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
0answers
42 views

Why are there no naturality condition in definition of exponential in a category?

In chapter 6 "Exponentials" of "Category Theory" Second Edition, Steve Awodey, 2010, there is no naturality requirement in the definition of exponential in a category. Exponential is an adjoint and ...
2
votes
0answers
68 views

Is the category of topological spaces coregular?

Everything is in the title. The category Top of topological spaces and continuous mappings is not regular, but is it coregular ? Furthermore, Top isn't cartesian closed, but does it satisfy the dual ...
2
votes
0answers
157 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
2
votes
0answers
44 views

Is the requirement that every morphism factors as an epi composed with a mono part of the definition of an abelian category?

Hilton and Stammbach require an abelian category to be an additive category in which 1) all kernels and cokernels exist 2) all monos are the kernel of their cokernel, all epis are the cokernel of ...
2
votes
0answers
130 views

Prove the FHHF theorem using as much abstract non-sense as possible

This is my second attempt to solve exercise 1.6H from Vakil: Assume $F$ is a covariant right-exact functor and show that we get a map $HF^i \rightarrow H^iF$. Attempt to a solution: Apply $F$ to the ...
2
votes
0answers
35 views

Do graded modules form an additive category?

I got two conflicting references. On page 466 of Shastri's Algebraic Topology, he claims that graded modules with homogeneous morphisms form an additive cat, whereas Hilton and Stammbach say on page ...
2
votes
0answers
60 views

About details of the Fakir theorem proof (associated idempotent triple)

On the ncatlab work http://ncatlab.org/toddtrimble/published/Associated+idempotent+monad+of+a+monad Todd Trimbe quote the Fakir theorem about the associated idempotent triple, and this is based on ...
2
votes
0answers
73 views

Is there such a thing as “combinatorial category theory”?

According to wikipedia, In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations Is there such a thing as ...
2
votes
0answers
98 views

Does $\mathsf{Man}$ possess countable products?

In our lecture we defined a category $\mathcal C$ to have arbitrary products, iff every diagram $A:\mathcal J \to \mathcal C$ with $\text{Morph}(\mathcal J)=\{\text{id}_j\}_{j \in Ob(\mathcal J)}$ has ...
2
votes
0answers
119 views

How to prove the equivalence of the two definitions of “equivalence of categories”?

As far as I know, there are two definitions of "equivalence of categories". The first definition of the equivalence of categories $A$ and $B$ is required that there exist functors $F\colon A \to B$ ...
2
votes
0answers
90 views

Is there a topos in which the natural numbers object are the finite dimensional vector spaces?

I recall reading somewhere that there is a topos in which the Dedekind reals are exactly the measurable functions. Now vector spaces are prominently characterised by dimensionality. This prompted the ...
2
votes
0answers
52 views

Stable fiber products of commutative rings

Let $R_1 \to T$ and $R_2 \to T$ be homomorphisms of commutative rings. Consider the fiber product $R=R_1 \times_T R_2$. Let $R \to R'$ be a homomorphism of commutative rings, and define $R'_i$ to be ...
2
votes
0answers
72 views

Tensors and Equalizers (or Symmetric tensors)

Quick question: Let $(\text{Sym}(V^{\otimes n}), e)$ be the equalizer of all the $S_n$ braidings (that is for $\sigma \in S_n$ we have $a_1\otimes\cdots\otimes a_n \mapsto a_{\sigma 1} \otimes ...
2
votes
0answers
49 views

Radical of Pullback

Let $Y$ be the pull-back of $M \to X \leftarrow N$ in mod$A$, where $A$ is a Artin algebra (or finite dimensional algebra). Is there any relationship between the rad$Y$, rad$M$ and rad$N$? ...
2
votes
0answers
52 views

Is $\Delta$ dense in $\bf Cat$?

I'm trying to prove that the functor $i\colon\Delta\to{\bf Cat}$ which regards $[n]\in\Delta$ as a category is dense, to deduce that the nerve $N\colon\bf Cat\to sSet$ is fully faithful: how can I do? ...
2
votes
0answers
26 views

Are lax kernel pairs stable under change of base?

