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)

4
votes
1answer
90 views

Recovering an object from its category

Consider the category of groups (but the question arises for any category of mathematical object, basically). It is easy to read off what the automorphism group of a group is or what its cardinality ...
2
votes
1answer
55 views

How to find a direct product in a category?

Having a category how to find a direct product in this category? Is it entirely a guesswork (try this function, try that one) or is there a method for this?
2
votes
1answer
67 views

Is a functor category of an $\mathbf{Ab}$-category an $\mathbf{Ab}$-category itself?

In Weibel's An introduction to homological algebra, exercise 2.6.4 reads Show that $\operatorname{colim}$ is left adjoint to $\Delta$. Conclude that $\operatorname{colim}$ is a right exact ...
1
vote
1answer
51 views

Morphisms in the category of group presentations

What are the morphisms in the category of group presentations?
3
votes
1answer
88 views

$F:C\to D$, $G:D\to E$, $G$ has an adjoint, $F$ is fully faithful and for each $Z$ there is $X$ s.t. $F(X) = H(G(Z))$: Does $F$ has an adjoint?

Suppose $F:C\to D$ and $G:D\to E$ are functors. Assume that $F$ is fully faithful, $G$ has a left adjoint $H:E\to D$, and for each $Z \in E$ there exists $X \in C$ such that $F(X) = H(Z)$. Does $F$ ...
0
votes
1answer
68 views

$M\cong_{R\operatorname{-Mod}}N$ if and only if $M\cong_{\operatorname{Ab}}N$ (Warning: This statement is FALSE!)

I was rather surprised by the fact that two modules are isomorphic if and only if their abelian group structures are isomorphic. I might just sketch the proof here. Given a ring homomorhpism ...
6
votes
1answer
293 views

Is it possible to define a ring as a category?

Is it possible to define a ring as a category? For example, a group can be defined as a category with just one objet and all morphisms being iso.
2
votes
0answers
59 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 ...
3
votes
1answer
90 views

When is the category of (quasi-coherent) sheaves of finite homological dimension?

Let say from the beginning that my background is category of modules over a ring. So I know that if we take a given (nice) scheme $X$, then category of sheaves on $X$ is Grothendieck, so it must have ...
7
votes
1answer
100 views

Can we really understand $R$ by studying $R$-modules? [duplicate]

According to Algebra: Chapter 0, the category $R\operatorname{-Mod}$ reveals a lot about $R$. However after completing the first eight chapters, I still found no examples where this happens. Can ...
5
votes
1answer
82 views

What is the cohomological explanation for the Condorcets voting paradox?

according to the nlab entry on the Condorcet Paradox in social choice (that is voting preferences may be circular even if voters preferences are not) has a cohomological explanation - what is it?
3
votes
1answer
79 views

Subcategory of Isomorphisms

There is a functor $\mathit{Iso} : \mathbf{Cat} \rightarrow \mathbf{Cat}$ which identifies the subcategory of a category in which only the isomorphisms appear as arrows — i.e. it strips off any ...
6
votes
2answers
634 views

Why is an empty set not a terminal object in categories $\mathsf{Top}$ and $\mathsf{Sets}$?

From Awodey: In any category $\mathsf{C}$, an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$, an object $1$ is terminal if for any object $C$ there is a unique ...
1
vote
3answers
99 views

Definition of limit in category theory - is $X$ a single object of $J$ or a subset of $J$?

Let $F : J → C$ be a diagram of type $J$ in a category $C$. A cone to $F$ is an object $N$ of $C$ together with a family $ψ_X : N → F(X)$ of morphisms indexed by the objects $X$ of $J$, such ...
2
votes
3answers
279 views

Can morphisms in the category Set be partial functions?

In Set, objects are taken to be sets, with morphisms as functions. There are two questions, both are possibly related: Are the morphisms in Set required to be total functions, as opposed to partial ...
0
votes
1answer
45 views

$F:\bf C\to\bf D$ a functor with a right adjoint $G$ and $\bf S$ a full subcat of $\bf C$: When does the inclusion have a right adjoint?

