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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16 views

Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
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0answers
9 views

Yoneda Lemma for 2-categories - lax version

Is there a sort of lax Yoneda Lemma for 2-categories? Here is what I seem to have proven (although I have not checked all the details): If $\mathcal{C}$ is a (weak) 2-category, $A$ is an object of ...
8
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0answers
64 views

Is it possible to develop differential geometry without points?

I read about pointless topology and locale theory, and become curious about this topic. For example, there is the concept "differential manifold" corresponds to "topological manifold". As this, are ...
3
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0answers
23 views

Heyting algebras and infinite distributive law

I want to prove that "a complete lattice satisfies the infinite distributive law $a\wedge(\vee{S})=\vee\{a\wedge s|s\in S\}$ iff it is a Heyting algebra". I proved "if" part but can't "only if" part. ...
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1answer
52 views

Is $\coprod \subseteq \prod$ true in any (complete cocomplete) Abelian category?

Consider $A_i, \; i \in I$ a collection of objects in an Abelian category with arbitrary products and coproducts $\mathcal{C}$. Is there always a functorial monomorphism $\coprod_{i}A_i ...
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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
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57 views

Pushout in $\mathsf{Set}$ where one of the maps is injective

From I.M. James' book General Topology and Homotopy Theory: Suppose we have a cotriad $$X \xleftarrow{\xi}W \xrightarrow{\eta} Y.$$ ... we might expect the pushout of the cotriad to be a ...
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1answer
42 views

Difficulties with right-adjoint-right-inverse in $\mathsf{Top}$

I'm having difficulties with section $9$ of chapter $\mathrm V$ of CWM. There is the following proposition: Proposition 1. If $G:\mathsf C\rightarrow \mathsf D$ is a faithful functor, if $\mathsf ...
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92 views

Showing a subcategory of $\mathbf{Top}$ is Cartesian-Closed

We start with some preliminary definitions (necessary because there is not much literature on this): a test map is a continuous function $\varphi:V\rightarrow X$ where $V$ is an open subspace of ...
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42 views

A monomorphism in the category of compact Hausdorff spaces is regular

Let $f \colon X \rightarrow Y$ be a monomorphism of compact Hausdorff spaces. This is just a continuous injection. I am trying to show that $f$ is regular, i.e. it is an equaliser. My first thought ...
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80 views

Motivation for the definition of an infinitesimal object

An infinitesimal object $D$ in a Cartesian closed category $\mathsf{C}$ is one for which the internal Hom functor $$(-)^D: \mathsf{C} \to \mathsf{C}$$ has a right adjoint. I am wondering what is the ...
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Some questions about synthetic differential geometry

I've been trying to read Kock's text on synthetic differential geometry but I am getting a bit confused. For example, what does it mean to "interpret set theory in a topos"? What is a model of a ...
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2answers
310 views

limits of finite dimensional vector spaces

Let $A$ be a finite dimensional vector space, and let \begin{equation*} \cdots \to A_{n+1} \xrightarrow{f_{n+1}} A_n \xrightarrow{f_{n}} A_{n-1}\rightarrow \cdots \end{equation*} be an inverse ...
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4answers
205 views

The free monoid functor is fully faithful?

For every set $A$, there is a free monoid $A^*$ and a function $i_A : A \rightarrow A^*$, such that for all monoids $Z$ and functions $j : A \rightarrow Z$, there is a unique monoid morphism $j^* : ...
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35 views

Why are morphisms of monads lax and not oplax natural transformations

Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them. A natural ...
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69 views

Proving that a category is cartesian closed

Let $Alg(1)$ be a category whose objects are sets with a unary operation, with no axioms. Morphisms of the category are functions of sets which preserve the operation. I need to show that this ...
3
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1answer
63 views

left adjointable functors

When a functor $F$ is left adjoint to some functor $G$, then one usually says that "$F$ is a left adjoint". Is this grammatically correct? Wouldn't it be more accurate to say that "$F$ is right ...
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73 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
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2answers
87 views

Is $\Bbb R$ the soberification of $\mathbb{Q}$?

