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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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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32 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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1answer
69 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?
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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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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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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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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 ...
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82 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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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
52 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
32 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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0answers
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?
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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 $\mathcal{C}'$? ...
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1answer
43 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 ...
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1answer
64 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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159 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 ...
2
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0answers
28 views

Is Tambara-Yamagami category admits a braiding when G is a nonabelian group?

Tambara-Yamagami category is a fusion category which its simple objects are elements of a group and one element out of group. i.e : $$simple\;objects = G \cup \{m\}$$ The fusion rule of this ...
2
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1answer
29 views

Pullbacks in the Ind-completion

Suppose we have a category $\mathcal{C}$, say finitely complete. Does the $\text{Ind}$-completion $\text{Ind}(\mathcal{C})$ (which informally is the completion of $\mathcal{C}$ under filtered colimits ...
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1answer
59 views

Categorical Foundations text

I've heard that someone's thought up a way of using category theory, involving something called topoi, as a foundation for mathematics. If this is true then are there any texts which explain such a ...
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1answer
38 views

What is the Eilenberg-Moore category of this diagonal-like monad?

The Eilenberg-Moore category of a monad $(T:C \to C, \eta, \mu)$ has as objects pairs $(x \in Ob(C), h:Tx \to x)$ such that $h \circ \mu_x = h \circ Th:T(Tx) \to x$ and $h \circ \eta_x = id_{x}$. A ...
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28 views

etale morphism between sheaves

We knoe that if $f$ and $ f\circ g$ are both etale morphisms between schemes, then so is $g$. Does this statement hold for etale morphisms between sheavs on etale site over a scheme? More generally, ...
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1answer
36 views

Is a Linear Transformation a Vector Space Homomorphism?

I see the terms linear transformation and (vector space) homomorphism used more or less interchangeably, and the set (space) of linear transformations from V to W referred to as Hom(V, W) or ...
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1answer
32 views

Universal Property of Natural Transformations Proof Verification/ Proof tips

Let $\phi$ be a natural transformation between functors $F, F':\mathscr{C} \to \mathscr{D}$, $\tau :Arr\mathscr{D} \to \mathscr{D}$ be the identity natural transformation between the objects of the ...
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2answers
93 views

Easy examples of dual objects in Category Theory??

could any of you provide me with 1) a definition of dual object within Category Theory which could be understood by, say, sophomores in college? 2) examples of dual objects which could be understood ...
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1answer
39 views

Minimality of field of fractions expressed by functor

I'm probably just below the needed amount of prominent examples to begin studying category theory, but first of all I can't hold back the intrigue, and second I might even benefit from having "arrow ...
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1answer
46 views

Definition of absolutely presentable functor

Let $C$ be a small category and $F \in \widehat{C}$. "$F$ is absolutely presentable" is defined as "the representable functor $C(F, -):C \rightarrow Sets$ preserves all small colimits". What is ...
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42 views

Equivalence of categories ($c^*$ algebras <-> topological spaces)

I try to use a littlebit category theory to have a better overview of the results in the theory of $c^*$-algebras, but I really have to read an introduction to category theory because I know almost ...
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42 views

Homotopy groups of mapping spaces

If I have an $\infty$-category $\mathcal{C}$ (AKA quasi-category), can I say anything about the homotopy groups of the mapping spaces $\mathrm{Hom}_\mathcal{C}(X,Y)$ for two objects $X$ and $Y$? ...
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83 views

How to read category diagrams?

I have problems with very basic categorial reasoning. Suppose we have a commutative "cone" diagram: $f:A \to B$, $g:B \to C$, $h:A \to C$ Is its "commutativity" equivalent of saying: $\forall x\in A ...
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1answer
37 views

Help for the proof of Lemma for pull-backs

I am learning category theory from the book by Steve Awodey, trying to complete all the proofs, and I got stuck at one. Lemma: Given the diagram above, if the square at the right and the ...
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1answer
77 views

When is a monoid contained in a group?