The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps ...
2
votes
0answers
69 views

Associated sheaf functor for sheaves over complete Heyting algebras

Suppose $A$ is a complete Heyting algebra and $a : \widehat{A} \to \text{Sh}(A)$ is the associated sheaf functor (where the topology on A is the usual sup topology). Is there a simpler description of ...
2
votes
0answers
53 views

When is Kleisli category regular?

Are there any known conditions on the monad $\mathbb{T}$, such that $\mathcal{C}_{\mathbb{T}}$ (in particular when $\mathcal{C} = \mathbf{Set}$) is regular?
2
votes
0answers
99 views

What is the coimage of a ring homomorphism?

Is there a general way to find the coimage of a ring homomorphism? For example, for the canonical injection $\mathbb{Z}\rightarrow\mathbb{Q}$, the image is $\mathbb{Z}$ but the coimage is ...
2
votes
0answers
84 views

How to denote the set of binary relations of which a particular ordered pair is a member?

Given a universe $U$ and two subsets $S$ and $T$ (also, both members of $U$), what is the name given to denote the set of all binary relations in $U$ where the ordered pair $(S,T)$ is a member? The ...
2
votes
0answers
85 views

When terminal objects are separators?

In category of Set the terminal object is also a separator (or generator). But this is not correct in general category. What condition would guarantee that the terminal object of a category if exists ...
2
votes
0answers
43 views

How to pose a decent counting problem of cells in higher category?

Suppose you are given a power series $$S = \Sigma_{i=0}^{\infty}a_ix^{i}$$ with coefficients in $\mathbb{N}$, and you are tasked with telling if there can not be a finite category (or any kind of ...
2
votes
0answers
47 views

Categories of sheaves of finite homological dimension revisited

This is the question When is the category of (quasi-coherent) sheaves of finite homological dimension? revisited. I made some mistakes: I posted the question before I sign in, I tried to make it more ...
2
votes
0answers
118 views

Pullbacks as manifolds versus ones as topological spaces

Let $Y_1\overset{f}{\longrightarrow}X\overset{f_2}{\longleftarrow} Y_2$ be smooth maps with a common target. Suppose that we have a pullback $Z$ of the diagram in (Mfd). Questions: Suppose that we ...
2
votes
0answers
246 views

Concrete category and “abstract category”

From Analysis and Its Foundations By Eric Schechter A concrete category consists of a collection of objects and a collection of morphisms. I am curious what an "abstract category" is? I found ...
2
votes
0answers
75 views

Picturing resolutions of complexes

I got a question about resolutions of complexes, I just wanted to make sure I'm looking at them the right way. Let $\cdots \rightarrow P^{-1} \rightarrow P^{0} \rightarrow X \rightarrow 0 ...
2
votes
0answers
66 views

Inductive vs projective limit of sequence of split surjections II

This question is a follow-up of this earlier question I asked. Let $$ A_1\twoheadrightarrow A_2\twoheadrightarrow A_3\twoheadrightarrow A_4\twoheadrightarrow \cdots $$ be an inductive sequence of ...
2
votes
0answers
40 views

Formulating Cauchy Completeness in the category of linear spaces using categorical limits and colimits

Formulating Cauchy Completeness in the category of linear spaces using categorical limits and colimits, can it be done? I've made some naive attempts replace cauchy sequence with sum and taking the ...
2
votes
0answers
41 views

Name for the initial object of a connected component of a category

I was just wondering whether there is already a name for an object of a category as described below. Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that ...
2
votes
0answers
56 views

Quivers with a binary operation on the arrows

A set with an arbitrary binary operation is called a magma. A set of dots with a set of arrows between them is called a quiver. A category is a quiver with a binary operation on the arrows obeying ...
2
votes
0answers
101 views

Involutive Functors

Let us call a (co)functor $F:\mathcal{C}\to\mathcal{C}$ on a category $\mathcal{C}$ involutive, if $F^2$ is naturally isomorphic to $1_\mathcal{C}$. For example, if ...
2
votes
0answers
39 views

Fibered Product of Subcategories

Is there a general construction or existence theorem for the fibered product of two subcategories of some ambient category? What sort of problems might one run into? Does this require a 2-categorical ...
2
votes
0answers
81 views

