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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The product as the space of functions to the coproduct

I recently had one of my "sun rises over Marblehead" moments when I realized that the elements of an arbitrary (Cartesian) product $\prod_{\alpha \in I}X_\alpha$ of sets can be thought as the set of ...
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On the correct definition of a *triangle* in an additive category

Given that we are in some additive category with "shift functors" (i.e., additive auto-equivalences) $A \mapsto A[n]$, Mark Haiman defines a triangle to be a sequence $A \rightarrow B \rightarrow C ...
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2answers
126 views

Simple question on exact sequences.

I am just learning about abelian categories and I would like to hear some advice on how to think about the concepts of kernel, cokernel, image etc correctly; I am a bit confused. I am trying to prove ...
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180 views

Category theory for sensorimotor learning?

Robotics, machine learning, inference, control, decision theory, system identification. There are many different views on how the information would flow from the environment into a robot, and the ...
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Direct limit of $CW$ complex and infinite Stiefel manifold

Let $V_{n}(\mathbb{R}^k)$ be the Stiefel manifold of ortogonal $n$-frames in $\mathbb{R}^k$ and $G$ a compact Lie group. A classifyng space for group $G$ is a connected topological space $BG$, ...
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Properties of quotient categories.

Let $\mathcal{A}$ be an abelian category and $\mathcal{C}$ a localizing subcategory in the sense of Gabriel. (A Serre subcategory or "thick" subcategory, such that the quotient functor $T\colon ...
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Adjoints preserve limits (resp. colimits) Do they preserves completeness (resp. cocompleteness)?

I know that left adjoints preserve colimits and right adjoints preserve limits. So clearly if the limits (resp. colimits) in both categories exist, the adjoints map them to each other. My question ...
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How does one show that two functors are *not* isomorphic?

Let $C$ be the category of finite-dimensional vector spaces over some field. It is easy to construct pairs of endofunctors $F, G$ of $C$, of the same variance, such that $F(V)$ and $G(V)$ have the ...
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Does taking the direct limit of chain complexes commute with taking homology?

Suppose I have a directed system $C_i$, $i\in\mathbb{N}$ of chain complexes over free abelian groups (bounded below degree $0$) $$C_i=0\rightarrow C^{0}_{(i)}\rightarrow ...
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207 views

Questions about epimorphisms and projectives in functor categories

Suppose $I$ is a small category, $R$ is a ring and $_R\mathrm{Mod}$ is the category of left $R$-modules. How do I show that the category $[I,~_R\mathrm{Mod}]$ of all functors from $I$ to ...
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Grothendieck topology on pre/sheaves

Given a Grothendieck topology $T$, say subcanonical, on a category $C$, we are able to talk about sheaves in $(C, T)$. Since pre/sheaves can be viewed as generalized objects of $C$ (via Yoneda ...
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What are morphisms of functors

I am not been able to understand, what is a morphism between two functors. I have gone through the formal definition involving a commutative diagram. Can someone explain that to me in a bit more ...
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a group is not the union of two proper subgroups - how to internalize this into other categories?

A well-known fact from group theory is that a group cannot be the union of two proper subgroups. I wonder, does this statement internalize into other categories than the category of sets? That is, is ...
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1answer
212 views

Cocartesian squares in the category of abelian groups.

Recently, I've been doing a recap of (basic) category theory and found an old exercise I seem to be unable to solve. The question is as follows. Let $A, B$ be abelian groups, $A'<A$ and $B'<B$ ...
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487 views

Do the adjoint functor theorems usefully dualise?

The special and general adjoint functor theorems exist to construct left adjoints to particular functors given certain conditions on them. However, I've not been able to find much mention – at least, ...
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Definition of Exact functors [duplicate]

As we know by definition, a functor for example $T\colon R\textrm{-}\mathsf{Mod}\to\mathsf{Ab}$ (from the category of $R$-modules to the category of abelian groups) is "exact" precisely when for any ...
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3answers
174 views

Does every category have a functor?

Is there any one (or more) categories that doesn't have a functor? Functors go between categories, so is there any category that only has an identity functor but no other functor that maps it to ...
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4answers
295 views

Preserving structures

Category theory abstracts the notion of the preservation of structure by means of morphisms. Is there a description of what it means to preserve structure of different types of mathematical structures ...
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What's more robust than a structural homomorphisms?

This question isn't category theory; but, category theoreticians tend to be interested in mathematical structure, so I thought the answer might exist within that knowledge base. Given two ...
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621 views

Is “cofunctor” an accepted term for contravariant functors?

People are used to the prefix co- flipping arrows in a concept1, and I have seen people using cofunctor to mean a functor that flips arrows, i.e. that takes $A \to B$ to $FB \to FA$. I know this ...
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4answers
509 views

What does a proof in an internal logic actually look like?

The nLab has a lot of nice things to say about how you can use the internal logic of various kinds of categories to prove interesting statements using more or less ordinary mathematical reasoning. ...
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1answer
286 views

Borceux. Handbook of Categorical Algebra I. Proposition 3.4.2.

I'm trying to understand proposition 3.' HoCA (vol. I). Proposition 3.4.2 Consider a functor $F\colon \mathcal A \to \mathcal B$ with both a left adjoint functor $G$ and a right adjoint functor $H$. ...
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Do these definitions of congruences on categories have the same result in this context?