I'm a beginner. I read about soberification of topological space and thought that if I soberificate $\mathbb{Q}$, for any $x \in \mathbb{R}$, the neighbourhood filter of $x$ in $\mathbb{Q}$ ...
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1answer
56 views

Proving that the free group on two generators is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$

I want to show that the free group on two elements $F(\{x,y\})$ is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$. The idea is to use the universal property of free groups to prove that ...
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217 views

Equivalent characterizations of faithfully exact functors of abelian categories

Let $F\colon \mathcal{A} \rightarrow \mathcal{B}$ be a functor of abelian categories. We will define some properties of $F$ before we state a question. Let $X \rightarrow Y \rightarrow Z$ be a ...
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2answers
35 views

Construction of the “swap map” via universal property for products in $Set$

Lemma (Universal Property for Products) let $X,Y$ be sets and let $A$ be a set along with functions $f:A\rightarrow{X}$ and $g:A\rightarrow{Y}$ then there exists a unique function ...
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1answer
27 views

Is any reflector from a presheaf category $PSh(K)$ to a topos $C$ necessarily left exact?

Let $C$ be a topos, $K$ a small category and $$ PSh(K) \leftrightarrows C $$ a reflective subcategory with inclusion $i\colon C\hookrightarrow PSh(K)$ and reflector $T$. Is $T$ left exact? $D$ ...
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140 views

Endomorphic Function Definition

I need to confirm my thinking on endomorphic functions. Since an endomorphism is just a surjective morphism on an object to itself in a category, can I alter the usual definition of a surjective ...
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603 views

Various definitions of group action

Sorry for the long post but this is a personal piece of maths, and I needed to be more precise as possible. There exists a well known equivalence between the category of $G$-sets and the category of ...
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1answer
66 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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172 views

Constructing a semigroup from a small category

The following was given as an example for a semigroup without an identity: Finite sets of matrices of varying dimensions, where the product $A*B=\{PQ \mid P \in A, Q \in B \text{ and } ...
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1answer
44 views

How to prove that $\mathrm{Proj}\left(B/J\right)$ is isomorphic to $\mathrm{Proj}\left(A/J\right)$ if $I\subset J$?

Let $B$ be a graded ring with positive degrees, and let $I$ and $J$ be homogeneous ideals of $B$. We suppose that there exists $N$ such that $I\cap B_{n}=J\cap B_{n}$ for all $n\ge N$. How to show ...
3
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1answer
43 views

If the counits of an adjunction are epimorpihsms then the right adjoint reflects monomorphisms?

Given an adjunction $F \dashv G$, I need to show that if the counits $\varepsilon_Y: FG(Y) \rightarrow Y$ are epimorphisms then $G$ reflects monomorphisms. I am completely stuck on what properties of ...
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52 views

What is identity arrow in the category Set? [duplicate]

Given is category Set Given two objects from this category, $A$, and $B$, which are sets without any other structure, there is an arrow $f: A \to B$, from $A$ to $B$, which is any total function from ...
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48 views

Is the maximum cardinality of a hom-set $2$? ($\emptyset$ and $1$)

After reading this: A set of morphisms from object $a$ to object $b$ in a category $C$ is called a hom-set and is written as $C(a, b)$ (or, sometimes, Hom$C(a, b)$). So every hom-set in a ...
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0answers
43 views

Why is it an equivalent definition of a triangulated full subcategory?