Suppose a functor $F:\bf C\to\bf D$ has a right adjoint $G$, let $\bf S$ be a full subcategory of $\bf C$, and denote by $I$ the inclusion of $\bf S$ into $\bf C$. What are non-trivial assumptions ...
4
votes
1answer
244 views

Representable functors

Is the fact of being "representable" only defined for functors $\mathcal{C}\rightarrow\mathbf{Sets}$, or is there some similar concept for other kinds of functors? For example in an exercise sheet ...
4
votes
3answers
180 views

How to show two functors form an adjunction

Say I have two functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G:\mathcal{D}\rightarrow\mathcal{C}$. How can I show they form an adjunction without writing explicitly the natural transformations ...
3
votes
0answers
54 views

Covariant functors represented by schemes

Let $S$ be a nice base scheme and let $F : \mathsf{Sch}/S \to \mathsf{Set}$ be a functor. Are there necessary and sufficient conditions that $F$ is represented by some scheme $X$, i.e. $F \cong ...
4
votes
0answers
77 views

Products of sites

Does the category of sites (i.e. small categories equipped with a Grothendieck topology) has products? Is there a connection to the product of locales (as discussed in Johnstone's Stone spaces, ...
6
votes
3answers
1k views

Right-adjoint functors are left-exact?

As a final exercise to VIII.1 in Algebra: Chapter 0, we are asked to prove If $\mathcal{F}\colon\operatorname{R-Mod}\to\operatorname{S-Mod}$ is a right-adjoint operator, then $\mathcal{F}$ is ...
1
vote
1answer
32 views

Generators of equivalent rings

Let $A, B$ two rings. I know that $G \in \operatorname{mod}-A$ is a generator for $\operatorname{mod}-A$ if and only if $\operatorname{Hom}(G,-)$ is a faithful functor from $\operatorname{mod}-A$ to ...
0
votes
1answer
61 views

Is there a systematic account of the number systems in the following 3x3 grid?

Consider the following sets of numbers, viewed as number systems with signature $(+,\times,\leq)$. Let $\mathbb{X} = \{1,2,3,\cdots\}$ denote the nonzero natural numbers. Let the completion of ...
2
votes
1answer
49 views

What is the name for the intermediary object(s) of functional composition?

Consider two morphisms: $f : X \to Y$ and $g : Y \to Z$ , and their composition: $g \circ f : X \to Z$. What is the name given to the role of $Y$ with respect to $g \circ f$? Is there a naming ...
9
votes
4answers
658 views

Calculus and Category theory

Quick question: Is it possible to differentiate a function with respect to another function, or is it limited to a particular variable? I tried thinking around how to make this question make sense, ...
4
votes
1answer
138 views

Quotient group as colimit

I have been wondering for a while about the following question without getting anywhere: Let $G$ be a group, $N$ a normal subgroup. Can the quotient group $G/N$ be seen as the (category theoretical) ...
2
votes
1answer
82 views

Joyal-Tierney definition of locally isomorphic objects

I am struggling with Joyal-Tierney's paper Strong stacks and classifying spaces, (appeared in "Category Theory (Como, 1990)", volume 1488 of LNM, pp. 213–236, Springer 1991). In particular one of the ...
5
votes
0answers
110 views

Directed and projective limit in Rel

I'm looking for the directed (or equivalently projective) limit of a directed family of relations in the category of sets with relations between them. Consider a family $\{ R_{ij} \subseteq U_i ...
5
votes
1answer
190 views

Additive category and zero map

Let $A$ be an additive category. Namely $A$ has a zero object, $A$ has finite products and coproducts, and Every Hom-set is an Abelian group such that composition of morphisms is bilinear. ...
3
votes
2answers
70 views

If $\mathcal F$ is a sheaf, then is $\mathcal F (- \times X)$ a sheaf?

Let $\mathcal F$ be a sheaf of sets on a site. Fix an object $X$ of the underlying category of the site, which is assumed to contain a final object and have products. Define a presheaf $\mathcal G$ ...
10
votes
2answers
264 views

Natural numbers objects in topoi: Recursion in a parameter

I am currently trying to prove an exercise from Sheaves in Geometry and Logic: A First Introduction to Topos Theory by Mac Lane and Moerdijk about natural numbers objects. First, we have the ...
0
votes
1answer
40 views

Paths between 0-cells in a classifying space. II

Let $\mathcal{C}$ be a small category and $X,Y$ objects within. If $X$ and $Y$ (as $0$-cells) lie within the same path-component in $B\mathcal{C}$, can one say anything in general on how $X$ and ...
1
vote
1answer
46 views

Paths between 0-cells in a classifying space.