As stated in the answer of Is the forgetful functor from groups to monoids right adjoint? , the forgetful functor $U:\mathbf{Grp}\rightarrow\mathbf{Mon}$ has a left adjoint $G$, and Grothendieck's ...
3
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2answers
62 views

An example of a coproduct of sheaves in the category of presheaves that is not a sheaf

For a site $C$ there is an adjunction $a\colon PSh(C)\leftrightarrows Sh(C)\colon \iota$ with sheafification $a$ and inclusion $i$. $a$ preserves finite limits. The inclusion $\iota$ does not preserve ...
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36 views

Continuous functors

Consider the category $\mathfrak{Top}$ of all small topological spaces. Let $C$ be any category and let $F:\mathfrak{Top}\longrightarrow C$ be any functor between them. Can a subcategory $D$ of ...
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2answers
60 views

Proving associativity in monoidal category: Free Monoid construction.

I am filling in the details of Mac Lane's proof of the following: If monoidal category $B$ has countable coproducts, and if the functors $-\square a$ and $a\square -$ preserve them, then the evident ...
3
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1answer
58 views

Question on the definition of a locally presentable category

According to nlab, a category $C$ is called locally presentable if it is accessible and has all small colimits. Moreover, one can show, that this conditions are equivalent to the condition of $C$ ...
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28 views

Regular epimorphisms in the category of simple, undirected graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...
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1answer
26 views

Quotients and regular epimorphism

In category theory, is a quotient the same as a regular (or extremal?) epimorphism? (Just like a subobject corresponds to a regular mono.)
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57 views

“Stable model categories are categories of modules” - Clarification about a few things

I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things: 1 - When they say that stable model categories are categories of ...
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1answer
112 views

Reference request: (categorical) commutative algebra text

I'd like a text that puts commutative algebra in a categorical framework. I'm wondering if anybody has any recommendations.
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1answer
20 views

Is the skeleton-coskeleton adjunction $sSet$-enriched?

Let $n\geq 0$ be an integer. Is the adjunction $$ \mathbf{sk}_k\colon sSet \leftrightarrows sSet\colon\mathbf{cosk}_k $$ of the skeleton and coskeleton an $sSet$-enriched adjunction?
2
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0answers
43 views

Example of a monomorphism and epimorphism that is not isomorphism. [duplicate]

I'm starting with a course of Introduction to Category Theory, and perhaps is dumb what I'm asking but I'm looking for an example of a monomorphism and epimorphism that is not isomorphism. Can you ...
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53 views

Is there a “properly categorical” description of Eilenberg-Moore algebras on a relative monad?

A relative monad is defined in Altenkirch et al. essentially as a pair of functors $J,F:\mathcal{C}\to\mathcal{D}$, a natural transformation $\eta:J\to F$, and an operation ...
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1answer
25 views

Covariant and Contravariant Functor of Fixed Set Question - Category of Sets

I am very new to Category Theory and am currently working on a simple question, I know I'm wrong, just wanted to know HOW wrong I am in my answer: Question: "Verify for Fixed set A, the operations ...
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1answer
24 views

Elements and arrows in an abelian category.

Suppose to work in an abelian category $\mathcal{A}$, so in particular for every objects $A$ and $B$, we have that $Hom(A,B)$ is an abelian group - in particular a set. My questions are: Does it ...
12
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2answers
144 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
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4answers
220 views

Elements and arrows in a category.

Suppose to have two objects $A$, $B$ in a fixed category and an arrow $\eta : A \to B$. Has an object "elements"? In the sense does the symbol $a \in A$ have sense? (in the most generic context, ...
5
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1answer
107 views

Stable epimorphisms of commutative rings

Recall that an epimorphism $f : A \to B$ in a category with fiber products is called stable (or universal) if for every morphism $C \to B$ the base change $A \times_B C \to C$ is an epimorphism. ...
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1answer
42 views

when a presheaf is a sheaf

I've seen a very natural definition when a presheaf $F:C^{op}\rightarrow Set$ is actually a sheaf. This definition used the functors $hom(-,-)$ and $F$ and notions of injective and surjective maps ...