Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebraic theories in homotopy theory   An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for
each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict
$T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed
that for a suitable choice of T giving such an algebra amounts to providing the
space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie
algebra...
  Given a functor $X: T\to Spaces$ which preserves products up to a weak
equivalence we show that $X$ is more or less canonically weakly equivalent to a
strict $T$-algebra $LX$. Thus any `homotopy' algebraic structure on the space
$X(t_1)$ can be rigidified to a strict algebraic structure on a space weakly
equivalent to $X(t_1)$. This fact can be interpreted as a generalization of the
results establishing equivalence of homotopy theories of loop spaces and
simplicial groups, products of Eilenberg-Mac Lane spaces and abelian monoids
etc.

31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph