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)

2
votes
1answer
31 views

How does the functor $\text{op}$ assign values to maps?

This has confused me before and now that I'm studying it again it still confuses me. There is a functor $\text{op}: C \to C^{op}$ for any category $C$. I have $\text{op} : \text{Hom}_C(X,Y) \to ...
0
votes
0answers
36 views

Sub(P) is complete?

Let $\mathcal{C}$ be some abstract category, and $P$ be any set valued presheaf on $\mathcal{C}$. I want to show that the set of subfunctors of $P$, $Sub(P)$, is actually a Heyting algebra. For two ...
1
vote
0answers
51 views

Isomorphic as sets. Do they mean bijective? (Kashiwara's Categories & Sheaves)

Here's the book. On page 10 it says: A set is called $\mathcal{U}$-small if it is isomorphic to a set belonging to $\mathcal{U}$. and on page 11 it says: A category $C$ is called a ...
2
votes
1answer
64 views

Going from Closed Categories to Monoidal Categories

EDIT: I trimmed down the exposition a bit. I really just wanted everyone to know what my approach has been, but what I had was a bit bloated. Suppose we have a closed category $V$ as defined here, ...
-1
votes
0answers
54 views

how to show naturally isomorphic

I have a homological exam on Saturday , and I have some problem to understand of naturally isomorphic.my problem . the end of this theorem must proof naturally isomorphic $T_n $and ...
3
votes
1answer
78 views

Iterating until a diagram commutes

I'm coming across the following 'commuting' diagram a lot in my work, and I think it should have a neat categorical formulation. But I can't work it out for myself, and don't know what too google for. ...
0
votes
1answer
25 views

In an SES of chain complexes in an abelian category two of complexes exact implies the third is exact.

Consider a short exact sequence of chain complexes: $$0_{\cdot} \rightarrow A_{\cdot} \xrightarrow{f} B_{\cdot} \xrightarrow{g} C_{\cdot} \rightarrow 0_{\cdot}$$ If any two of ...
5
votes
1answer
62 views

Does the ring of continuous functions determine $\mathbb R^n$?

I have two related questions which are just making the question asked in the title more specific: (a) Is every ring homomorphism (or maybe $\mathbb R$-algebra homorphism) between rings of the form ...
1
vote
1answer
35 views

Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories $\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over ...
4
votes
1answer
77 views

Why do dagger categories supposedly capture the structure of a Hilbert space?

A dagger functor is a contravariant endofunctor $(\;)^\dagger$ satisfying $X^\dagger = X$ on objects and $f^{\dagger\dagger}$ on morphisms. It is supposed to model adjoint maps on Hilbert spaces, and ...
1
vote
1answer
42 views

Weibel's book, Page 8. $\text{Tot}(C)$. What is the sum of the horizontal and vertical differentials in a bicomplex?

... define the total complexes $\text{Tot}(C) = \text{Tot}^{\Pi}(C)$ and $\text{Tot}^{\oplus}(C)$ by $\prod_{p+q = n} C_{p,q}$, and $\bigoplus_{p + q = n}C_{p,q}$. The formula $d = d^h + d^v$ ...
0
votes
1answer
39 views

Relation between fixed point and retraction theorem

There is this particular exercise in Lawvere/Schanuels book "Conceptual Mathematics: A first introduction to categories" that I've worked on, but I'm not entirely sure if I'm correct. Plus, I'm a ...
4
votes
0answers
47 views

Weibel exercise 1.2.2.: kernels, monics, and monomorphisms are the same in $R$-Mod.

See image below. I just want help proving that all kernels in $R$-Mod are monics. My attempt: Let $f : A \to B$ be a map in $R$-Mod. Suppose $i$ is a kernel of $f$, that is: $fi = 0$ and ...
1
vote
1answer
30 views

What does “universal w.r.t. this property” mean? (kernel of a morphism in an additive category)

In an additive category $\mathcal{A}$ a kernel of a morphism $f: B\to C$ is defined to be a map $i : A \to B$ such that $fi = 0$ and that is universal with respect to this property. This is ...
1
vote
1answer
42 views

Bridgeland stability conditions: The heart satisfies the Harder-Narasimhan property

Given a stability condition $(Z,\mathcal{P})$ on a triangulated category $\mathcal{D}$. Take $\mathcal{A}=\mathcal{P}((0,1])$. Then $\mathcal{A}$ is the heart of a bounded t-structure on ...
3
votes
1answer
83 views

Category theory: Enough that polygonal diagrams commute

I've read somewhere that for a categorical diagram to commute, it is enough that all its polygonal subdiagrams commute. I want a reference and a detailed proof of this. Please also give a formal ...
1
vote
1answer
37 views

Labeled commutative diagram

Consider a commutative diagram. For example the following diagram in $\mathbf{Set}$: $$ \begin{array}{ccc} & \overset{+1}{\longrightarrow} &\\ \mathbb{Z} & & \mathbb{Z} \\ & ...
2
votes
1answer
25 views

Algebraic theories and canonical algebras

I am reading Bodo Pareigis-Categories and functors (Pure and Applied Mathematics, Vol. 39). Let $\mathcal{U}$ be an algebraic theory (in the sense of Lawvere theories). A product-preserving functor ...
2
votes
0answers
75 views

Is this a better way to think about Groups as Categories?

I asked a bit ago how to reconcile the category theoretic and set theoretic definitions of groups (groupoid which is a monoid vs the set theoretic definition), and I got the answer I was looking ...
0
votes
1answer
110 views

Properties of Pushouts in the category of topological spaces

I have following question: $X, X_0, X_1$ and $X_2$ are topological spaces. Furthermore, $\mu_i:X_0\rightarrow X_i$ and $\alpha_i:X_i\rightarrow X$ morphisms in the category of topological spaces, ...
1
vote
2answers
70 views

A misleading commutative diagram

Let $U$ be a set, let $\phi$ be an involutive bijection of $U$ with itself. Let $A$, $B$ be subsets of $U$. Consider the commutative diagram $A \overset{\phi}{\leftrightarrow} B$ describing a ...
3
votes
1answer
93 views

Prove that all cycles are identities

In the following commutative diagram: $$ \begin{array}{ccc} A & \longleftrightarrow & B \\ \updownarrow & & \updownarrow \\ C & \longleftrightarrow & D \end{array} $$ all ...
2
votes
1answer
64 views

Inclusions of Vector Spaces vs Sets

I have a conjecture relating statements about inclusions of sets to corresponding statements about inclusions of linear subspaces. More specifically, consider a formula \begin{equation*} \phi \equiv ...
1
vote
0answers
53 views

An endomorphism $f$ such that $f\circ f=1$

What is the name for an endomorphism $f$ of a category such that $f\circ f=1$? Note that I work with category $\mathbf{Set}$.
3
votes
1answer
50 views

Is a locally $\mathcal{C}$ space a direct limit of $\mathcal{C}$ spaces?

Let $\mathcal{C}$ be a class of topological spaces (for example Hausdorff spaces, compact spaces, connected spaces, finite spaces, etc...), and let $X$ be a topological space. Is the following true? ...
6
votes
1answer
57 views

Binomial rings closed under colimits?

A binomial ring is a ring (for the purposes of this question all rings are commutative and unital) which is torsion-free and has, for each $n$, a binomial function $\binom{x}{n}$ satisfying ...
2
votes
0answers
87 views

properties of pullback diagrams

Suppose you have a commutative diagram: $\require{AMScd}$ $\begin{CD} A @>>> B\\ @VVV @VVV \\ C @>>> D \\ @VVV @VVV \\ E @>>> F \end{CD}$ Let $T$ be the top "square", $B$ ...
4
votes
1answer
37 views

The identity elements of two multiplication satisfying interchange law.

Let $M$ be a set, and $\circ_1,\circ_2$ two binary operations defined on $M$ satisfying that both $(M,\circ_1),(M,\circ_2)$ are semigroups and $(a\circ_1 b)\circ_2(c\circ_1 d)=(a\circ_2 ...
2
votes
1answer
63 views

Universal Property for isomorphic objects

Suppose $A,B$ are isomorphic objects of some category $C$. Suppose that A satisfies a given universal property does B satisfy the same? If not can you provide a counterexample for an elementary ...
3
votes
1answer
62 views

Why should we expect duality to give useful concepts in category theory?

Why should we expect the abstract notion of flipping arrows in a category to generate useful concepts from other useful ones? What exactly does flipping the direction of arrows mean and why is it ...
6
votes
1answer
54 views

How to construct a tensor product of two preadditive categories in pure categorical fashion?

Let $\mathsf C$ and $\mathsf D$ be two preadditive categories (by an preadditive category I mean a category together with compatible abelian group structure on every hom-set). I thought of ...
1
vote
1answer
56 views

Category theory with objects as logical expressions over $\{\vee,\wedge,\neg\}$ and morphisms as?

