#
Enriched algebraic theories and monads for a system of arities

##
Rory B. B. Lucyshyn-Wright

Under a minimum of assumptions, we develop in generality the basic
theory of universal algebra in a symmetric monoidal closed category
$V$ with respect to a specified system of arities $j:J
\hookrightarrow V$. Lawvere's notion of *algebraic theory*
generalizes to this context, resulting in the notion of
*single-sorted $V$-enriched $J$-cotensor theory*, or
$J$-*theory* for short. For
suitable choices of $V$ and $J$, such $J$-theories include the
enriched algebraic theories of Borceux and Day, the enriched Lawvere
theories of Power, the equational theories of Linton's 1965 work, and
the $V$-theories of Dubuc, which are recovered by taking $J = V$
and correspond to arbitrary $V$-monads on $V$. We identify a modest
condition on $j$ that entails that the
$V$-category of $T$-algebras exists
and is monadic over $V$ for every $J$-theory $T$, even when $T$ is
not small and $V$ is neither complete nor cocomplete. We show that
$j$ satisfies this condition if and only if $j$ presents $V$ as a
free cocompletion of $J$ with respect to the weights for left Kan
extensions along $j$, and so we call such systems of arities
*eleutheric*. We show that $J$-theories for an eleutheric
system may be equivalently described as (i) monads in a certain
one-object bicategory of profunctors on $J$, and (ii) $V$-monads on
$V$ satisfying a certain condition. We prove a characterization
theorem for the categories of algebras of $J$-theories, considered as
$V$-categories $A$ equipped with a specified $V$-functor $A
\rightarrow V$.

Keywords:
algebraic theory; Lawvere theory; universal algebra; monad; enriched
category theory; free cocompletion

2010 MSC:
18C10, 18C15, 18C20, 18C05, 18D20, 18D15, 08A99, 08B99,
08B20, 08C05, 08C99, 03C05, 18D35, 18D10, 18D25, 18A35

*Theory and Applications of Categories,*
Vol. 31, 2016,
No. 5, pp 101-137.

Published 2016-01-31.

http://www.tac.mta.ca/tac/volumes/31/5/31-05.pdf

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