Various structures are studied in category theory using properties of objects and morphisms between them. Many construction 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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45 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
58 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, ...
4
votes
3answers
124 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
21 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
46 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
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1answer
31 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 ...
4
votes
2answers
123 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
62 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
35 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 ...
4
votes
0answers
57 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
50 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
57 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 $\mathcal G$ ...
9
votes
2answers
104 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
33 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
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1answer
37 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
83 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 ...
8
votes
8answers
288 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 ...
3
votes
3answers
81 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
$$
...
2
votes
0answers
35 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 ...
3
votes
0answers
33 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
75 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 ...
6
votes
2answers
82 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
33 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
65 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
63 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
46 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
67 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 ...
3
votes
1answer
84 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 ...
4
votes
2answers
77 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 ...
3
votes
1answer
38 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
46 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 ...
7
votes
2answers
137 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 ...
3
votes
1answer
93 views
Reference for forcing using topos theory
I've just saw in Maclane and Moerdijik's book ("Sheaves in Geometry and Logic: A First Introduction to Topos Theory") about the Cohen forcing viewed in a categorical way using Topos theory. Is there ...
1
vote
1answer
59 views
If we have an equivalence relation on a class, is it possible to define what it means for the collection of equivalence “to be a set”?
My background in set theory is that of a casual acquaintance that I would like to know become friends with (I am not sure set theory feels the same way). For my question, I would like to stay within ...
6
votes
0answers
85 views
Is there a useful Galois connection between Languages and Grammars?
I've just beginning to learn logic and proof theory - and the following rather vague and perhaps ill-formed question occurred to me.
Given an alphabet it's straightforward to construct the Language, ...
2
votes
1answer
94 views
Algebraic definition of ringoid
Category theory has the concept of a groupoid and this is a different concept from the use of the word groupoid to refer to a magma.
Wikipedia gives an algebraic definition of this concept of ...
5
votes
1answer
80 views
A new(?) partial order on the set of continuous maps
Let $X,Y$ be topological spaces. Define a partial order on $\hom(Y,X)$ as follows: $f \leq g$ if $f^{-1}(U) \subseteq g^{-1}(U)$ for all open subsets $U \subseteq X$. Equivalently, $f(y)$ is a ...
9
votes
2answers
135 views
Fiber products of manifolds
Let $\mathsf{Man}$ be the category of smooth manifolds. Denote by $|~|$ the forgetful functor to $\mathsf{Top}$. If $X \to S$ and $Y \to S$ are morphisms in $\mathsf{Man}$, then $X \times_S Y$ exists ...
4
votes
3answers
57 views
If $C'$ is a subcategory of $C$, why can $\mathrm{Hom}_C(X, Y)$ and $\mathrm{Hom}_{C'}(X, Y)$ be different?
Let $C$ be a category and $C'$ a subcategory of $C$. Then by the definition of a subcategory, $$\mathrm{Hom}_{C'}(X, Y) \subseteq \mathrm{Hom}_C(X, Y)$$ for $X, Y$ in $C'$. My question is why it is ...
1
vote
2answers
49 views
When are zero morphisms preserved?
I came across a problem where I want to say the fuctor I am working with preserves zero morphisms and I did some quick internet searching to try and figure out when this is actually true and came up ...
4
votes
1answer
50 views
When is a fully faithful functor an equivalant functor?
Let $\mathcal{A},\mathcal{B} $ be Abelian categories, and let $R:\mathcal{A}\to \mathcal{B}$, $L:\mathcal{B} \to \mathcal{A}$ be fully faithful functors. Moreover, we assume $R$ is right adjoint to ...
2
votes
1answer
51 views
An example of a map that has no section but each of its fibers are not empty
"Conceptual mathematics" by Lawvere and Schanuel, 2nd ed. on page 82 says:
... If one fiber is empty, the map has no sections. Furthermore, for
maps between finite sets the converse is also ...
4
votes
1answer
61 views
An existence of a right Kan extension
Let $M$, $C$ and $A$ be categories. The following statement is well known.
If $M$ is small and $A$ is complete, then any functor $T:M \to A$ has
a right Kan extension along any $K:M \to C$.
...
2
votes
1answer
101 views
Is this square diagram cocartesian for every regular local ring?
Let $K$ be a field and $R=\{f\in K[X]\mid f(0)=f(1)\}$ the $K$-algebra obtained by pulling back $K[X]\to K\times K$, $X\mapsto (0,1)$ along the diagonal.
Is the induced square
\begin{eqnarray}
...
2
votes
2answers
85 views
Examples of Morita equivalent rings
Can someone give some examples of Morita equivalent rings different from the classical one? (i.e. that a ring $R$ is Morita equivalent to the ring $M_n(R)$)
5
votes
2answers
144 views
Monad = Reflective Subcategory?
After answering this question here about Kleisli triples, I realized that this whole Kleisli triple construction:
$T:{\rm Ob}\mathcal C\to{\rm Ob}\mathcal C$, $\ \eta_A:A\to TA$ for all $A\in ...
3
votes
1answer
58 views
Analysis and categories
Can the concept of topology be generalized to deal with categories instead of sets such that one can define continuous mappings between categories possibly using categories with countably or ...
3
votes
3answers
106 views
Does a Kleisli triple need naturality conditions?
I'm reading a paper by Eugenio Moggi entitled "Notions of Computation and Monads". It introduces the concept of a “Kleisli triple” on a category $\mathcal C$, which is $(T, \eta, -^*)$, where:
$T$ ...
8
votes
0answers
101 views
What are the “correct” modules over locally ringed spaces?
$$\begin{array}{ccccc} \text{schemes} & \longrightarrow & \text{locally ringed spaces} & \longrightarrow & \text{ringed spaces} \\ | && | && | \\ \text{quasi-coherent ...
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vote
2answers
67 views
Objects in a category [duplicate]
My friends and I are trying to structure the proof of this question but we are stump. Can anyone help us?
Prove that a final object in a category C is initial in the opposite category C(op).
