All questions with tag [math: category-theory]


Interpretation of a Category theory question

A problem I'm attempting says Let $p:A \to B$ be a map of sets and $p^*: \mathcal{P}B \to \mathcal{P}A$ be the induced map of power sets sending $X \subseteq B$ to $p^*(X) = \{a \in A: p(a) \in X\}$. Exhibit left and right adjoints to $p^*$ but I can't quite work out what it's saying the functor is: namely, are we $(i)$ defining $p^*$ to be ...
2022-07-25 17:47:21

Composing covers with epis

I am beginner of sheaf-theory and beg your pardon for this maybe silly question. Let $\mathcal{C}$ be a Grothendieck site and $T$ the category of sheaves on $\mathcal{C}$ and let $f:X\rightarrow Y$ be an epic morphism in $T$ into a representable sheaf $Y$. I have a general lack of understanding how such epic morphisms look like and this leads to...
2022-07-25 17:47:06

Relationship between functors

You will have to forgive me as I am very new to category theory - fifth of the way through Categories for a working mathematician. I'm interested in the following; Let $F:A \to B$ and $G:A \to C$ be full functors, where $A$,$B$ and $C$ are groupoids and $B$ has a single object. Let $G$ be such that $Gf=Gf'$ if and only if $Ff=Ff'$ where $f$ and ...
2022-07-25 17:46:33

coproducts of structures

Suppose $S$ is a family members of $L$ - frameworks where $L$ is some collection of constant icons, relationship icons, and also function icons. Does the coproduct of components of $S$ exist? Otherwise, just how does one confirm it? If of course, just how is the coproduct specified? Are the maps from components of S to the coproduct all monic?...
2022-07-25 17:44:43

Balanced Categories, Full/Faithful Functors and Monomorphic Units/Counits

I wish to show that for an adjunction $F: \mathcal{C} \to \mathcal{D} \dashv G: \mathcal{D} \to \mathcal{C}$, if both the unit $\eta$ and the counit $\epsilon$ are monomorphisms and $\mathcal{C}$ is balanced (i.e. a monomorphic epimorphism is an isomorphism), then $F$ is full and faithful. I have completed 2 preceding parts to this question: fir...
2022-07-25 17:43:59

Adjoints and commutative triangles

I'm working through a proof that specifying a left adjoint for a functor $G: \mathcal{D} \to \mathcal{C}$ is equivalent to specifying, for each object $A \in Ob( \mathcal{C})$, an initial object of $(A \downarrow G)$. Here $(A \downarrow G)$ represents the category whose objects are pairs $(B,f)$ with $B \in Ob(\mathcal{D})$, $f: A \to GB$, and ...
2022-07-25 17:41:29

Exact sequences and proving the five lemma

I am currently trying to understand a proof of the Five lemma. For reference, the Five lemma is as follows: In an abelian category, consider the commutative diagram $A \longrightarrow B \longrightarrow C \longrightarrow D \longrightarrow E$ $\downarrow \,\,\,\,\,\,\,\,\,\,\,\,\, \downarrow \,\,\,\,\,\,\,\,\,\,\,\,\, \downarrow \,\,\,\,\...
2022-07-25 17:23:25

Trying to find a left adjoint

Let $\mathsf{Idem}$ be the category of sets that come with an idempotent endomorphism, i.e. the objects are pairs $(A,e)$ where $A$ is a set and $e:A \to A$ is an idempotent. The morphisms $f:(A,e) \to (B,d)$ are morphisms $f: A \to B$ in $\mathsf{Set}$ such that $df = fe$. Let $U\colon \mathsf{Idem} \to \mathsf{Set}$ be the forgetful functor. I...
2022-07-25 17:20:11

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of . Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by which I mean a non-trivial commutative ring with unit such that the zero ideal is prime. Let $\textbf{Fld}$ be the category of fields – by which I mean an integral domain s...
2022-07-25 17:14:32

Why aren't there any coproducts in the category of $\bf{Fields}$?

