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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Homology functors defined on $\mathsf{Top} \times \mathsf{Top}$ in Eilenberg-Steenrod axioms?

The Eilenberg-Steenrod axioms state that a homology theory is a sequence of functors $H_n : \mathsf{Top} \times \mathsf{Top} \to \mathsf{Ab}$ satisfying some additional properties. What I don't ...
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3answers
66 views

An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
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13 views

Relationship between differential cohesion and synthetic differential geometry

I am wondering what is the relationship between differential cohesion and synthetic differential geometry? Is the latter a vertical categorification of the theory of synthetic differential geometry? ...
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35 views

Associative Law for a Monoid in a Monoidal Category.

Let $B$ be a monoidal category, and $c$ a monoid in $B$. Powers of $c$ are defined by taking $c^{n}$ to be the $\otimes $-string of length $n$ of $c$ in which the parentheses are all in front. We ...
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1answer
41 views

Generalize a result to any category.

Consider two categories $\mathscr{C}$ and $\mathscr{D}$ where $\mathscr{C}= Grp$ and $\mathscr{D}= \textbf{Set}$, then we are taking the forgetful and faithful functor $p$ (this is, we have a group ...
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1answer
32 views

Why is the the double dual functor on finite-dimensional vector spaces naturally isomorphic to the identity?

$\require{AMScd}$ Note: I have already seen this question, which asks about a specific aspect of the construction - here I am trying to construct this functor and failing at a very different stage. ...
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22 views

How to show the homotopy category is not abelian [duplicate]

Suppose $K^+(M)$ is the category, whose objects are bounded below complex of abelian groups, morphisms are chain maps modulo homotopy equivalence. How to show the category is not abelian? Exercise ...
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26 views

Representation of functions in the simplex category

I am using the following characterization of the simplex category $\Delta$: The objects are finite ordinals and the arrows are weakly monotone functions $f:n\rightarrow m$. $0$ is initial and $1$ is ...
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26 views

Pro-completion of finite algebras as Stone algebras

Recall that a profinite algebra (e.g. group, monoid, or whatsoever) is a cofiltered/inverse limit of finite algebra. In Johnstone's Stone space, he showed that finite discrete algebras are finitely ...
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1answer
58 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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77 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 ...
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128 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 ...
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37 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
58 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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45 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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1answer
68 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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79 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
89 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 $\mathbb{Q}$ ...
6
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1answer
60 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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1answer
29 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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36 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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3answers
141 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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2answers
54 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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1answer
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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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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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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1answer
88 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 ...
2
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1answer
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 ...
2
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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 ...
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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 ...
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0answers
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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31 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 ...
5
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1answer
68 views

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?
2
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1answer
31 views

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 ...
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118 views

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 ...
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 ...
3
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2answers
111 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 ...
3
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1answer
78 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 ...
6
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1answer
75 views

What are the best topics to learn for a first (and second) course in Category Theory?

I am a mathematics student in my last year of undergraduate studies and I'm taking a first Course in Category Theory. The professor that is giving the course is not a category theorist and because of ...
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2answers
52 views

Uses of Universal Properties

So in reading about category theory I'm starting to see this picture that it is just a higher level of abstraction where we consider similarities between mathematical structures by way of morphisms ...
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1answer
50 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 ...
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1answer
28 views

Prove that a functor $F:C_{(X,\leq)}\to C_{(Y,\leq)}$, being $(X,\leq),(Y,\leq)$ partially ordered sets, is just an application that is monotone.

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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35 views

Do combinatorial species have adjoints?

A combinatorial species is a functor $F$ from the category $\mathbb{B}$ of finite sets and bijections to itself. What (if anything) can be said about adjunctions of species?
3
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1answer
49 views

Abelian subcategory generated by a full subcategory.

If $\mathcal{C}$ is a full subcategory of an abelian category $\mathcal{C}'$ to what extent does the abelian subcategory generated by $\mathcal{C}$ depend on the ambient category $\mathcal{C}'$? ...
1
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1answer
42 views

Prove that every continuous mapping between $\omega$-complete partially ordered sets is monotone.

I'm a physicist trying to understand rigorous proofs in the very basics of category theory and I'm having difficulties seeing things that perhaps are trivial but just can't see them, so I need some ...
2
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0answers
32 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E ...
4
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1answer
62 views

Iterated Coproduct in a Monoidal Category; finding the unit of a monoid.

Suppose $B$ is a monoidal category and further that the functors $-\bigotimes a:B\rightarrow B$ and $a\bigotimes -:B\rightarrow B$ preserve coproducts. The we have $\theta :\coprod _{b} a\bigotimes ...
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156 views

Textbooks on higher category theory

What textbooks on higher category theory are there? What books do you recommend? I am looking for self-contained introductions, no research reports. There are lots of informal summaries and arXiv ...