We know that an additive full subcategory S of a triangulated category T is called a triangulated subcategory if it is closed under isomorphism, shift and if any two objects in a distinguished ...
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86 views

Epimorphisms of locally compact spaces

Let $LCH$ be the category of locally compact Hausdorff spaces with proper continuous maps. Question. What are the epimorphisms in $LCH$? I suspect them to be surjective, but I haven't been able to ...
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+50

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where ...
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5answers
2k views

What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other ...
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32 views

Understanding `contramap` of a Functor [closed]

Given a Functor, which has the fmap function: fmap :: Functor f => (a -> b) -> f a -> f b that must follow Functor ...
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2answers
59 views

Right adjoint of covariant hom functor

I've constructed the left adjoint of the functor $\mathbf{Hom(A, -)}: \mathbf{Sets} \to \mathbf{Sets}$. Now I'm trying to prove that the functor does not have a right adjoint, but I'm not sure how to ...
2
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0answers
30 views

Proof of theorems in the field of banach-and $c^*$-algebras in a categorial language

At the moment I'm studying the basics in the theory of banach- and $c^*$-algebras. There are many results in the theory of $c^*$-algebra which you first prove in the unital case and then in the ...
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0answers
45 views

Evaluating composition of functors

Let $R$ be a ring and $S$ its $n \times n$ matrix ring. We consider the categories $_R Q$ and $_S Q$ of their respective left modules. We define a functor $F \colon _R M \to _S M$ by $$ F(M) = M^n $$ ...
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1answer
39 views

What does the category of $G$-set look like when $G = C_p$?

Let $G$ be a finite group. The category of $G$-set consists of finite $G$-sets as objects and $G$-equivariant maps as morphisms. Each finite $G$-set is isomorphic to a disjoint union of $G/H$'s, where ...
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2answers
60 views

What is the inverse limit?

In his book of Differential Topology, Hirsch starts a little detour in his theory in order to present a way to see things in a general perspective. The precise fragment I'm referring to is the ...
6
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1answer
360 views

Structuralist slogans

I am afraid to make a bad impression by misusing this forum but I am looking for as-many-as-possible mathematically inspired formulations and references to one (sometimes vague) idea. The idea is ...
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34 views

In a closed monoidal category, is $[-,-]$ always a bifunctor?

We say a monoidal category $\mathcal V=(\mathcal V_0,\otimes,I,a,l,r)$ is closed if the endofunctor $-⊗Y$ has a right adjoint $[Y,-]$, called the exponential, for every $Y$. The object $[Y,Z]$ for ...
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Mac Lane's Coherence Theorem: Why not just use the functors themselves?

I have a softball question on Mac Lane's proof. Suppose $B=\left ( B, \square , \alpha ,\rho ,\lambda \right )$ is a monoidal category. Fix $b\in B$. Define $W$, the (monoidal) category of binary ...
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1answer
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How general $ [X,[Y,Z]] \cong [X \times Y, Z] $ is?

In some algebraic categories (abelian groups, modules on a ring) and in pointed spaces categories the following holds: $$ [X,[Y,Z]] \cong [X \times Y, Z] $$ In wich generality this lemma holds?
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1answer
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If a composition of two maps is smooth, as well as one of the maps, then so is the other.

Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth. If any of $f$ and $g$ is smooth can we conclude that the ...
3
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1answer
56 views

Is there a name of the dual of quotient?

If $\mathcal{C}$ is an abelian category, we can consider the quotient $B/A$ when $A$ is a subobject of $B$ (i.e. there is a mono from $A$ to $B$.) It satisfies following universal property: For ...
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1answer
85 views

Showing $\lambda_1=\rho_1$ in monoidal category

For a monoidal category $\mathcal{C}$ with $\alpha_{a,b,c}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c$, $\rho_a : a \otimes 1 \rightarrow a$, and $\lambda_a: 1 \otimes a ...
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2answers
109 views

Naturality in linear algebra

Question. How can we formalize these intuitions about predicates on matrices? Let $P$ denote a predicate on matrices, so that $P(A)$ is true for some choices of matrix $A$ and false for all ...
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1answer
49 views

Suppose $\mathcal{C}$ is a category, is it true that if a composition $f\circ g$ of two morphisms is an epimorphism, then $f$ is an epimorphism?

In my "Introduction to Category Theory" class, my teacher wrote on the board something like this: "... due to the fact that if a composition $f\circ g$ of two morphisms is an epimorphism, then ...