Well the inquiry is mentioned in the title. I do not recognize much concerning field theory and also i was suprised when i read it on wikipedia please give some instances many thanks beforehand
2022-07-25 17:12:35

Counterexample for a pullback-pushout situation

Suppose you are in the category of sets or more generally in a topos (i.e. sheaf topos) and $f:A\rightarrow C$, $g:B\rightarrow C$ are two morphisms. There is a canonical map $u:D\to C$ from $D$ (defined as the pushout of the diagram $A\leftarrow A\times_C B\rightarrow B$ consisting of the two projections) into $C$. Presumably $u$ doesn't have t...
2022-07-25 17:01:20

Koszul Complex Homology

I'm attempting to understand Eisenbud's proof that: If $x_1,x_2,\ldots,x_i$ is an $M$-sequence, then $H^i(M\otimes K(x_1,...,x_n))=((x_1,\ldots,x_i)M:(x_1,\ldots,x_n))/(x_1,\ldots,x_i)M$. Here $M$-sequence means $M$-regular, i.e. $(x_1,\ldots,x_i)M\neq M$ and for each $j \in {1,\ldots,i}, x_j$ is not a zero divisor on $M/(x_1,\ldots,x_{j-1}...
2022-07-25 16:42:37

Coproduct in the category of (noncommutative) associative algebras

For the instance of commutative algebras, I recognize that the coproduct is offered by the tensor item, yet just how is the scenario in the basic instance? (for associative, yet not always commutative algebras over a ring $A$). Does the coproduct also exist as a whole or otherwise, when does it exist? If it aids, we might think that $A$ itself i...
2022-07-25 13:24:56

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 show that $G$ has a left adjoint, and I have been given a clue that this is the functor $F : \operatorname{Set} \to \mathcal{C}$ where on objects, ...
2022-07-25 13:19:35

How much is a topological space $X$ determined by the category of sheaves of abelian groups on $X$?

Well, the title pretty much says it all. We have a functor $$\mathsf{Sch}_{Ab} : \mathsf{Top} \to \mathsf{Cat}$$ which takes a topological space $X$ to the category $\mathsf{Sch}_{Ab}(X)$ of sheaves of abelian groups on $X$. Every continuous map $f:X \to Y$ gives rise to a functor $f_*:\mathsf{Sch}_{Ab}(X) \to \mathsf{Sch}_{Ab}(Y)$, which takes ...
2022-07-25 13:11:18

Codomains, products and limits

I have a proof in front of me of the theorem that if a category $\mathcal{C}$ has equalisers and all small/finite products, then it has all small/finite limits. I'm not sure how standard the proof is (it's in some notes I've taken rather than a book) but I know the result is quite standard so perhaps you'll be familiar enough with it to help me ...
2022-07-25 12:58:46

Relation between inductive and projective limits

just how to confirm that: $\mathrm{Hom}(\varinjlim X_i,Y) \cong \varprojlim \mathrm{Hom}( X_i,Y)$? Included (05/10/12) It need to be $Y$, as opposed to $Y_i$, and also many thanks for the solutions.
2022-07-25 12:57:04

Adjoints to the forgetful functor $U:C^M\to C$

I'm trying to get my head around adjoints to the forgetful functor $U:\bf{C^M}\to \bf {C}$ where $M$ is a monoid interpreted as a category. My current line of thinking is that the left adjoint $L:\bf C\to C^M$ is given by $L(A)=M\times A$ where $M$ acts on $M\times A$ by $m(M\times A)=(mM)\times A$, the unit $u:A\to M\times A$ is given by the in...
2022-07-25 12:53:38

what is the difference between functor and function?

As it is, what is the distinction in between functor and also function? Regarding I recognize, they look actually comparable. And also is functor made use of in set theory? I recognize that function is made use of in set theory. Many thanks.
2022-07-25 12:46:46

Formal Definition/counter part in mathematics for "Objects" of Object Oriented Models

I'm a rookie in both official maths and also academic computer technology, so please bear with me if you locate my inquiry is not effectively mounted. Object Oriented Modeling appears really valuable in specifying intricate communications when imitating real life. Yet it is primarily made use of in shows. I was asking yourself if we have a compa...
2022-07-25 07:57:06