Let $\mathcal{D}$ be a small category and let $A=A\left(\mathcal{D}\right)$ be its set of arrows. Define $P$ on $A$ by: $fPg\Leftrightarrow\left[f\text{ and }g\text{ are parallel}\right]$ and let ...
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1answer
46 views

When is $T$-Alg monoidal closed?

Given a category $\mathcal{V}$ and a monad $(T,\eta,\mu)$, what would be the sufficient conditions on $\mathcal{V}$ and $T$, for the category of $T$ algebras to be monoidal closed? (I'm pretty sure ...
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240 views

Functoriality of the Fundamental group

The fundamental group is a functor from the category of pointed topological spaces to the category of groups. Therefore every base-point preserving continuous function $f$ between pointed ...
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Do we ever study “mixed” categories?

Consider a category whose object class includes the class of all topological spaces and the class of all topological groups. Furthermore, let the hom-sets between any two objects be the usual hom-sets ...
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Category of adjunctions inducing a particular monad

Every pair $F \dashv G$ of adjoint functors $F: \mathcal C \to \mathcal D$, $G: \mathcal D \to \mathcal C$ induces a monad $\mathbb T = (T,\eta,\mu)$ on $\mathcal C$. Given a monad $\mathbb T = ...
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1answer
402 views

Representable functors preserve limits. [duplicate]

Theorem : Representable functors preserve limits. I'm struggling to see why this is true. It's not obvious to me where I should be actually using the fact that functor in question is representable. ...
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378 views

how many empty sets are there?

Would I be correct in saying that in the category of sets, the "class of sets that are isomorphic to the empty set is a proper class"? In other words, there are LOTS of initial objects in the ...
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1answer
47 views

Dagger category generated by $\mathsf{Set}$ viewed as a subcategory of $\mathsf{Rel}$.

Whenever a category $\mathcal{C}$ is being viewed a subcategory of a dagger category $\mathcal{D}$, define that the dagger category generated by $\mathcal{C}$ is the least subcategory of $\mathcal{D}$ ...
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272 views

Does every smooth surjective function have a smooth right inverse?

If you feel this question might be too broad, let me know and I’ll try to get more specific. If $r \colon I → J$ is a smooth surjective function between perfect subspaces $I$ and $J$ of $ℝ$, can ...
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1answer
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Commuting square of functors

Let $\mathcal{E}$ be a complete and cocomplete category. Given a functor $i: \mathcal{C} \to \mathcal{D}$ between small categories, there is a triple of adjoint functors between their respective ...
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1answer
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Whether the limit on representable functors be non-representable?

I'm looking for examples of the following situation: Let $A$ be a complete and cocomplete category, $B$ is a small category and $T\colon B\to\mathbf{Set}^{A^{op}}$ be a functor, such that for any ...
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Group actions and natural isomorphisms

Let $G$ be a group-as-category, and let $S$ be the image of $G$ by the $Hom(G,-)$ functor. The $Hom$ functor defines a bijection $\chi: G \to S$ between elements of $S$ and morphisms of $G$, and thus ...
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Uniqueness of adjoint functors up to isomorphism

Suppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are ...
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1answer
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Viewing groups as objects of the concrete category $\mathsf{Grp}$

Sometimes I ask questions about how structures (groups, topological spaces etc.) ought to be defined, and oftentimes a categorial solution is suggested. Here is a recent example. Now from my ...
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2answers
311 views

Basic categories cheat sheet

Has anyone come across a cheat sheet containing basic properties of the most well-known categories (i.e. does it have (co)products, (co)equalizers, (co)limits, etc?)?
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Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof

What if the set of all well-formed statements in ZFC formed a kind of pseudo-category where a morphism f between objects A, B represented a formal proof that A implied B? What if that category could ...
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1answer
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Awodey - A question about Remark 1.7

At pag 15. Theorem 1.6 Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions. The following Remark 1.7 is: "This shows us what is ...
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Question about bifunctors/bimodules

I'm wondering how a functor $C \to D$ induces a bifunctor $C \times C^{op} \to D$ (it's an "example"). Am I downright stupid not seeing this? Second problem I have: For categories $C$ and $D$: A ...
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1answer
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A natural example in category theory

I'm looking for a natural example of a category $\mathcal{C}$ with finite limits (or just finite products) wherein some object $X$ is not isomorphic to a subobject of an inhabited object. In other ...
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1answer
141 views

Sheaf as a functor

Let $X$ be any topological space, $S$ - any category (e.g. of sets). Consider a new category $C$: its objects are only open subsets of $X$ and a set of morphisms from $U$ to $V$ is nonempty if and ...
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447 views

Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
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2answers
178 views

Existence of not locally small categories

I had a strange remark answered to one of my questions some time ago. My question was involving "locally small categories", and that comment was saying that the existence of not locally small ...
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Free objects in $\mathrm{Set}(G).$

What are the free objects in the category of $G$-sets for a group $G$? After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that ...
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Examples of epimorphisms which are not split epimorphisms?

Are there some examples of epimorphisms which are not split epimorphisms? Thank you very much.
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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 ...
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About the category $\mathrm{Set}(G)$

I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I ...
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1answer
62 views

Has every set a universal map with respect to the “squaring functor”?

Exercise $26C$ of Herrlich & Strecker Category theory asks to show the following: Show that every set has a universal map with respect to the "squaring functor". Recall that a universal map ...
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What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...