Thompson's group F and monoidal categories

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category generated by one object $A$ such that there exists an isomorphism $\alpha: A \otimes A \to A$, then for every object $X ...
2
votes
0answers
87 views

Morphisms with an arbitrary number of objects

Is this structure familiar for you? It consists of a category $C$ a set $M$ a function ``$\operatorname{arity}$'' defined on $M$ a function $\operatorname{Obj}_m$ defined for every ...
2
votes
0answers
83 views

The Two Eilenberg-Moores

So, there is the Eilenberg-Moore spectral sequence, and there is (for any monad $(T,\mu,\eta)$ on a category $C$) the Eilenberg-Moore Category $C^T$ of $T$-algebras. The silly question, is the ...
2
votes
0answers
64 views

Having trouble finding where a functor sends morphisms.

Suppose $\mathcal{C}$ is a locally small category with coproducts, and there is a functor $G: \mathcal{C} \to \operatorname{Set}$ which is representable, with representation $(A, x)$. I am trying to ...
2
votes
0answers
91 views

Subgroup which “becomes normal in” a particular category

Is there a name for a subgroup $H < G$ which is not necessarily normal, but such that, for any $f : G \rightarrow Y$ where $Y$ belongs to a particular category (e.g. finite groups, linear groups, ...
2
votes
0answers
113 views

Associative Product Bifunctor

Let $\mathcal{C}$ be a category with finite products. Under the assumption of the Axiom of Global Choice it is posible to define the functor $$\mathbf{Prod}\colon \mathcal{C} \times ...
2
votes
0answers
109 views

A natural homomorphism of dual modules which is not a monomorphism?

I've been mixing up my reading with a little category theory. When thinking about dual modules, this idea popped up. Suppose $M$ and $N$ are modules over some commutative ring $A$. Suppose $M^\vee$ ...
2
votes
0answers
91 views

Any amalgam of a set of generators is a generator, in the category of $S$-sets.

Let $S$ be a monoid and denote by $Act-S$ the category of $S$-sets. I am having a problem understanding a step in the proof of the fact that if $\left\{G_i\right\}_{i \in I}$ is a set of generators ...
2
votes
0answers
83 views

Image of projective objects.

Let $A$ and $B$ be two abelian categories. Assume that there exist a functor $F$ between them which is exact, full and essencially surjective. If $x$ is a projective object in $A$, then $F(x)$ is a ...
2
votes
0answers
165 views

Universal property - inverse limit in the category of sets

Let $\mathscr{J}$ be a small category and $\mathscr{C}$ the category of sets. Suppose we have a functor $F:\mathscr{J} \to \mathscr{C}$ (so that there is an $A_i \in \mathscr{C}$ for each $i \in ...
2
votes
0answers
152 views

Constructing a semigroup from a small category

The following was given as an example for a semigroup without an identity: Finite sets of matrices of varying dimensions, where the product A*B={PQ|P in A & Q in B & dim(Q)=codim(P)}, and ...
2
votes
0answers
76 views

Making a Family of Problems into a Category and defining the Morphisms

This question relates to my previous question found here: Defining Category of Problems Let $\left\{\Pi_i \right\}_{i \in I}$ be a family of problems. Let the solution $u_i$ of $\Pi_i$ lie in some ...
2
votes
0answers
60 views

Choosing a specific model of a Lawvere Theory

Models for Lawvere theories are functors that preserve products. I'm a little confused about this detail: suppose you have a Lawvere theory for groups. If you have a specific group in mind, is there ...
2
votes
0answers
104 views

Field reductions. part three

Follow-up to Field reductions. part two For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting ...
1
vote
0answers
28 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
1
vote
0answers
41 views

Examples and definition of cocompact objects

An object $X$ of a locally small category $C$ that admits filtered colimits is called compact if $$ \operatorname{Hom}_{C}(X,-) $$ preserves filtered colimits. Let $C$ be a locally small category ...
1
vote
0answers
56 views

Characterizing the real numbers as a dense complete monoidal poset

Let $P$ be a poset with a monoidal structure respecting the poset structure. This means there is an operation $P \times P \to P$ such that $a \leq b \implies ac \leq bc$. As usual, call a poset ...
1
vote
0answers
57 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...