I am wondering if there is a standard definition for a category with objects as first order logical (FOL) expressions e.g. $\neg x \vee y$. It seems to me that these logical expressions would be part ...
0
votes
1answer
59 views

Definition of a Cartesian Closed Category

I would like to check with someone that the following of a Cartesian Closed Category is correct and that I did not make a mistake when translating to a notation with predicate quantifiers. Let ...
1
vote
2answers
48 views

Category of torsion free abelian groups not abelian

In this article it says that the category of torsion free abelian groups is not abelian since the map $\mu: \mathbb Z \to \mathbb Z: k \mapsto 2k$ is not a kernel. I have trouble showing this: If ...
2
votes
1answer
37 views

Categorical sum of maps

I was reading that the product and the sum of two open maps are again open (and the sum of closed maps is closed too). Since the product of ${\rm id} : \Bbb R \to \Bbb R$ with itself is not open, I ...
5
votes
0answers
65 views

Do torsionfree abelian groups form a (possibly many-sorted) algebraic category?

By "(possibly many-sorted) algebraic category", I mean "category of models of a (possibly many sorted) Lawvere theory. I have arguments for both affirmative and negative answers, and I can't find a ...
0
votes
1answer
38 views

What is a Monad in the two category $Rel$?

The 2-category $Rel$ is a category with sets as $0$-cells, relations as $1$-cells (with relation composition as composition), and inclusions as $2$-cells (with vertical composition being the fact that ...
2
votes
2answers
64 views

Is the infinite dihedral group an inverse limit of the finite dihedral groups?

The p-adic numbers are the inverse limit of the rings $\mathbb Z / p^n \mathbb Z$. Can the infinite dihedral groups be construed as some sort of inverse limit of finite dihedral groups?
2
votes
1answer
50 views

Algebraic theories and forgetful functor

Let $\bf{N}$ be the category whose objects are all the nonegative integers while the morphisms $m\longrightarrow n$ are the mappings from $m$ to $n$, considered as finite sets (so $\bf{N}$ is the ...
4
votes
1answer
224 views

When is a category isomorphic to its opposite?

I could verify that if Morph$(A, B)$ is in bijective correspondence with Morph$(B, A)$ for all objects $A, B$ in a category then one shall construct isomorphism between that category and it's ...
2
votes
3answers
62 views

Given a subset $S$, is there any universal construction (property) of $L(S)$, the linear span of $S$?

Given a subset $S$ of a vector space $V$, is there any universal construction (property) of $L(S)$, the linear span of $S$ (like the way we can construct the fraction field of an integral domain)?
2
votes
0answers
33 views

How does a symmetric graph indexing category work?

In Category theory for the sciences, section 4.2.1.20 it is explained how a graph is a functor from an indexing category to a set. I think I understand the basic concept: Ar is mapped to a set of ...
1
vote
2answers
79 views

Show that the categories $G$-mod and $\mathbb{Z}G$-mod are equivalent.

I have another basic question inspired from reading the sixth chapter of Weibel's "An Introduction to Homological Algebra". First version of the question: a bit ambiguous At the first paragraph, ...
4
votes
1answer
51 views

How to prove adjunctions compose (via units and counits)?

Familiar background (partly to fix notation). Suppose we have functors $F\colon \mathscr{A} \to \mathscr{B}$, $G\colon \mathscr{B} \to \mathscr{A}$ such that $F \dashv G$, and functors $F'\colon ...
0
votes
1answer
40 views

Posets as Categories and Direct Systems of Objects?

My algebra textbook brings the following example of category: Every poset $(X, \preceq)$ might be seen as a category as follows: Objects: Elements of $X$; Morphisms: $\textrm{Hom}(a, ...
0
votes
1answer
133 views

canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization ...
1
vote
1answer
41 views

Associativity of the additive law in a category with finite biproducts

It is well known that if $\mathcal{C}$ is a category with finite biproducts, then we can define a binary operation "$+$" on every set of morphisms $Hom_{\mathcal{C}}(X,Y)$ using the diagonal ...
7
votes
2answers
243 views

Category of binomial rings

A binomial ring is a commutative ring $R$ such that (1) the additive group of $R$ is torsionfree and (2) $n!$ divides $x(x-1)\dotsc(x-n+1)$ for all $n \in \mathbb{N}$ and $x \in R$. We may then define ...
2
votes
1answer
55 views

Sheafification as a Kan Extension of the Identity?

How can the sheafification functor be described in terms of a Kan extension of the identity on the category of $\mathsf{Set}$-valued sheaves (over some topological space)? How about general $\mathsf ...
2
votes
1answer
90 views

How is the free group on $S$ generators a cogroup?

According to nLab: Cogroup objects in the category of groups are free groups, and to give a free group the structure of a cogroup object is the same a choosing a generating set. This is an old ...