{C}\in\textbf{Cat}$. I want to construct a functor $$[\mathcal{C},-]:\textbf{Cat}\rightarrow\textbf{...
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Brouwer's fixed point theorem and the one-point topology

In the 2-dimensional case, Brouwer's fixed point theorem (BFPT) says that every continuous function $D^2\to D^2$ has a fixed point, where $D^2$ is the disk. Now fix a particular topology: pick some ...
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is it because of the factorization of a morphism?

From P309 of Rotman's Intro to Homological Algebra Thm 5.91. If A is an abelian category, then Sh(X, A) is an abelian category. The proof boils down to showing that monomorphisms $\varphi$ are kernels ...

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