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 ...

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What functors are these?

The category of "just arrow" categories is equivalent to Cat as we see here. If we have a "just arrow" category $C$, I think we can also have the set of equations, $EQ_C$, over the (compositions of) ...
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157 views

Is the category of monoids cartesian closed? Why?

Is the category of monoids cartesian closed? Why? I read Steve Awodey's "Category Theory", and could not solve the exercise in chapter 6, stated above. Here I speak of the "category of monoids" ...
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41 views

Calculating (co)limits of ringed spaces in $\mathbf{Top}$

Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces. There are forgetful functors $$ ...
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30 views

Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has ...
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41 views

Understanding the Gluing axiom of the Structure Sheaf on $Spec(R)$

Let $X = Spec(R)$ be an affine scheme for some commutative ring $R$. The structure sheaf $\mathscr{O}_{X}$ is a contravariant functor (I think) $\text{Open}(X) \leadsto \text{Ring}$ from the category ...
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28 views

How do dependent products in category theory relate to type theory?

I feel like I understand the construction of dependent product types relatively well, it makes sense to me how the introduction and elimination rules work together to create the concept of a function ...
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44 views

Can we define structures like groups or monoids in the context of pure category theory?

In a category $\mathcal C$ with terminal object $1$ and objects $A$, $B$, $C$ we have $\quad$ $A\times (B \times C) \cong (A\times B) \times C$; $\quad$ $1 \times A \cong A \cong A\times 1$; ...
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52 views

Correct definition of model category

When answering this question, In a model category, is the full subcategory of fibrant objects a reflective subcategory? I realized that I wasn't even sure what the correct definition of a model ...
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Is every monomorphism an injection?

We say a morphism is a monomorphism if $fg=fh$ implies $g=h$. So if $f$ is a monomorphism, is it necessarily an injection? i.e. $f(x)=f(y)$ implies $x=y$. My approach is to consider a specific ...
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41 views

Turning Preordered Sets into Preordered Monoids (Constructing Preordered Monoids from Preordered Sets)

Question: Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not ...
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33 views

Is there a reasonable Grothendieck topology on the category of modules over a ring?

How about over a field (i.e. f.d vector spaces)? Can these categories be considered as a site?
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Does $\operatorname{Spec}$ preserve pushouts?

The spectrum-functor $$ \operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set} $$ sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a ...
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49 views

motivation for the direct limit [on hold]

I know just the very basics on Category Theory and that's why I'm going to ask a stupid question. I'm trying to get an intuition for direct limits for my course on Commutative Algebra. All the books ...
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2answers
41 views

Morphisms of a category with one object, which is a group

I've read two articles(1,2), but still have a question about structure of group, say $G$, as an one-object category. Let's call this category $C$. I understand that morphisms of $G$, which is ...
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69 views

limits' definition

Please, can somebody help me? I was given the following definition of the LIMIT: Let $I$ be a small category and $F:I \to C$ a covariant functor ( where $C$ is a category), $K \in Ob(C)$, ...
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20 views

When does the first cohomology group commute with inverse limit?

Let $M_i,i\in\mathbb{N}$ be an inverse system of continous, discrete G-modules and let $M=\varprojlim M_i$. Under what conditions on $M$ and $M_i$ do we have $\varprojlim H^1(G, M_i) \cong H^1(G, M)$? ...
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30 views

cokernel in the pointed set category $Set.$ [on hold]

Please, can someone give me an example of cokernel in the pointed set category? Thanks!
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33 views

Quasicoherent sheaf on the functor of points is the same as on the scheme itself

I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ ...
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Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
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2answers
36 views

Defintion of the $\mathrm{hom}$ functor in category theory

I am going through some notes 'Physics, Topology, Logic and Computation: A Rosetta Stone', on category theory. We first define the opposite category: Given a category $C$, we define the opposite ...
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40 views

Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
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1answer
42 views

Tensor of cocomplete categories

Let $C$, $D$ and $E$ be cocomplete categories. Is there a construction $C \otimes D$ such that there is a correspondence between functors $C \otimes D \to E$ preserving colimits and functors $C ...
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32 views

an example of direct product in Ab

Can someone help me with this? Are there direct products in the category $Ab$ (the category of the abelian groups)? If yes, please, can you give me an example? Thank you?
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1answer
51 views

“Coforgetful” functors?

Let $F : \mathbf{Sets} \to \mathbf{Cat}$ be the free functor that takes the elements of a set to the objects of a discrete category. Does it has a left adjoint? Awodey (2010 p.249 ex.8) says it does, ...
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983 views

Which is the most powerful language, set theory or category theory? [on hold]

As far as I know, mathematics is written based on a language which can be for example set theory or category theory. My concern is about the power of these languages. How can we realize which language ...
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78 views

Is there any comprehensive definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$?

In the definition of the opposite category $\mathcal C^{op}$ of a category $\mathcal C$, it is said that : objects remain the same and arrows' directions are changed (that is, and arrow $f:A \to B$ in ...
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62 views

Order-Preserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
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1answer
39 views

Exactness of a right adjoint functor

Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair. I want to ...
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84 views

Category Theory Zoo

There are a few very useful websites when it comes to either finding a specific object with certain properties (and maybe lacking other properties) or finding out which properties a certain object ...
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83 views

Name for categories with a certain property on coproducts

Is there a name for categories with the following property: The category has zero morphisms, coproducts, and for each family $(X_i)_{i \in I}$ of objects the natural map $$\hom(Y,\bigoplus_{i \in I} ...
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Example of a category without product

A question was raised in our class about the non-existence of product in a category. The two examples that came up in the discussion was the category of smooth manifolds with boundary and the category ...
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27 views

Is the braid category biclosed and bicomplete?

