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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65 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 ...
11
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176 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
30 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
44 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 ...
3
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1answer
69 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
41 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 ...
2
votes
1answer
63 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, ...
2
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1answer
44 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
33 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
100 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
44 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 ...
3
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1answer
48 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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45 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 ...
2
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0answers
44 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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votes
3answers
85 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
38 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 ...
7
votes
1answer
95 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
votes
2answers
66 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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0answers
37 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 ...
3
votes
2answers
62 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
votes
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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0answers
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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62 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
121 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
21 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?
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0answers
44 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 ...
3
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0answers
57 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
30 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
votes
2answers
161 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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votes
4answers
247 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
votes
1answer
109 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. ...
2
votes
1answer
44 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 ...
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0answers
53 views

Classifying space infinite totally ordered set contractible

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I ...
2
votes
1answer
33 views

free product with amalgamation is correspondingly a pushout

I'm trying to proof that the following diagram in the category of groups with $i_1$ and $i_2$ being inclusions is a pushout iff $G$ is the free product with amalgamation (up to isomorphism). It should ...
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0answers
68 views

Proof of the Coherence of Monoids in a monoidal category (Final Edit)

This is not Maclane's Coherence theorem; rather, a variant. I would like a critique of step 2 of my attempted proof. Let $B=\left ( B,\square, \alpha ,\lambda ,\varrho , \right )$ be a moinodal ...
4
votes
1answer
49 views

Concrete category of topological spaces over preordered sets: what are the initial morphisms?

I am reading The Joy of Cats to become more familiar with category theory and I came upon the following question on concrete categories. Let Top be the category of topological spaces and let Prost be ...
3
votes
1answer
54 views

Why is a cartesian morphism called cartesian?

I am reading about fibred categories. After reading the definition of "vertical" morphism, I can imagine why they are named like that. What about "cartesian" morphisms? What is cartesian about them? I ...
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votes
0answers
57 views

Adjunction between topological and simplicial presheaf categories

For a small discrete category $C$, the singularization/realization adjunction $Top \leftrightarrows sSet$ induces an adjunction $Top^C \leftrightarrows sSet^C$. If I have a small topological category ...
2
votes
0answers
30 views

Characterization of epic morphisms in the category of rings.

Let $C$ denotes the category of rings (with identities). It's easy to prove that for any $R\in \text{ob }C$, any multiplicative set $S\subset R$ and any ideal $I\leqslant R$, both $R\to S^{-1}R$ and ...
2
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1answer
94 views

How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
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votes
2answers
69 views

Properties of the category of smooth vector bundles over a smooth manifold

I am wondering if there are any sources that discuss the properties of the category of vector bundles over a smooth manifold. It seems that most differential geometry texts I've looked at avoid ...
4
votes
2answers
48 views

Some lengthy question on natural transformations, category theory, and dual objects

So I am fairly new to algebra, and my instructor often uses terms from category theory (such as Hom-set, natural transformations, etc) that I am not familiar with. As an attempt to have a better ...
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Three theorems for the price of one? (like duality)

Why the notion of "duality" (when we get two theorems for the price of one) are ubiquitous in mathematics (order and lattice theory, category theory, group theory), but "triality" (three theorems for ...
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1answer
72 views

An equivalence of categories

Let $F: \Pi(X) \to \text{SET}$ be a functor, where $\Pi(X)$ is the fundamental groupoid of $X$. I have shown earlier that we can construct a covering $p: Y = \bigsqcup_{x\in X} F(x) \to X$ from the ...
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48 views

Choice of a skeleton

Suppose we are in presence of a strong enough axiom of choice (e.g., choice for conglomerates). I know that any category has a skeleton, but I would like to know if I can choose a skeleton which ...
5
votes
1answer
64 views

Correct meaning of two spaces being homotopy equivalent under a space

Let $p_0 : A \to X_0 $ and $p_1 : A \to X_1$ be two maps. I am confused about what does it mean to say that '$X_0$ and $X_1$ are homotopy equivalent under $A$'. Which of the following statements is ...
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0answers
26 views

Definition of Schur Functors on morphisms

I've been learning about Schur functors on nLab: http://ncatlab.org/nlab/show/Schur+functor A definition is given, for $R$ some finite dimensional representation of $S_n$, by the formula $S_R(-)=R ...
4
votes
1answer
78 views

Adjoints functors in scheme theory

What's useful information available to us, when we state that : If $ f: X \to Y $ a morphism of schemes, and if $ \mathcal{F} $ denotes $ \mathcal{ O }_X $ - modules, and $ \mathcal { G} $ denotes $ ...