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8 votes
2 answers
303 views

When is a locally presentable category (locally) cartesian-closed?

Let $\kappa$ be a regular cardinal. A category $\mathscr C$ is locally $\kappa$-presentable iff it is the free completion of a small $\kappa$-cocomplete category under $\kappa$-filtered colimits. Is t …
9 votes
1 answer
301 views

Internal logic of locally strongly finitely presentable categories

There is a duality between locally strongly finitely presentable (LSFP) categories and (Cauchy complete) cartesian categories, i.e. multisorted algebraic theories [1]. The internal logic of cartesian …
11 votes
0 answers
410 views

A right adjoint preserves Phi-colimits if and only if the left adjoint does what?

Let $\Phi$ be a class of categories (e.g. filtered categories), and consider an adjunction $L : \mathbf C \rightleftarrows \mathbf D : R$. A $\Phi$-colimit is a colimit whose diagram is in $\Phi$. We …
1 vote

The coEilenbeg-Moore category of an Eilenberg-Moore category

This situation is studied in detail in Pavlovic–Hughes's The nucleus of an adjunction and the Street monad on monads and applied to study the Dedekind–MacNeille completion in Tight limits and completi …
varkor's user avatar
  • 10.7k
5 votes

Is there a "duality involution" on presentable categories?

$\newcommand\Pr{\mathit{Pr}}\newcommand\Pres{\mathit{Pres}}$Not an answer, but too long for a comment. Gabriel–Ulmer duality lends some intuition here. For simplicity I consider the finitely presentab …
varkor's user avatar
  • 10.7k
6 votes
0 answers
137 views

Characterisation of essentially algebraic theories with a fixed set of sorts

It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / …
2 votes
0 answers
92 views

Coslices of $\mathbb D$-presentable categories

Let $\mathbb D$ be a sound limit doctrine. When $\mathbb D$ is the doctrine of $\lambda$-small limits, then for any $\mathbb D$-presentable category $\mathscr C$ and $X \in \mathscr C$, the coslice $X …
2 votes
1 answer
212 views

When is a finitary functor induced by Ind (co)continuous

Let $\mathbf C$ and $\mathbf D$ be small categories. $\mathrm{Ind}(\mathbf C)$ is an accessible category (by definition), and is locally finitely presentable (i.e. cocomplete, or equivalently complete …
3 votes
Accepted

Characterisation of essentially algebraic theories as monads

I'm going to give a partial answer to my question, which addresses a misconception I had and illustrates why many of the existing generalisations of theory–monad correspondence are not sufficient to p …
varkor's user avatar
  • 10.7k
7 votes
1 answer
551 views

Characterisation of essentially algebraic theories as monads

The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, Algebraic Theories: A Categorical Introduction to General Algebra). The catego …