Let $\mathcal{C}$ be a small category. Giving a morphism $u: X \to Y$ in $\mathcal{C}$ is equivalent to giving a functor $f: \{0 < 1 \} \to \mathcal{C}$ (with $f(0) = X$, $f(1) = Y$, and $f(0 \to ...
2
votes
1answer
127 views

Sequence of vectors spaces of linear transformations of vector spaces.

Let $V_0$ be a vector space over some field k. Then the set of linear transformations $V_1 = \{T:V_0\rightarrow V_0\mid T\text{ is linear}\}$ is a vector space. Now, let $V_{n+1}= \{T:V_n\rightarrow ...
14
votes
9answers
1k views

Reference to self-study Abstract Algebra and Category Theory

I'm very interested in learning abstract algebra and category theory on my own. It seems a very powerful tool in math and it seems worthwile to take a time and learn about it. I just don't know even ...
4
votes
3answers
155 views

Does the inclusion from affine schemes into schemes preserve pushouts?

Let $K$ be a field. What is an example of two $K$-algebra morphisms $R\to T$ and $S\to T$ such that $\operatorname{Spec}(R\times_T S)$ is not the pushout of the diagram $$ ...
3
votes
1answer
102 views

What is the (propositional) logic associated with an orthomodular lattice?

In Quantum Mechanics the space of projections on the associated Hilbert Space of States forms an Orthomodular Lattice. Von Neumann calls this a Quantum Logic. When projections commute they generate a ...
4
votes
0answers
63 views

Equivalence of categories of coalgebras

I'm studying monadicity and comonadicity and I´m stuck with the following: Let $L\dashv R:X\rightarrow Y$ be an adjunction with unit $\eta$ and counit $\varepsilon$. The induced monad on $Y$ is ...
3
votes
1answer
205 views

Forgetful Functors Create Limits

I'm working on the following problem but I can't seem to make any headway. A widget is a set $A$ with elements $0,1 \in A$, a ternary operation $[-,-,-]: A^3 \to A$, and for each rational number ...
7
votes
3answers
141 views

What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$?

When dealing with a space $X$, that posses a lot of structure (complete lattice, complete metric space, vector space), what can be said about the cartesian exponentiation $X^Y=\{f \mid f:Y\rightarrow ...
1
vote
2answers
73 views

Groupoids isomorphism

Let $G, G'$ be two groups and $X=\{x,y\}$ be a set of two elements. Consider a groupoid $\mathcal{G}$ with objects from $X$ such that Hom$(x,x)=G$ and Hom$(y,y)=G'$. Suppose Hom$(x,y) \neq ...
4
votes
1answer
115 views

Algebra book recommendation inverse limit, universal property

Could you recommend me a book in which I can read about inverse limit, universal property and things like that? I'd really appreciate all your help. I'd prefer something as elementary as possible (I ...
2
votes
0answers
136 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 ...
0
votes
2answers
64 views

When two morphisms are dual to each other in a 2 category?

Here is one famouse approach of defining Adjoint functors: We say $F: D\rightarrow C$ is left adjoint to $G:C \rightarrow D$ or equivalently $G$ is right adjoint to $F$ if $$ C\left(FY,X\right) ...
3
votes
1answer
120 views

Properties of the Category of topological spaces with $n$ basepoints.

I've recently encounted a problem in my reading which would seem to be more naturally phrased if the category we work in shifted from the category $\textbf{Top}^*$ of pointed topological spaces, to ...
5
votes
2answers
307 views

Why do we quotient by chain homotopy in the derived category.

Let $\mathcal A$ be an abelian category. To define the derived category ${\tt D}(\mathcal A)$ of $\mathcal A$ we take the category ${\tt Ch}(\mathcal A)$ of chain complexes in $\mathcal A$, quotient ...
5
votes
3answers
326 views

Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
4
votes
1answer
100 views

Which categories correspond to the untyped and typed lambda calculus?

Simply typed lambda calculus is the internal language of Cartesian Closed Categories. What category has its internal language the typed lambda calculus? And the untyped lambda calculus? Can we in ...
3
votes
2answers
244 views

What is a monad in a $2$-category?

The wikipedia article on monads somewhat mysteriously notes that Monads can be defined in any 2-category ${\mathfrak C}$. The monads defined above are for ${\mathfrak C}$ = Cat. where Cat is the ...
8
votes
2answers
234 views

Does induction for a functor algebra imply it is initial?

By "induction" I mean "no proper subalgebras". My thinking goes like this: For natural numbers, recursion and induction are in some sense the same thing. In particular, given a recursive definition ...