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 ...

learn more… | top users | synonyms (2)

6
votes
1answer
89 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 ...
1
vote
2answers
56 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 ...
2
votes
1answer
53 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 ...
0
votes
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 ...
1
vote
0answers
42 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
votes
1answer
56 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
vote
1answer
44 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
votes
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
votes
1answer
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
votes
0answers
177 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
votes
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
votes
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 ...
4
votes
1answer
70 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 ...
4
votes
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
votes
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 ...
1
vote
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 ...
1
vote
2answers
101 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 ...
1
vote
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
votes
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 ...
4
votes
0answers
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
votes
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 $Y$? ...
2
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 ...
1
vote
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
96 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 ...
0
votes
0answers
38 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 $C$ ...
0
votes
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 ...
0
votes
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.)
1
vote
0answers
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 ...
3
votes
1answer
124 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.
0
votes
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?
2
votes
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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
5
votes
4answers
248 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 ...
0
votes
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 ...
1
vote
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 ...
0
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
31 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
votes
1answer
97 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 ...