# Questions tagged [ct.category-theory]

Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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### Which direction does a lax dinatural transformation go?

In 2-category theory we often encounter notions with noninvertible coherence 2-cells, in which case there is a choice about which way the 2-cell should point. Generally, one of these directions is ...
• 61.4k
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### Small complete categories in HoTT+LEM

Freyd's theorem in classical category theory says that any small category $\mathcal{C}$ admitting products indexed by the set $\mathcal{C}_1$ of all its arrows is a preorder. I'm interested in whether ...
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### Can the Haskell kind system be interpreted as a category?

Background I consider myself fairly knowledgeable about the parts of category theory that concerns computer science, and while I have seen many examples of how Haskell kind constructors can be thought ...
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• 1,686
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### Is the category of projections interesting?

Let $C$ be a category and let $C'$ be the wide subcategory whose maps are projections, that is maps in $C$ which belong to some limiting cone (over a discrete base). Since limiting cones compose, $C'$ ...
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• 12k
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### Kleisli is initial, a coherence issue

Usually, when proving that the Kleisli adjunction of a monad $M$ is initial in the category of all adjunctions splitting the monad, we hide coherences under the carpet. Or so did I, until I got ...
• 12k
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### "Quasi-coherent" vector spaces in Sch/S

$\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A_S$-vector space objects ...
• 353
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### Category of multisorted Lawvere's theories

Multisorted Lawvere's theory consists of sets of sorts $S$ small category $T$ a preserving-product essentially bijective functor $(\mathrm{finSet}/S)^{\mathrm{op}} \to T$ How is category of ...
• 4,360
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### Enriched categories from Lawvere theories

Given a Lawvere theory $\mathbb{T}$, is there a convenient way (e.g. a functorial construction) that freely enriches a category $\mathcal{C}$ to a $\mathbb{T}$-enriched category? For example, given ...
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### $(\infty,n)$-operads?

I wonder whether there is (or should be) a theory of colored $(\infty,n)$-operads or multicategories? We know that multicategories are generalizations of categories, and nonsymmetric colored $\infty$-...
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### Decategorifying Grothendieck topoi and categorifying topological spaces

(This is in a sense a follow-up to this question.) I was under the impression these days that Grothendieck topoi were also¹ analogous to topological spaces in that the former were left exact ...
• 1,143
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Suppose I have a monad $M_S = \langle S , \eta_S, \mu_S \rangle$. I want to map this monad to another monad, $M_Q = \langle Q , \eta_Q, \mu_Q \rangle$. What is the minimum I have to define to give ...
• 363
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### Is every filtration on an abelian category strict?

It's well-known that the category of filtered objects in an abelian category is generally not an abelian category. One must generally add extra structure to ensure that all morphisms are strict for ...
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### Is there a canonical product on the category of monads on Set?

I would like to know if there is a partial monoidal product on the category of monads on Set. I want this partial monoidal product to "handle" monad composition which we understand exists ...
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### It's there a way to take a composite monad and a monad map to create a map of the composite?

Let us suppose you start with two monads, $M_S = \langle S , \eta_S, \mu_S \rangle$ and $M_T = \langle T , \eta_T, \mu_T \rangle$ and suppose you have a distributive law, $\lambda: ST \rightarrow TS$ ...
• 363
1 vote