Let $\mathcal{B}$ be the braid category, as in Categories for the Working Mathematician §XI.4 p.262 (objects are natural numbers and morphisms are the braids $n\to n$). Then this can be given a ...
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How does Hartshorne's definition of group schemes encode the law for the neutral element?

Hartshorne's Algebraic Geometry says A scheme $X$ with a morphism to another scheme $S$ is a group scheme over $S$ if there is a section $e\colon\;S\to X$ (the identity) and a morphism ...
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43 views

Confusion regarding definition of adjoint functor - Hilton and Stammbach

While defining Adjoint functors in their book A Course in Homological Algebra, Hilton and Stammbach said the follwing: Let $F:\mathfrak{S}\rightarrow \mathfrak{M}_{\Lambda}$ be the free functor ...
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Recovering $\mathsf{Ab}$ from $\mathbb{Z}$-mod in a closed symmetric monoidal category

Given the usual definition of a module over a ring it is trivial to show that a $\mathbb{Z}$-module is an abelian group (it is just by definition). But my question concerns recovering this idea in a ...
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1answer
64 views

Categorification of algebra structures

This might be a bit of a soft question. Take a $\mathbb{C}$-linear category. Form the complex vector space spanned by its objecs modulo exact sequences. This construction is, as far as I know, the ...
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Complete atomic boolean algebras as coalgebras of some endofunctor on Set

I was hoping to use the fact that CABAs are powersets with extra structure on the morphisms to find an endofunctor $F:\text{Set}\to\text{Set}$ with $\text{Set}^{op}\simeq\text{Coalg}F$. I started by ...
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1answer
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Hartshorne's notation $s: U \to \coprod_{\mathfrak{p} \in U} A_{\mathfrak{p}}$

I understand that Hartshorne is defining the sections on an open set of $\operatorname{Spec} A$ as functions from the points $\mathfrak{p} \in U$ into their localizations $A_{\mathfrak{p}}$ that are ...
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1answer
61 views

How do you know two morphisms are equal (without using elements)

Given two morphisms in some category, which is to say that you are told that $f$ and $g$ are in the cat $C$ and nothing more, how can you know if they are equal? Normally we appeal to the elements of ...
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1answer
36 views

Is the associated bundle construction a bifunctor?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and ...
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1answer
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Is this definition of $(c\downarrow G)$ the slice category or is it something else?

I'm learning about the slice and coslice category constructs, and I think I understand the basics from Wikipedia. However, in this lecture script (in German), there's another definition given, which ...
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1answer
25 views

In the coslice category, why are the morphisms from the terminal category element inclusions?

Wikipedia's definition of the coslice category uses the symbols $i_B$ for the objects $(B,i_B)\in(A\downarrow\mathcal C)$. Similarly, the definition of the slice category uses the symbol $\pi_B$. ...
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49 views

Presheaves,simplicial sets, evaluation functor, Yoneda lemma,hom-functor

In the bottom of page 8 in this paper, how it follows from the Yoneda lemma, that the evaluation functors $E_C$ are precisely of the form $\mathrm {hom(hom(-,}C),-)$ ?
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How to simulate power sets in structural set theory (ETCS)?

How to simulate power sets in structural set theory (ETCS)? (nlab) It turns out that one of the primary attributes of a structural set theory is that the elements of a set have no “internal” ...
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Defining natural transformations based on generalized elements?

Let $F : \mathbf{C} \to \mathbf{D} : G$ be two functors between categories $\mathbf{C}$ and $\mathbf{D}$. A natural transformation $\eta$ from $F$ to $G$ is a collection of morphisms $\eta : FC \to ...
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38 views

Is there a connection between local soundness and completeness in proof theory, and free objects in category theory?

I was watching Frank Pfenning's lecture series on proof theory, where he described the notions of local soundness, and local completeness. He described local soundness of a logical connective as, ...
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1answer
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Proving a set is open in a locally ringed space $(X,\mathscr O_X)$

Let $(X, \mathscr O_X)$ be a locally ringed space and let $A = \Gamma(X,\mathscr O_X)$ be the global sections. For $f\in A$, define the "distinguished open base" as: $$D(f) = \{x\in X : \pi_x(f) ...
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Factorization to Prove Properties Shared between Categories

Question: Let $[n]$ be the finite chain category $[0\to1\to\cdots\to n]$. Then define an infinite sequence $$\mathcal{F}: \text{Seq}\to C$$ in $C$ to be essentially finite if $\mathcal{F}$ factors ...
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1answer
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Name of a category constructed from the action of a group on a category

Let $G$ be a group acting on a category $C$. That is we have a morphism of groups $G \to Aut(C)$. We can now form a new category as follows: Its objects are tuples consisting of an object $x$ of ...
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Data about a morphism and Data about a category [closed]

I have been trying to develop or find theorems about probabilities over categories. This would include probabilistic categories, where morphisms and equations over morphisms are assigned ...